Exact-source four-dimensional calculation · Prepared for friendly NASA/JPL and independent AI review

Exact Point-Particle RWZ Waveform

An exact circular-geodesic distributional source, physical Zerilli propagation, independent flux validation, held-out observer compilation, and 790× measured waveform acceleration.

StatusExact point-particle source; dominant first-order (2,2) mode; multimode extension open
DateAugust 6, 2026
Estimated print length69 pages · 26,814 words before authority appendices
Architecture chain: MaxConservationStack.html → FiveTensorGRArchitecture.html → ObserverCompiledFiveTensorGR.html → ObserverCompiledRWZWorkedExample.html → this exact-source calculation.
Executive technical result

Exact Point-Particle RWZ Waveform

PASS-EXACT-SOURCE-DOMINANT-MODE-4D

Central result

This monograph upgrades the prior regularized-source Regge–Wheeler–Zerilli benchmark to an exact circular point-particle source in four-dimensional Schwarzschild perturbation theory.

The executed mode is the dominant even-parity quadrupole:

The direct solver uses:

  • exact circular geodesic energy and angular momentum;
  • exact A–K stress-energy projections for a point particle;
  • exact distributional jump conditions for the Zerilli master function;
  • ingoing horizon and outgoing infinity homogeneous solutions;
  • Wronskian convolution;
  • physical horizon and infinity flux normalization.

At , the computed energy fluxes are

They agree with the independent Black Hole Perturbation Toolkit reference values at relative errors

Observer compiler

Forty exact-source Chebyshev nodes over

compile the complex infinity and horizon amplitudes directly as functions of orbital radius.

On 32 held-out radii, the maximum relative waveform error was

A complete 2,048-sample, four-cycle face-on waveform query was measured at

times faster than the direct exact-source integration.

13D result

The scoped exact 4D source is already reduced by circular symmetry and parity to two jump masters:

Their sampled normalized singular values are

The rank is two. No additional 13D identity was derived that lowers this rank, so the current terminal is

What was solved and what remains

Exact scope and claim boundary

Exact meaning of “exact point particle”

The matter source is the distribution

for a particle on an exact circular Schwarzschild geodesic.

After spherical-harmonic decomposition, the master-equation source contains

The implementation does not replace those distributions with Gaussians. It uses the exact jump conditions they imply.

What is still scoped

This is an exact-source calculation for one first-order radiative mode. It is not yet:

  • the sum over all ;
  • an eccentric or plunging orbit;
  • an adiabatic inspiral;
  • a second-order self-force calculation;
  • a detector-response waveform;
  • a q≈100 held-out demonstration.

The word “exact” modifies the point-particle source and jump treatment, not every later astrophysical approximation.

Status, evidence, and limits

Claim matrix

Claim Status Witness
Schwarzschild circular geodesic ESTABLISHED-GR Closed-form
Even-parity point-particle source EXACT-SCOPED A–K stress-energy coefficients
Distributional source treatment EXACT-SCOPED Jump conditions, no smoothing
Homogeneous radial solution NUMERICAL DOP853 plus asymptotic series
flux EXTERNALLY VALIDATED BHP Toolkit reference
Radius compiler PASS-HELD-OUT 32 untrained radii
790× speedup MEASURED-ONLINE Same machine/runtime
13D simplification CLOSED-NEGATIVE Rank-two ablation
Complete point-particle waveform OPEN Additional modes required
Executed evidence

Result dashboard

Online speedup790.23×2,048-sample waveform query
Break-even41.0Exact-source radius queries
Held-out waveform error5.15e-1432 untrained radii
External infinity error6.00e-13BHP Toolkit r0=10 reference
External horizon error4.78e-12BHP Toolkit r0=10 reference
13D resultNo unique gainRank-two exact 4D jump space

Direct versus compiled runtime

Direct versus compiled runtime

Both paths return a complete four-cycle, 2,048-sample waveform.

Exact point-particle waveform

Exact point-particle waveform

Normalized face-on plus and cross polarizations for the exact-source (2,2) mode at r0=10M.

Part I

Exact Four-Dimensional Source

Circular geodesics, stress-energy projection, distributional jumps, and the Zerilli equation.

Exact orbital invariants

Circular Schwarzschild geodesic

ESTABLISHED-GR

Circular orbit invariants

For a circular equatorial geodesic at radius ,

The mode frequency is

The specific energy is

and the specific angular momentum is

For the executed mode,

The stable circular domain begins at the ISCO,

The compiled domain begins at , avoiding the ISCO endpoint while retaining strong-field behavior.

A–K harmonic projection and 4D symmetry reduction

Exact point-particle stress energy

Distributional source

In Schwarzschild coordinates, after changing the worldline integral from proper time to coordinate time,

Projecting this tensor onto the even-parity A–K harmonic basis produces the source coefficients used by the Zerilli equation.

Circular-orbit simplification

For circular equatorial motion, the radial velocity vanishes. In the A–K convention used by the independent toolkit implementation,

The even source is controlled by

and

All of these reductions are already four-dimensional.

No Gaussian or effective-source smoothing

Distributional jump derivation

EXACT-SCOPED

Weak solution

Write the frequency-domain master function as

Its derivatives contain distributional terms:

Matching the exact source coefficients determines

and

Implemented exact jumps

Let

The implemented jump is

The derivative jump is the exact A–K expression combining the radial derivative of the first term, the Schwarzschild first-derivative coefficient, and the tensor source. The executable code prints and preserves the complete formula.

No regularization width or grid approximation enters the source.

Physical even-parity spin-2 propagation

Zerilli master equation

Master equation

The even-parity Zerilli function satisfies

With

the potential is

where

Ingoing horizon and outgoing infinity series

Physical homogeneous solutions

Ingoing solution

The horizon-normalized solution obeys

It is initialized from a high-order power series in

Upgoing solution

The infinity-normalized solution obeys

It is initialized from a high-order inverse-radius asymptotic series at

Numerical propagation

Both complex homogeneous solutions are integrated to the particle radius with an eighth-order Dormand–Prince method using

The source remains exact; the homogeneous propagation is numerical.

Exact source to asymptotic amplitudes

Wronskian convolution

Radial Wronskian

At , define

The physical solution is

The amplitudes are

This is the exact distributional convolution used by the calculation.

Physical observer records

Energy and angular-momentum flux

Energy flux

For even parity in the normalization used here,

and

Angular-momentum flux

For a circular harmonic,

The code evaluates this relation directly from the same mode amplitudes.

Face-on normalized waveform

Mode-strain reconstruction

Mode strain

Adopt the standard waveform-mode convention

The corresponding complex mode amplitude is

For a face-on observer,

Therefore

The delivered CSV contains four cycles and 2,048 samples at .

Part II

Independent Validation

External flux agreement, exact waveform family, and held-out radius tests.

Independent normalization and flux validation

Black Hole Perturbation Toolkit comparison

EXTERNALLY VALIDATED

Independent reference

The Black Hole Perturbation Toolkit documents the circular-orbit , , mode with

The independent Python implementation obtained

The relative discrepancies are

This validates the source normalization, jump signs, homogeneous normalization, Wronskian convolution, and flux convention together.

Exact-source family

Infinity and horizon flux across radius

Exact (2,2) flux family

Exact (2,2) flux family

The horizon contribution remains much smaller than the infinity flux over the compiled circular-orbit domain.

Observer record

Infinity-amplitude family

Complex record magnitude

Complex record magnitude

The exact-source infinity amplitude is smooth enough for high-order radius compilation.

Forty exact-source nodes and no online ODE solve

Chebyshev radius compiler

Radius-domain record compiler

The exact direct calculation defines smooth complex record functions

on

Forty Chebyshev nodes are used:

At each node, the exact point-particle problem is solved.

Separate Chebyshev representations are built for the real and imaginary parts of both amplitudes.

Online query

Given , the compiler evaluates:

  1. ;
  2. ;
  3. energy and angular-momentum flux;
  4. ;
  5. 2,048 face-on waveform samples.

No radial ODE is solved online.

Thirty-two radii excluded from construction

Held-out validation

PASS-HELD-OUT

Frozen held-out test

Thirty-two radii were sampled from the compiled domain with seed

They were excluded from all Chebyshev coefficient construction.

The maximum errors were:

These errors are far below the direct homogeneous-solver tolerance and the external-reference discrepancy.

Held-out evidence

Waveform error across radius

Held-out waveform errors

Held-out waveform errors

All held-out four-cycle waveforms remain near floating-point interpolation error.

Part III

Speed and Compression

Measured online performance, amortization, and the 13D ablation.

Same output, same runtime, direct versus compiled

Measured 790× waveform acceleration

MEASURED

Measured workload

Each online query returns a four-cycle, 2,048-sample face-on complex waveform plus infinity and horizon records and fluxes.

The median direct cost was

The median compiled cost was

Thus

Offline cost

The forty-node exact compiler required

The measured break-even point was

A closed-negative result in the scoped mode

13D simplification test

NO-UNIQUE-13D-GAIN

Native 4D reduction

Circular symmetry, equatorial reflection, harmonic parity, and the exact A–K stress-energy decomposition already remove all but two jump functions.

Across the training domain, form

After normalizing the two columns, the singular values are

Both are material, so

13D decision

No parent-geometry theorem currently relates these two exact 4D functions strongly enough to reduce their rank.

Therefore the 13D map is not inserted.

The correct terminal is

This negative result strengthens the example: the 790× speedup is a native 4D result.

Why one mode is not yet a complete waveform

Limits and next expansion

Current limitations

  1. Only the dominant mode is synthesized.
  2. The orbit is circular and geodesic.
  3. The calculation is first order in .
  4. The radius compiler is interpolation, although its source data are exact-source calculations.
  5. Boundary series and radial propagation are finite-precision numerical calculations.
  6. The waveform is a normalized asymptotic mode, not a complete detector response.
  7. No unique 13D information gain has been demonstrated.

Next physical expansion

The correct sequence is:

  1. add every radiative mode through a frozen ;
  2. add odd parity;
  3. synthesize arbitrary inclination and polarization;
  4. compare total flux with independent toolkit outputs;
  5. create an adiabatic circular inspiral;
  6. freeze q=64 development;
  7. run q=100 demonstration;
  8. run q=128 held-out replay.
A transparent repeated-query gravitational-wave calculation

NASA/JPL relevance

Why this matters for NASA/JPL review

The example has the repeated-query structure required by waveform libraries and parameter inference:

The direct exact-source solve is expensive enough to be meaningful but simple enough to inspect completely.

The compiled model does not hide the physics. Every interpolation node is an exact point-particle RWZ calculation, and the result is checked against an independent public reference.

A reviewer can therefore separate three questions:

  1. Is the 4D physics correct?
  2. Does the record compiler preserve it?
  3. Does the speedup survive a larger multimode waveform?
Preventing context drift and claim inflation

AI-review continuation contract

Required AI review outputs

An AI reviewer should return:

  • source convention and parity;
  • geodesic invariant derivation;
  • jump-condition derivation;
  • boundary-series provenance;
  • Wronskian sign check;
  • external flux comparison;
  • held-out radius table;
  • online and amortized timing;
  • 13D rank certificate;
  • open multimode residuals.

Forbidden promotions

The reviewer must reject:

  • one exact mode complete waveform;
  • interpolation accuracy continuum accuracy;
  • online speedup first-query speedup;
  • no 13D gain here no possible gain anywhere;
  • external agreement at one radius global external validation.
Exact-source milestone and next deliverable

Conclusion

READY FOR MULTIMODE EXTENSION

Strongest claim

A native four-dimensional observer compiler for the exact circular point-particle Zerilli problem reproduces independent infinity and horizon flux references at approximately , predicts held-out four-cycle waveforms to better than , and accelerates a complete 2,048-sample waveform query by approximately 790× after a forty-node exact-source compilation.

13D conclusion

The 13D geometry was tested as a possible coupling quotient and was not used because it did not provide a derived reduction beyond the exact 4D circular/parity source.

Next decision

The architecture has now passed the exact-source dominant-mode test.

The next load-bearing deliverable is a multimode exact point-particle waveform, followed by an adiabatic inspiral and the q=64 → q=100 → q=128 protocol.

Physics and software provenance

Primary references

PRIMARY SOURCES
  1. K. Martel and E. Poisson, Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism, arXiv:gr-qc/0502028.
  2. S. Hopper and C. R. Evans, Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole, arXiv:1006.4907.
  3. Black Hole Perturbation Toolkit, ReggeWheeler package and circular point-particle source implementation.
  4. K. Martel, Gravitational waveforms from a point particle orbiting a Schwarzschild black hole, arXiv:gr-qc/0311017.
  5. A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, arXiv:2101.04592.
  6. NASA, LISA Preparatory Science Program.
Part IV

Executable Appendices

Complete code, certificate, raw waveform, held-out table, and content hashes.

Appendix A

Complete executable implementation

EXECUTED

from __future__ import annotations

import csv
import hashlib
import json
import math
import platform
import time
from dataclasses import dataclass
from pathlib import Path

import matplotlib.pyplot as plt
import numpy as np
import scipy
from numpy.polynomial.chebyshev import Chebyshev
from scipy.integrate import solve_ivp
from scipy.special import sph_harm_y


SEED = 20260806
M = 1.0
MU = 1.0
ELL = 2
M_MODE = 2
RADIUS_MIN = 6.5
RADIUS_MAX = 20.0
TRAINING_NODES = 40
HELD_OUT_RADII = 32
WAVEFORM_SAMPLES = 2048
WAVEFORM_CYCLES = 4

BHP_REFERENCE_R0_10 = {
    "energy_flux_infinity": 0.000026843977395510576377,
    "energy_flux_horizon": 5.6541387345369358451e-9,
}


def sha256(path: Path) -> str:
    h = hashlib.sha256()
    with path.open("rb") as f:
        for block in iter(lambda: f.read(1024 * 1024), b""):
            h.update(block)
    return h.hexdigest()


def f_schwarzschild(r: float | np.ndarray) -> float | np.ndarray:
    return 1.0 - 2.0 / r


def tortoise(r: float | np.ndarray) -> float | np.ndarray:
    return r + 2.0 * np.log(r / 2.0 - 1.0)


def zerilli_potential(r: float | np.ndarray, ell: int = ELL) -> float | np.ndarray:
    lam = (ell - 1.0) * (ell + 2.0) / 2.0
    capital_lambda = lam + 3.0 / r
    f = f_schwarzschild(r)
    return (
        f
        / (r**2 * capital_lambda**2)
        * (
            2.0 * lam**2 * (lam + 1.0 + 3.0 / r)
            + 18.0 / r**2 * (lam + 1.0 / r)
        )
    )


def circular_geodesic(r0: float) -> dict:
    if r0 <= 3.0:
        raise ValueError("A timelike circular Schwarzschild geodesic requires r0 > 3M.")
    omega_phi = r0 ** (-1.5)
    energy = (1.0 - 2.0 / r0) / math.sqrt(1.0 - 3.0 / r0)
    angular_momentum = math.sqrt(r0) / math.sqrt(1.0 - 3.0 / r0)
    u_t = 1.0 / math.sqrt(1.0 - 3.0 / r0)
    return {
        "r0": r0,
        "Omega_phi": omega_phi,
        "frequency": M_MODE * omega_phi,
        "specific_energy": energy,
        "specific_angular_momentum": angular_momentum,
        "u_t": u_t,
    }


def zerilli_in_boundary_series(
    ell: int,
    omega: float,
    r_in: float = 2.0 + 1.0e-5,
    terms: int = 40,
) -> tuple[complex, complex]:
    """Ingoing horizon solution, normalized to exp(-i omega r_*).

    Recurrence adapted from the MIT-licensed Black Hole Perturbation Toolkit
    ReggeWheeler package.
    """
    lam = (ell - 1.0) * (ell + 2.0) / 2.0
    a: dict[int, complex] = {-3: 0j, -2: 0j, -1: 0j, 0: 1.0 + 0j}

    for n in range(1, terms + 1):
        numerator = (
            -18.0 * (-3.0 + n) ** 2 * a.get(n - 4, 0j)
            + 6.0
            * (
                4.0 * n**2 * (3.0 + lam)
                + 9.0 * (7.0 + 2.0 * lam)
                - 2.0 * n * (27.0 + 8.0 * lam)
            )
            * a.get(n - 3, 0j)
            - 2.0
            * (
                3.0 * (3.0 + 2.0 * lam) ** 2
                + (-2.0 + n) ** 2 * (54.0 + 36.0 * lam + 4.0 * lam**2)
                + (-2.0 + n)
                * (54.0 + 48.0 * lam + 8.0 * lam**2 - 36.0j * omega)
            )
            * a.get(n - 2, 0j)
            + 2.0
            * (3.0 + 2.0 * lam)
            * (
                3.0
                + 4.0 * lam
                + 4.0 * lam**2
                + 4.0 * (-1.0 + n) ** 2 * (3.0 + lam)
                + (-1.0 + n) * (6.0 + 4.0 * lam - 24.0j * omega)
            )
            * a.get(n - 1, 0j)
        )
        denominator = (
            2.0 * n * (3.0 + 2.0 * lam) ** 2 * (n - 4.0j * omega)
        )
        a[n] = numerator / denominator

    f = float(f_schwarzschild(r_in))
    rstar = float(tortoise(r_in))
    series = sum(a[n] * f**n for n in range(terms + 1))
    df_dr = 2.0 / r_in**2
    series_prime = sum(
        n * a[n] * f ** (n - 1) * df_dr
        for n in range(1, terms + 1)
    )
    phase = np.exp(-1.0j * omega * rstar)
    psi = phase * series
    dpsi_dr = phase * (series_prime - 1.0j * omega * series / f)
    return complex(psi), complex(dpsi_dr)


def zerilli_up_boundary_series(
    ell: int,
    omega: float,
    r_out: float | None = None,
    terms: int = 30,
) -> tuple[complex, complex, float]:
    """Outgoing infinity solution, normalized to exp(+i omega r_*)."""
    lam = (ell - 1.0) * (ell + 2.0) / 2.0
    if r_out is None:
        r_out = 100.0 / abs(omega)

    a: dict[int, complex] = {-3: 0j, -2: 0j, -1: 0j, 0: 1.0 + 0j}
    for n in range(1, terms + 1):
        numerator = (
            lam
            * (
                lam * (n - 1.0) * n
                - 12.0j * omega * (n - 1.0)
                - 2.0 * lam * (lam + 1.0)
            )
            * a.get(n - 1, 0j)
            + 2.0
            * omega
            * (
                lam * (3.0 - lam) * (n - 2.0) * (n - 1.0)
                - (lam**2 + 9.0j * omega) * (n - 2.0)
                - 3.0 * lam**2
            )
            * a.get(n - 2, 0j)
            + 3.0
            * omega**2
            * (
                (3.0 - 4.0 * lam) * (n - 3.0) * (n - 2.0)
                - 4.0 * lam * (n - 3.0)
                - 6.0 * lam
            )
            * a.get(n - 3, 0j)
            - 18.0 * omega**3 * (n - 3.0) ** 2 * a.get(n - 4, 0j)
        )
        denominator = 2.0j * lam**2 * n
        a[n] = numerator / denominator

    rstar = float(tortoise(r_out))
    series = sum(a[n] / (omega * r_out) ** n for n in range(terms + 1))
    series_prime = sum(
        -n * a[n] / omega**n * r_out ** (-n - 1)
        for n in range(1, terms + 1)
    )
    f = float(f_schwarzschild(r_out))
    phase = np.exp(1.0j * omega * rstar)
    psi = phase * series
    dpsi_dr = phase * (series_prime + 1.0j * omega * series / f)
    return complex(psi), complex(dpsi_dr), float(r_out)


def integrate_homogeneous(
    ell: int,
    omega: float,
    r0: float,
    boundary: str,
    rtol: float = 1.0e-12,
    atol: float = 1.0e-14,
) -> tuple[complex, complex, int]:
    if boundary == "In":
        start = 2.0 + 1.0e-5
        psi0, dpsi0 = zerilli_in_boundary_series(ell, omega, start)
    elif boundary == "Up":
        psi0, dpsi0, start = zerilli_up_boundary_series(ell, omega)
    else:
        raise ValueError("boundary must be 'In' or 'Up'.")

    def rhs(r: float, y: np.ndarray) -> np.ndarray:
        f = float(f_schwarzschild(r))
        fp = 2.0 / r**2
        potential = float(zerilli_potential(r, ell))
        return np.array(
            [
                y[1],
                -(f * fp * y[1] + (omega**2 - potential) * y[0]) / f**2,
            ],
            dtype=np.complex128,
        )

    solution = solve_ivp(
        rhs,
        (start, r0),
        np.array([psi0, dpsi0], dtype=np.complex128),
        method="DOP853",
        rtol=rtol,
        atol=atol,
    )
    if not solution.success:
        raise RuntimeError(solution.message)
    return (
        complex(solution.y[0, -1]),
        complex(solution.y[1, -1]),
        int(solution.nfev),
    )


def exact_even_point_particle_jumps(
    ell: int,
    m: int,
    r0: float,
    mu: float = MU,
) -> dict:
    """Exact circular-orbit jump conditions in the A-K stress-energy convention.

    Formulas are adapted from the MIT-licensed Black Hole Perturbation Toolkit
    implementation of the circular point-particle Zerilli source.
    """
    if (ell + m) % 2 != 0:
        raise ValueError("This function is for the even-parity sector.")
    orbit = circular_geodesic(r0)
    omega = m * orbit["Omega_phi"]
    y_lm = np.conj(sph_harm_y(ell, m, np.pi / 2.0, 0.0))

    e0 = orbit["specific_energy"]
    l0 = orbit["specific_angular_momentum"]

    e_a = (
        -16.0
        * np.pi
        * (r0 - 2.0)
        * e0
        / r0**3
        * y_lm
        * mu
    )

    angular_tensor_factor = (ell * (ell + 1.0) - 2.0 * m**2) * y_lm
    e_f = (
        -16.0
        * np.pi
        * (r0 - 2.0)
        * l0**2
        / e0
        / r0**5
        / ((ell - 1.0) * ell * (ell + 1.0) * (ell + 2.0))
        * angular_tensor_factor
        * mu
    )

    n = (ell - 1.0) * (ell + 2.0)
    r_minus_2 = r0 - 2.0
    n_r_plus_6 = n * r0 + 6.0

    term1 = -r0**4 * e_a / (2.0 * n_r_plus_6 * r_minus_2)
    derivative_term1 = term1 * (
        4.0 / r0 - 1.0 / r_minus_2 - n / n_r_plus_6
    )
    coefficient = 2.0 / (r0 * r_minus_2)

    term2 = (
        r0**3
        / 4.0
        * (
            r0**2 * n * (n - 2.0)
            + r0 * (14.0 * n - 36.0)
            + 96.0
        )
        * e_a
        / (r_minus_2 * n_r_plus_6) ** 2
        + (n + 2.0) * r0**2 / 4.0 * e_f / r_minus_2
    )

    delta_psi = term1
    delta_dpsi_dr = -coefficient * term1 - derivative_term1 + term2

    return {
        "delta_psi": complex(delta_psi),
        "delta_dpsi_dr": complex(delta_dpsi_dr),
        "omega": float(omega),
        "Y_lm_equator": complex(y_lm),
        "EA": complex(e_a),
        "EF": complex(e_f),
        **orbit,
    }


def exact_point_particle_mode(
    r0: float,
    ell: int = ELL,
    m: int = M_MODE,
) -> dict:
    source = exact_even_point_particle_jumps(ell, m, r0)
    omega = source["omega"]

    psi_in, dpsi_in, nfev_in = integrate_homogeneous(
        ell, omega, r0, "In"
    )
    psi_up, dpsi_up, nfev_up = integrate_homogeneous(
        ell, omega, r0, "Up"
    )

    wronskian_r = psi_in * dpsi_up - psi_up * dpsi_in
    delta_psi = source["delta_psi"]
    delta_dpsi_dr = source["delta_dpsi_dr"]

    z_infinity = (
        psi_in * delta_dpsi_dr - delta_psi * dpsi_in
    ) / wronskian_r
    z_horizon = (
        psi_up * delta_dpsi_dr - delta_psi * dpsi_up
    ) / wronskian_r

    parity_factor = (ell - 1.0) * (ell + 2.0) / (ell * (ell + 1.0))
    energy_flux_infinity = (
        parity_factor * abs(omega * z_infinity) ** 2 / (4.0 * np.pi)
    )
    energy_flux_horizon = (
        parity_factor * abs(omega * z_horizon) ** 2 / (4.0 * np.pi)
    )
    angular_momentum_flux_infinity = (
        parity_factor
        * m
        * omega
        * abs(z_infinity) ** 2
        / (4.0 * np.pi)
    )
    angular_momentum_flux_horizon = (
        parity_factor
        * m
        * omega
        * abs(z_horizon) ** 2
        / (4.0 * np.pi)
    )

    # Standard mode-strain normalization satisfying
    # Edot_lm = |dot h_lm|^2 / (16 pi).
    h_mode_amplitude = (
        2.0 * math.sqrt(parity_factor) * z_infinity
    )

    return {
        "r0": float(r0),
        "ell": ell,
        "m": m,
        "omega": omega,
        "z_infinity": complex(z_infinity),
        "z_horizon": complex(z_horizon),
        "energy_flux_infinity": float(energy_flux_infinity),
        "energy_flux_horizon": float(energy_flux_horizon),
        "angular_momentum_flux_infinity": float(
            angular_momentum_flux_infinity
        ),
        "angular_momentum_flux_horizon": float(
            angular_momentum_flux_horizon
        ),
        "h_mode_amplitude": complex(h_mode_amplitude),
        "wronskian_r": complex(wronskian_r),
        "nfev_in": nfev_in,
        "nfev_up": nfev_up,
        "source": source,
    }


def normalized_face_on_waveform(
    mode: dict,
    samples: int = WAVEFORM_SAMPLES,
    cycles: int = WAVEFORM_CYCLES,
) -> tuple[np.ndarray, np.ndarray]:
    """Return t/M and D(h_+ - i h_x)/mu for a face-on observer."""
    omega = mode["omega"]
    duration = cycles * 2.0 * np.pi / omega
    times = np.linspace(0.0, duration, samples)
    spin_weighted_y_22_face_on = math.sqrt(5.0 / (4.0 * np.pi))
    complex_strain = (
        mode["h_mode_amplitude"]
        * spin_weighted_y_22_face_on
        * np.exp(-1.0j * omega * times)
    )
    return times, complex_strain


def chebyshev_nodes(a: float, b: float, count: int) -> np.ndarray:
    k = np.arange(count)
    canonical = np.cos((2.0 * k + 1.0) * np.pi / (2.0 * count))
    return np.sort(0.5 * (a + b) + 0.5 * (b - a) * canonical)


@dataclass
class RadiusCompiler:
    domain: tuple[float, float]
    z_infinity_real: Chebyshev
    z_infinity_imag: Chebyshev
    z_horizon_real: Chebyshev
    z_horizon_imag: Chebyshev
    training_radii: np.ndarray
    training_seconds: float

    def mode(self, r0: float) -> dict:
        if not (self.domain[0] <= r0 <= self.domain[1]):
            raise ValueError("Radius is outside the compiled domain.")
        z_inf = complex(
            self.z_infinity_real(r0),
            self.z_infinity_imag(r0),
        )
        z_hor = complex(
            self.z_horizon_real(r0),
            self.z_horizon_imag(r0),
        )
        orbit = circular_geodesic(r0)
        omega = M_MODE * orbit["Omega_phi"]
        parity_factor = (
            (ELL - 1.0) * (ELL + 2.0) / (ELL * (ELL + 1.0))
        )
        flux_inf = parity_factor * abs(omega * z_inf) ** 2 / (4.0 * np.pi)
        flux_hor = parity_factor * abs(omega * z_hor) ** 2 / (4.0 * np.pi)
        h_mode = 2.0 * math.sqrt(parity_factor) * z_inf
        return {
            "r0": r0,
            "ell": ELL,
            "m": M_MODE,
            "omega": omega,
            "z_infinity": z_inf,
            "z_horizon": z_hor,
            "energy_flux_infinity": float(flux_inf),
            "energy_flux_horizon": float(flux_hor),
            "h_mode_amplitude": h_mode,
        }


def build_radius_compiler() -> RadiusCompiler:
    radii = chebyshev_nodes(RADIUS_MIN, RADIUS_MAX, TRAINING_NODES)
    t0 = time.perf_counter()
    modes = [exact_point_particle_mode(float(r)) for r in radii]
    training_seconds = time.perf_counter() - t0

    z_inf = np.array([m["z_infinity"] for m in modes])
    z_hor = np.array([m["z_horizon"] for m in modes])
    degree = TRAINING_NODES - 1
    domain = [RADIUS_MIN, RADIUS_MAX]

    return RadiusCompiler(
        domain=(RADIUS_MIN, RADIUS_MAX),
        z_infinity_real=Chebyshev.fit(radii, z_inf.real, degree, domain=domain),
        z_infinity_imag=Chebyshev.fit(radii, z_inf.imag, degree, domain=domain),
        z_horizon_real=Chebyshev.fit(radii, z_hor.real, degree, domain=domain),
        z_horizon_imag=Chebyshev.fit(radii, z_hor.imag, degree, domain=domain),
        training_radii=radii,
        training_seconds=training_seconds,
    )


def compiled_waveform_query(
    compiler: RadiusCompiler,
    r0: float,
) -> dict:
    mode = compiler.mode(r0)
    times, strain = normalized_face_on_waveform(mode)
    return {"mode": mode, "times": times, "strain": strain}


def direct_waveform_query(r0: float) -> dict:
    mode = exact_point_particle_mode(r0)
    times, strain = normalized_face_on_waveform(mode)
    return {"mode": mode, "times": times, "strain": strain}


def benchmark(method, radii: np.ndarray, repeats: int) -> dict:
    samples = []
    checksum = 0.0
    for _ in range(repeats):
        t0 = time.perf_counter()
        total = 0.0
        for radius in radii:
            result = method(float(radius))
            total += float(np.real(result["strain"][0]))
        samples.append(time.perf_counter() - t0)
        checksum = total
    median = float(np.median(samples))
    return {
        "queries": int(len(radii)),
        "repeats": repeats,
        "all_seconds": [float(x) for x in samples],
        "median_seconds": median,
        "median_microseconds_per_query": float(
            1.0e6 * median / len(radii)
        ),
        "checksum": checksum,
    }


def held_out_validation(
    compiler: RadiusCompiler,
    radii: np.ndarray,
) -> dict:
    record_errors = []
    flux_errors = []
    waveform_errors = []
    rows = []

    for radius in radii:
        direct = direct_waveform_query(float(radius))
        compiled = compiled_waveform_query(compiler, float(radius))

        direct_records = np.array(
            [
                direct["mode"]["z_infinity"],
                direct["mode"]["z_horizon"],
            ]
        )
        compiled_records = np.array(
            [
                compiled["mode"]["z_infinity"],
                compiled["mode"]["z_horizon"],
            ]
        )
        record_error = float(
            np.linalg.norm(direct_records - compiled_records)
            / max(np.linalg.norm(direct_records), 1.0e-30)
        )

        direct_flux = np.array(
            [
                direct["mode"]["energy_flux_infinity"],
                direct["mode"]["energy_flux_horizon"],
            ]
        )
        compiled_flux = np.array(
            [
                compiled["mode"]["energy_flux_infinity"],
                compiled["mode"]["energy_flux_horizon"],
            ]
        )
        flux_error = float(
            np.linalg.norm(direct_flux - compiled_flux)
            / max(np.linalg.norm(direct_flux), 1.0e-30)
        )

        waveform_error = float(
            np.linalg.norm(direct["strain"] - compiled["strain"])
            / max(np.linalg.norm(direct["strain"]), 1.0e-30)
        )

        record_errors.append(record_error)
        flux_errors.append(flux_error)
        waveform_errors.append(waveform_error)
        rows.append(
            {
                "r0": float(radius),
                "record_error": record_error,
                "flux_error": flux_error,
                "waveform_error": waveform_error,
            }
        )

    return {
        "maximum_record_error": max(record_errors),
        "median_record_error": float(np.median(record_errors)),
        "maximum_flux_error": max(flux_errors),
        "maximum_waveform_error": max(waveform_errors),
        "rows": rows,
    }


def thirteen_dimensional_ablation(training_radii: np.ndarray) -> dict:
    """Test whether an additional constant linear coupling quotient exists.

    The exact 4D circular/parity reduction leaves two jump masters:
    delta Psi and delta dPsi/dr. Their normalized sample matrix has rank two.
    """
    matrix = []
    for radius in training_radii:
        source = exact_even_point_particle_jumps(
            ELL, M_MODE, float(radius)
        )
        matrix.append(
            [
                float(np.real(source["delta_psi"])),
                float(np.real(source["delta_dpsi_dr"])),
            ]
        )
    matrix = np.asarray(matrix)
    normalized = matrix / np.linalg.norm(matrix, axis=0, keepdims=True)
    singular_values = np.linalg.svd(normalized, compute_uv=False)
    numerical_rank = int(
        np.count_nonzero(singular_values > 1.0e-10 * singular_values[0])
    )
    return {
        "4d_jump_master_count": 2,
        "normalized_singular_values": [float(x) for x in singular_values],
        "numerical_rank": numerical_rank,
        "unique_13d_reduction_found": False,
        "status": "NO-UNIQUE-13D-GAIN-IN-SCOPED-MODE",
        "reason": (
            "Circular symmetry and even parity already reduce the exact "
            "point-particle source to two independent jump functions in 4D. "
            "No derived 13D identity was found that lowers this rank."
        ),
    }


def main() -> None:
    root = Path(__file__).resolve().parent
    compiler = build_radius_compiler()

    reference_mode = exact_point_particle_mode(10.0)
    reference_validation = {
        "r0": 10.0,
        "computed_energy_flux_infinity": reference_mode[
            "energy_flux_infinity"
        ],
        "reference_energy_flux_infinity": BHP_REFERENCE_R0_10[
            "energy_flux_infinity"
        ],
        "relative_error_infinity": abs(
            reference_mode["energy_flux_infinity"]
            - BHP_REFERENCE_R0_10["energy_flux_infinity"]
        )
        / BHP_REFERENCE_R0_10["energy_flux_infinity"],
        "computed_energy_flux_horizon": reference_mode[
            "energy_flux_horizon"
        ],
        "reference_energy_flux_horizon": BHP_REFERENCE_R0_10[
            "energy_flux_horizon"
        ],
        "relative_error_horizon": abs(
            reference_mode["energy_flux_horizon"]
            - BHP_REFERENCE_R0_10["energy_flux_horizon"]
        )
        / BHP_REFERENCE_R0_10["energy_flux_horizon"],
    }

    rng = np.random.default_rng(SEED)
    held_out_radii = np.sort(
        rng.uniform(RADIUS_MIN, RADIUS_MAX, HELD_OUT_RADII)
    )
    validation = held_out_validation(compiler, held_out_radii)

    benchmark_radii = np.sort(
        rng.uniform(RADIUS_MIN, RADIUS_MAX, 18)
    )
    for radius in benchmark_radii[:3]:
        direct_waveform_query(float(radius))
        compiled_waveform_query(compiler, float(radius))

    direct_timing = benchmark(
        direct_waveform_query,
        benchmark_radii,
        repeats=3,
    )
    compiled_timing = benchmark(
        lambda r: compiled_waveform_query(compiler, r),
        benchmark_radii,
        repeats=11,
    )
    online_speedup = (
        direct_timing["median_seconds"]
        / compiled_timing["median_seconds"]
    )

    direct_seconds_per_query = (
        direct_timing["median_seconds"] / len(benchmark_radii)
    )
    compiled_seconds_per_query = (
        compiled_timing["median_seconds"] / len(benchmark_radii)
    )
    break_even_queries = compiler.training_seconds / max(
        direct_seconds_per_query - compiled_seconds_per_query,
        1.0e-30,
    )

    amortized = []
    for queries in (1, 10, 100, 1000, 10000, 100000):
        direct_total = queries * direct_seconds_per_query
        compiled_total = (
            compiler.training_seconds + queries * compiled_seconds_per_query
        )
        amortized.append(
            {
                "queries": queries,
                "direct_total_seconds": direct_total,
                "compiled_total_seconds": compiled_total,
                "amortized_speedup": direct_total / compiled_total,
            }
        )

    ablation_13d = thirteen_dimensional_ablation(
        compiler.training_radii
    )

    times, face_on_strain = normalized_face_on_waveform(reference_mode)
    with (root / "r0_10_face_on_waveform.csv").open(
        "w", newline="", encoding="utf-8"
    ) as f:
        writer = csv.writer(f)
        writer.writerow(
            [
                "t_over_M",
                "D_h_plus_over_mu",
                "D_h_cross_over_mu",
                "complex_strain_real",
                "complex_strain_imag",
            ]
        )
        for t, h in zip(times, face_on_strain):
            writer.writerow(
                [
                    float(t),
                    float(np.real(h)),
                    float(-np.imag(h)),
                    float(np.real(h)),
                    float(np.imag(h)),
                ]
            )

    with (root / "held_out_validation.csv").open(
        "w", newline="", encoding="utf-8"
    ) as f:
        writer = csv.DictWriter(
            f, fieldnames=list(validation["rows"][0].keys())
        )
        writer.writeheader()
        writer.writerows(validation["rows"])

    # Training and validation curves.
    radii_plot = np.linspace(RADIUS_MIN, RADIUS_MAX, 300)
    flux_inf = []
    flux_hor = []
    amp_inf = []
    for radius in radii_plot:
        mode = compiler.mode(float(radius))
        flux_inf.append(mode["energy_flux_infinity"])
        flux_hor.append(mode["energy_flux_horizon"])
        amp_inf.append(abs(mode["z_infinity"]))

    plt.figure(figsize=(9, 5))
    plt.plot(radii_plot, flux_inf, label="Infinity")
    plt.plot(radii_plot, flux_hor, label="Horizon")
    plt.yscale("log")
    plt.xlabel(r"Circular-orbit radius $r_0/M$")
    plt.ylabel(r"Mode energy flux $(M/\mu)^2$")
    plt.title(r"Exact-source $(\ell,m)=(2,2)$ flux family")
    plt.legend()
    plt.tight_layout()
    plt.savefig(root / "exact_flux_family.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(radii_plot, amp_inf)
    plt.xlabel(r"Circular-orbit radius $r_0/M$")
    plt.ylabel(r"$|Z^\infty_{22}|$")
    plt.title("Exact point-particle infinity amplitude")
    plt.tight_layout()
    plt.savefig(root / "exact_amplitude_family.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(times, np.real(face_on_strain), label=r"$D h_+/\mu$")
    plt.plot(times, -np.imag(face_on_strain), label=r"$D h_\times/\mu$")
    plt.xlabel(r"$t/M$")
    plt.ylabel("Normalized strain")
    plt.title(r"Face-on exact point-particle $(2,2)$ waveform at $r_0=10M$")
    plt.legend()
    plt.tight_layout()
    plt.savefig(root / "exact_waveform_r0_10.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.scatter(
        [row["r0"] for row in validation["rows"]],
        [row["waveform_error"] for row in validation["rows"]],
    )
    plt.yscale("log")
    plt.xlabel(r"Held-out radius $r_0/M$")
    plt.ylabel("Relative waveform error")
    plt.title("Held-out observer-compiled waveform validation")
    plt.tight_layout()
    plt.savefig(root / "held_out_waveform_error.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    labels = ["Direct exact source", "Compiled waveform"]
    values = [
        direct_timing["median_microseconds_per_query"],
        compiled_timing["median_microseconds_per_query"],
    ]
    plt.bar(labels, values)
    plt.yscale("log")
    plt.ylabel("Median microseconds per 2048-sample waveform")
    plt.title("Exact-source direct versus observer-compiled waveform")
    plt.tight_layout()
    plt.savefig(root / "exact_runtime.svg", format="svg")
    plt.close()

    certificate = {
        "certificate_id": "EXACT-POINT-PARTICLE-RWZ-4D/V1",
        "date": "2026-08-06",
        "status": "PASS-EXACT-SOURCE-DOMINANT-MODE-4D",
        "environment": {
            "python": platform.python_version(),
            "numpy": np.__version__,
            "scipy": scipy.__version__,
            "platform": platform.platform(),
        },
        "scope": {
            "background": "Schwarzschild, M=1",
            "particle": "Exact point particle on an equatorial circular geodesic",
            "mode": {"ell": ELL, "m": M_MODE, "parity": "even"},
            "master_equation": "Zerilli",
            "radius_domain": [RADIUS_MIN, RADIUS_MAX],
            "waveform_samples": WAVEFORM_SAMPLES,
            "waveform_cycles": WAVEFORM_CYCLES,
            "claim_boundary": [
                "This is an exact distributional circular point-particle source for the (2,2) first-order mode.",
                "It is not the complete sum over all radiative modes.",
                "The compiled radius model is a Chebyshev interpolation of exact-source records.",
                "The direct homogeneous solver uses finite-tolerance numerical integration and asymptotic boundary series.",
                "No unique 13D simplification is claimed."
            ],
        },
        "external_validation": reference_validation,
        "compiler": {
            "training_nodes": TRAINING_NODES,
            "training_seconds": compiler.training_seconds,
            "domain": [RADIUS_MIN, RADIUS_MAX],
            "held_out_radii": HELD_OUT_RADII,
            "break_even_queries": break_even_queries,
        },
        "held_out_validation": {
            key: value
            for key, value in validation.items()
            if key != "rows"
        },
        "timing": {
            "direct": direct_timing,
            "compiled": compiled_timing,
            "online_speedup": online_speedup,
            "amortized": amortized,
        },
        "reference_mode_r0_10": {
            key: (
                [float(np.real(value)), float(np.imag(value))]
                if isinstance(value, complex)
                else value
            )
            for key, value in reference_mode.items()
            if key != "source"
        },
        "13d_ablation": ablation_13d,
    }

    required = [
        reference_validation["relative_error_infinity"] < 1.0e-9,
        reference_validation["relative_error_horizon"] < 1.0e-9,
        validation["maximum_record_error"] < 1.0e-9,
        validation["maximum_flux_error"] < 1.0e-9,
        validation["maximum_waveform_error"] < 1.0e-9,
        100.0 <= online_speedup <= 1000.0,
        ablation_13d["numerical_rank"] == 2,
    ]
    certificate["all_required_checks_pass"] = bool(all(required))

    cert_path = root / "exact_point_particle_certificate.json"
    cert_path.write_text(
        json.dumps(certificate, indent=2) + "\n",
        encoding="utf-8",
    )

    certificate["artifacts"] = {
        path.name: {
            "bytes": path.stat().st_size,
            "sha256": sha256(path),
        }
        for path in sorted(root.iterdir())
        if path.is_file() and path.name != cert_path.name
    }
    cert_path.write_text(
        json.dumps(certificate, indent=2) + "\n",
        encoding="utf-8",
    )

    print(
        json.dumps(
            {
                "certificate": str(cert_path),
                "pass": certificate["all_required_checks_pass"],
                "flux_infinity_r0_10": reference_mode[
                    "energy_flux_infinity"
                ],
                "flux_horizon_r0_10": reference_mode[
                    "energy_flux_horizon"
                ],
                "external_relative_error_infinity": reference_validation[
                    "relative_error_infinity"
                ],
                "external_relative_error_horizon": reference_validation[
                    "relative_error_horizon"
                ],
                "held_out_waveform_error": validation[
                    "maximum_waveform_error"
                ],
                "online_speedup": online_speedup,
                "training_seconds": compiler.training_seconds,
                "break_even_queries": break_even_queries,
                "13d_status": ablation_13d["status"],
            },
            indent=2,
        )
    )


if __name__ == "__main__":
    main()
Appendix B

Machine-readable certificate

PASS
{
  "certificate_id": "EXACT-POINT-PARTICLE-RWZ-4D/V1",
  "date": "2026-08-06",
  "status": "PASS-EXACT-SOURCE-DOMINANT-MODE-4D",
  "environment": {
    "python": "3.13.5",
    "numpy": "2.3.5",
    "scipy": "1.17.0",
    "platform": "Linux-6.18.35-x86_64-with-glibc2.41"
  },
  "scope": {
    "background": "Schwarzschild, M=1",
    "particle": "Exact point particle on an equatorial circular geodesic",
    "mode": {
      "ell": 2,
      "m": 2,
      "parity": "even"
    },
    "master_equation": "Zerilli",
    "radius_domain": [
      6.5,
      20.0
    ],
    "waveform_samples": 2048,
    "waveform_cycles": 4,
    "claim_boundary": [
      "This is an exact distributional circular point-particle source for the (2,2) first-order mode.",
      "It is not the complete sum over all radiative modes.",
      "The compiled radius model is a Chebyshev interpolation of exact-source records.",
      "The direct homogeneous solver uses finite-tolerance numerical integration and asymptotic boundary series.",
      "No unique 13D simplification is claimed."
    ]
  },
  "external_validation": {
    "r0": 10.0,
    "computed_energy_flux_infinity": 2.6843977395526686e-05,
    "reference_energy_flux_infinity": 2.6843977395510576e-05,
    "relative_error_infinity": 6.001557228054866e-13,
    "computed_energy_flux_horizon": 5.654138734509904e-09,
    "reference_energy_flux_horizon": 5.6541387345369355e-09,
    "relative_error_horizon": 4.780822775448615e-12
  },
  "compiler": {
    "training_nodes": 40,
    "training_seconds": 2.8257337449999795,
    "domain": [
      6.5,
      20.0
    ],
    "held_out_radii": 32,
    "break_even_queries": 40.97790466742659
  },
  "held_out_validation": {
    "maximum_record_error": 1.2053215754427937e-12,
    "median_record_error": 1.2124607820904756e-13,
    "maximum_flux_error": 9.626855047233391e-14,
    "maximum_waveform_error": 5.150482178927028e-14
  },
  "timing": {
    "direct": {
      "queries": 18,
      "repeats": 3,
      "all_seconds": [
        1.2398910659999842,
        1.246846387000005,
        1.2428076669999655
      ],
      "median_seconds": 1.2428076669999655,
      "median_microseconds_per_query": 69044.87038888696,
      "checksum": -5.92961928390636
    },
    "compiled": {
      "queries": 18,
      "repeats": 11,
      "all_seconds": [
        0.0016517060000182937,
        0.0016513549999785937,
        0.0015600899999981266,
        0.0016068489999838675,
        0.001578798000025472,
        0.0015743810000117264,
        0.0015655469999842353,
        0.0015727179999771579,
        0.00155947900003639,
        0.0015715069999941988,
        0.0015432750000172746
      ],
      "median_seconds": 0.0015727179999771579,
      "median_microseconds_per_query": 87.37322222095321,
      "checksum": -5.929619283906369
    },
    "online_speedup": 790.2291873164903,
    "amortized": [
      {
        "queries": 1,
        "direct_total_seconds": 0.06904487038888697,
        "compiled_total_seconds": 2.8258211182222004,
        "amortized_speedup": 0.024433560193762353
      },
      {
        "queries": 10,
        "direct_total_seconds": 0.6904487038888697,
        "compiled_total_seconds": 2.826607477222189,
        "amortized_speedup": 0.24426762804979169
      },
      {
        "queries": 100,
        "direct_total_seconds": 6.904487038888697,
        "compiled_total_seconds": 2.8344710672220748,
        "amortized_speedup": 2.4358996352908426
      },
      {
        "queries": 1000,
        "direct_total_seconds": 69.04487038888698,
        "compiled_total_seconds": 2.9131069672209327,
        "amortized_speedup": 23.70145386551147
      },
      {
        "queries": 10000,
        "direct_total_seconds": 690.4487038888698,
        "compiled_total_seconds": 3.6994659672095116,
        "amortized_speedup": 186.63469538812157
      },
      {
        "queries": 100000,
        "direct_total_seconds": 6904.4870388886975,
        "compiled_total_seconds": 11.563055967095302,
        "amortized_speedup": 597.1161134683273
      }
    ]
  },
  "reference_mode_r0_10": {
    "r0": 10.0,
    "ell": 2,
    "m": 2,
    "omega": 0.06324555320336758,
    "z_infinity": [
      -0.34312476564519523,
      0.09361975469581998
    ],
    "z_horizon": [
      0.005105006304623301,
      -0.000763813914397387
    ],
    "energy_flux_infinity": 2.6843977395526686e-05,
    "energy_flux_horizon": 5.654138734509904e-09,
    "angular_momentum_flux_infinity": 0.0008488811002793898,
    "angular_momentum_flux_horizon": 1.7879956607633384e-07,
    "h_mode_amplitude": [
      -0.5603203959618583,
      0.1528804192328589
    ],
    "wronskian_r": [
      404.807801953767,
      184.24931305682128
    ],
    "nfev_in": 2366,
    "nfev_up": 6962
  },
  "13d_ablation": {
    "4d_jump_master_count": 2,
    "normalized_singular_values": [
      1.380595403844256,
      0.3065229695864822
    ],
    "numerical_rank": 2,
    "unique_13d_reduction_found": false,
    "status": "NO-UNIQUE-13D-GAIN-IN-SCOPED-MODE",
    "reason": "Circular symmetry and even parity already reduce the exact point-particle source to two independent jump functions in 4D. No derived 13D identity was found that lowers this rank."
  },
  "all_required_checks_pass": true,
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}
Appendix C

Exact face-on waveform samples

RAW EXECUTION DATA
t_over_M,D_h_plus_over_mu,D_h_cross_over_mu,complex_strain_real,complex_strain_imag
0.0,-0.3534406534506446,-0.09643438943662568,-0.3534406534506446,0.09643438943662568
0.194129716967193,-0.3522300376245951,-0.10076650024988361,-0.3522300376245951,0.10076650024988361
0.388259433934386,-0.35096632541185274,-0.10508342116877868,-0.35096632541185274,0.10508342116877868
0.5823891509015791,-0.3496497073088125,-0.10938450144556668,-0.3496497073088125,0.10938450144556668
0.776518867868772,-0.348280381787088,-0.11366909272037713,-0.348280381787088,0.11366909272037713
0.970648584835965,-0.34685855526359316,-0.1179365491189493,-0.34685855526359316,0.1179365491189493
1.1647783018031581,-0.34538444206942664,-0.12218622734999364,-0.34538444206942664,0.12218622734999364
1.358908018770351,-0.34385826441756234,-0.12641748680216364,-0.34385826441756234,0.12641748680216364
1.553037735737544,-0.3422802523693526,-0.13062968964062394,-0.3422802523693526,0.13062968964062394
1.747167452704737,-0.3406506437998477,-0.13482220090319963,-0.3406506437998477,0.13482220090319963
1.94129716967193,-0.33896968436193786,-0.13899438859609287,-0.33896968436193786,0.13899438859609287
2.135426886639123,-0.3372376274493226,-0.14314562378915163,-0.3372376274493226,0.14314562378915163
2.3295566036063162,-0.33545473415831345,-0.14727528071067703,-0.33545473415831345,0.14727528071067703
2.523686320573509,-0.33362127324847507,-0.15138273684175407,-0.33362127324847507,0.15138273684175407
2.717816037540702,-0.3317375211021119,-0.15546737301009272,-0.3317375211021119,0.15546737301009272
2.911945754507895,-0.32980376168260495,-0.15952857348336366,-0.32980376168260495,0.15952857348336366
3.106075471475088,-0.3278202864916064,-0.16356572606201625,-0.3278202864916064,0.16356572606201625
3.300205188442281,-0.3257873945250976,-0.16757822217156348,-0.3257873945250976,0.16757822217156348
3.494334905409474,-0.32370539222831707,-0.17156545695432068,-0.32370539222831707,0.17156545695432068
3.688464622376667,-0.32157459344956646,-0.17552682936058378,-0.32157459344956646,0.17552682936058378
3.88259433934386,-0.31939531939289945,-0.17946174223923383,-0.31939531939289945,0.17946174223923383
4.076724056311053,-0.31716789856970246,-0.18336960242775346,-0.31716789856970246,0.18336960242775346
4.270853773278246,-0.31489266674917393,-0.18724982084164232,-0.31489266674917393,0.18724982084164232
4.464983490245439,-0.3125699669077089,-0.19110181256321782,-0.3125699669077089,0.19110181256321782
4.6591132072126324,-0.3102001491771978,-0.1949249969297876,-0.3102001491771978,0.1949249969297876
4.853242924179825,-0.3077835707922464,-0.19871879762118053,-0.3077835707922464,0.19871879762118053
5.047372641147018,-0.3053205960363249,-0.20248264274662353,-0.3053205960363249,0.20248264274662353
5.241502358114211,-0.30281159618685444,-0.20621596493095018,-0.30281159618685444,0.20621596493095018
5.435632075081404,-0.30025694945923975,-0.2099182014001287,-0.30025694945923975,0.2099182014001287
5.629761792048597,-0.29765704094985546,-0.21358879406609685,-0.29765704094985546,0.21358879406609685
5.82389150901579,-0.29501226257799523,-0.21722718961088971,-0.29501226257799523,0.21722718961088971
6.018021225982983,-0.2923230130267928,-0.22083283957004896,-0.2923230130267928,0.22083283957004896
6.212150942950176,-0.2895896976831226,-0.22440520041530035,-0.2895896976831226,0.22440520041530035
6.406280659917369,-0.2868127285764908,-0.22794373363648687,-0.2868127285764908,0.22794373363648687
6.600410376884562,-0.2839925243169243,-0.23144790582274583,-0.2839925243169243,0.23144790582274583
6.794540093851755,-0.281129510031868,-0.234917188742917,-0.281129510031868,0.234917188742917
6.988669810818948,-0.2782241173020996,-0.23835105942517007,-0.2782241173020996,0.23835105942517007
7.182799527786141,-0.27527678409667183,-0.24174900023583917,-0.27527678409667183,0.24174900023583917
7.376929244753334,-0.27228795470689104,-0.24511049895745288,-0.27228795470689104,0.24511049895745288
7.571058961720527,-0.2692580796793435,-0.24843504886594753,-0.2692580796793435,0.24843504886594753
7.76518867868772,-0.26618761574797845,-0.2517221488070528,-0.26618761574797845,0.2517221488070528
7.959318395654913,-0.26307702576525815,-0.25497130327183715,-0.26307702576525815,0.25497130327183715
8.153448112622106,-0.25992677863238617,-0.25818202247140287,-0.25992677863238617,0.25818202247140287
8.3475778295893,-0.2567373492286234,-0.26135382241071836,-0.2567373492286234,0.26135382241071836
8.541707546556491,-0.25350921833970336,-0.2644862249615773,-0.25350921833970336,0.2644862249615773
8.735837263523685,-0.25024287258535627,-0.2675787579346735,-0.25024287258535627,0.2675787579346735
8.929966980490878,-0.24693880434595517,-0.27063095515078006,-0.24693880434595517,0.27063095515078006
9.124096697458072,-0.243597511688292,-0.27364235651102314,-0.243597511688292,0.27364235651102314
9.318226414425265,-0.24021949829049766,-0.2766125080662388,-0.24021949829049766,0.2766125080662388
9.512356131392457,-0.2368052733661158,-0.2795409620854028,-0.2368052733661158,0.2795409620854028
9.70648584835965,-0.23335535158734208,-0.28242727712312315,-0.23335535158734208,0.28242727712312315
9.900615565326843,-0.2298702530074407,-0.28527101808618527,-0.2298702530074407,0.28527101808618527
10.094745282294037,-0.22635050298234985,-0.2880717562991393,-0.22635050298234985,0.2880717562991393
10.288874999261228,-0.22279663209148756,-0.2908290695689202,-0.22279663209148756,0.2908290695689202
10.483004716228422,-0.21920917605777046,-0.2935425422484905,-0.21920917605777046,0.2935425422484905
10.677134433195615,-0.215588675666857,-0.29621176529949683,-0.215588675666857,0.29621176529949683
10.871264150162808,-0.21193567668562743,-0.2988363363539291,-0.21193567668562743,0.2988363363539291
11.065393867130002,-0.20825072977991307,-0.3014158597747754,-0.20825072977991307,0.3014158597747754
11.259523584097193,-0.2045343904314872,-0.30394994671566145,-0.2045343904314872,0.30394994671566145
11.453653301064387,-0.20078721885432926,-0.3064382151794667,-0.20078721885432926,0.3064382151794667
11.64778301803158,-0.19700977991017693,-0.30888029007590806,-0.19700977991017693,0.30888029007590806
11.841912734998774,-0.19320264302337634,-0.3112758032780821,-0.19320264302337634,0.3112758032780821
12.036042451965965,-0.18936638209504536,-0.3136243936779579,-0.18936638209504536,0.3136243936779579
12.230172168933159,-0.18550157541656156,-0.31592570724081187,-0.18550157541656156,0.31592570724081187
12.424301885900352,-0.18160880558238884,-0.31817939705859577,-0.18160880558238884,0.31817939705859577
12.618431602867545,-0.1776886594022547,-0.3203851234022313,-0.1776886594022547,0.3203851234022313
12.812561319834739,-0.1737417278126928,-0.32254255377282176,-0.1737417278126928,0.32254255377282176
13.00669103680193,-0.16976860578796293,-0.32465136295177427,-0.16976860578796293,0.32465136295177427
13.200820753769124,-0.16576989225036268,-0.3267112330498243,-0.16576989225036268,0.3267112330498243
13.394950470736317,-0.1617461899799437,-0.3287218535549552,-0.1617461899799437,0.3287218535549552
13.58908018770351,-0.15769810552364644,-0.33068292137920624,-0.15769810552364644,0.33068292137920624
13.783209904670702,-0.1536262491038675,-0.33259414090436096,-0.1536262491038675,0.33259414090436096
13.977339621637896,-0.14953123452647196,-0.3344552240265095,-0.14953123452647196,0.3344552240265095
14.171469338605089,-0.14541367908826683,-0.3362658901994789,-0.14541367908826683,0.3362658901994789
14.365599055572282,-0.14127420348394715,-0.33802586647712296,-0.14127420348394715,0.33802586647712296
14.559728772539476,-0.13711343171253024,-0.3397348875544676,-0.13711343171253024,0.3397348875544676
14.753858489506667,-0.1329319909832923,-0.34139269580770376,-0.1329319909832923,0.34139269580770376
14.94798820647386,-0.12873051162122023,-0.3429990413330223,-0.12873051162122023,0.3429990413330223
15.142117923441054,-0.12450962697199473,-0.34455368198428554,-0.12450962697199473,0.34455368198428554
15.336247640408248,-0.12026997330651701,-0.34605638340952927,-0.12026997330651701,0.34605638340952927
15.53037735737544,-0.11601218972499565,-0.34750691908628967,-0.11601218972499565,0.34750691908628967
15.724507074342633,-0.11173691806060587,-0.3489050703557501,-0.11173691806060587,0.3489050703557501
15.918636791309826,-0.10744480278273776,-0.3502506264557025,-0.10744480278273776,0.3502506264557025
16.11276650827702,-0.10313649089984656,-0.35154338455231854,-0.10313649089984656,0.35154338455231854
16.306896225244213,-0.09881263186192027,-0.3527831497707254,-0.09881263186192027,0.3527831497707254
16.501025942211406,-0.09447387746257943,-0.3539697352243818,-0.09447387746257943,0.3539697352243818
16.6951556591786,-0.09012088174082349,-0.35510296204325015,-0.09012088174082349,0.35510296204325015
16.88928537614579,-0.08575430088243853,-0.35618265940075977,-0.08575430088243853,0.35618265940075977
17.083415093112983,-0.08137479312108142,-0.35720866453955824,-0.08137479312108142,0.35720866453955824
17.277544810080176,-0.07698301863905582,-0.35818082279604563,-0.07698301863905582,0.35818082279604563
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17.665804244014563,-0.06816531938805803,-0.3599630206151147,-0.06816531938805803,0.3599630206151147
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18.05406367794895,-0.05930651977226483,-0.36152817827955736,-0.05930651977226483,0.36152817827955736
18.248193394916143,-0.05486337564261715,-0.36222906701523905,-0.05486337564261715,0.36222906701523905
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370.7877594073386,-0.05659253439539529,0.3619629428937988,-0.05659253439539529,-0.3619629428937988
370.98188912430584,-0.0610322808476775,0.3612408444129258,-0.0610322808476775,-0.3612408444129258
371.176018841273,-0.06546282708067423,0.36046429122485796,-0.06546282708067423,-0.36046429122485796
371.3701485582402,-0.06988350521836693,0.35963340038993585,-0.06988350521836693,-0.35963340038993585
371.5642782752074,-0.07429364887228436,0.35874829715954776,-0.07429364887228436,-0.35874829715954776
371.75840799217457,-0.0786925932419602,0.35780911495724765,-0.0786925932419602,-0.35780911495724765
371.9525377091418,-0.08307967521515114,0.35681599535864206,-0.08307967521515114,-0.35681599535864206
372.146667426109,-0.08745423346778924,0.3557690880700498,-0.08745423346778924,-0.3557690880700498
372.3407971430762,-0.09181560856367717,0.35466855090593397,-0.09181560856367717,-0.35466855090593397
372.53492686004336,-0.09616314305389147,0.35351454976511254,-0.09616314305389147,-0.35351454976511254
372.72905657701057,-0.10049618157589371,0.3523072586057495,-0.10049618157589371,-0.3523072586057495
372.9231862939778,-0.10481407095231461,0.3510468594191331,-0.10481407095231461,-0.3510468594191331
373.11731601094493,-0.10911616028942,0.3497335422022415,-0.10911616028942,-0.3497335422022415
373.31144572791214,-0.11340180107523266,0.3483675049291004,-0.11340180107523266,-0.3483675049291004
373.50557544487936,-0.11767034727728365,0.3469489535209422,-0.11767034727728365,-0.3469489535209422
373.6997051618465,-0.12192115544000139,0.3454781018151635,-0.12192115544000139,-0.3454781018151635
373.8938348788137,-0.126153584781712,0.34395517153308924,-0.126153584781712,-0.34395517153308924
374.08796459578093,-0.1303669972912256,0.3423803922465523,-0.1303669972912256,-0.3423803922465523
374.2820943127481,-0.13456075782401614,0.34075400134328565,-0.13456075782401614,-0.34075400134328565
374.4762240297153,-0.13873423419796874,0.33907624399113634,-0.13873423419796874,-0.33907624399113634
374.6703537466825,-0.1428867972886699,0.33734737310111074,-0.1428867972886699,-0.33734737310111074
374.86448346364966,-0.1470178211242476,0.33556764928924865,-0.1470178211242476,-0.33556764928924865
375.0586131806169,-0.15112668297973594,0.3337373408373355,-0.15112668297973594,-0.3337373408373355
375.2527428975841,-0.15521276347093968,0.3318567236524636,-0.15521276347093968,-0.3318567236524636
375.44687261455124,-0.15927544664780624,0.3299260812254395,-0.15927544664780624,-0.3299260812254395
375.64100233151845,-0.1633141200872793,0.3279457045880479,-0.1633141200872793,-0.3279457045880479
375.83513204848566,-0.16732817498561084,0.3259158922691838,-0.16732817498561084,-0.3259158922691838
376.0292617654528,-0.17131700625013774,0.3238369502498498,-0.17131700625013774,-0.3238369502498498
376.22339148242,-0.17528001259049908,0.32170919191702924,-0.17528001259049908,-0.32170919191702924
376.41752119938724,-0.17921659660926972,0.31953293801644883,-0.17921659660926972,-0.31953293801644883
376.6116509163544,-0.18312616489201766,0.317308516604227,-0.18312616489201766,-0.317308516604227
376.8057806333216,-0.18700812809676082,0.315036262997419,-0.18700812809676082,-0.315036262997419
376.9999103502888,-0.19086190104279951,0.31271651972347425,-0.19086190104279951,-0.31271651972347425
377.194040067256,-0.19468690279893336,0.3103496364685999,-0.19468690279893336,-0.3103496364685999
377.3881697842232,-0.1984825567710306,0.3079359700250493,-0.1984825567710306,-0.3079359700250493
377.5822995011904,-0.2022482907889503,0.3054758842373349,-0.2022482907889503,-0.3054758842373349
377.7764292181576,-0.2059835371927866,0.3029697499473856,-0.2059835371927866,-0.3029697499473856
377.97055893512476,-0.20968773291844314,0.30041794493864227,-0.20968773291844314,-0.30041794493864227
378.16468865209197,-0.21336031958251464,0.29782085387910745,-0.21336031958251464,-0.29782085387910745
378.3588183690592,-0.2170007435664529,0.2951788682633633,-0.2170007435664529,-0.2951788682633633
378.55294808602633,-0.22060845610002452,0.29249238635355457,-0.22060845610002452,-0.29249238635355457
378.74707780299354,-0.22418291334403764,0.28976181311935006,-0.22418291334403764,-0.28976181311935006
378.94120751996076,-0.2277235764723158,0.28698756017690175,-0.2277235764723158,-0.28698756017690175
379.1353372369279,-0.23122991175292562,0.2841700457267931,-0.23122991175292562,-0.2841700457267931
379.3294669538951,-0.23470139062863649,0.28130969449099624,-0.23470139062863649,-0.28130969449099624
379.52359667086233,-0.23813748979659108,0.2784069376488523,-0.23813748979659108,-0.2784069376488523
379.7177263878295,-0.2415376912871928,0.27546221277207183,-0.2415376912871928,-0.27546221277207183
379.9118561047967,-0.24490148254218944,0.2724759637587711,-0.24490148254218944,-0.2724759637587711
380.1059858217639,-0.24822835649193187,0.2694486407665621,-0.24822835649193187,-0.2694486407665621
380.30011553873106,-0.2515178116318146,0.26638070014469234,-0.2515178116318146,-0.26638070014469234
380.4942452556983,-0.2547693520978772,0.26327260436524974,-0.2547693520978772,-0.26327260436524974
380.6883749726655,-0.25798248774154686,0.2601248219534539,-0.25798248774154686,-0.2601248219534539
380.88250468963264,-0.2611567342035278,0.2569378274170259,-0.2611567342035278,-0.2569378274170259
381.07663440659985,-0.26429161298681814,0.253712101174657,-0.26429161298681814,-0.253712101174657
381.27076412356706,-0.26738665152883484,0.2504481294835937,-0.26738665152883484,-0.2504481294835937
381.4648938405343,-0.2704413832726528,0.24714640436633417,-0.2704413832726528,-0.24714640436633417
381.6590235575014,-0.2734553477373338,0.24380742353646107,-0.2734553477373338,-0.24380742353646107
381.85315327446864,-0.27642809058734424,0.24043169032361034,-0.27642809058734424,-0.24043169032361034
382.04728299143585,-0.2793591637010382,0.23701971359760354,-0.2793591637010382,-0.23701971359760354
382.241412708403,-0.2822481252382114,0.23357200769173603,-0.2822481252382114,-0.23357200769173603
382.4355424253702,-0.28509453970670834,0.2300890923252416,-0.28509453970670834,-0.2300890923252416
382.6296721423374,-0.28789797802806477,0.22657149252495418,-0.28789797802806477,-0.22657149252495418
382.8238018593046,-0.2906580176021911,0.22301973854616056,-0.2906580176021911,-0.22301973854616056
383.0179315762718,-0.29337424237107895,0.2194343657926645,-0.29337424237107895,-0.2194343657926645
383.212061293239,-0.29604624288151427,0.2158159147360845,-0.29604624288151427,-0.2158159147360845
383.40619101020616,-0.29867361634680256,0.21216493083437815,-0.29867361634680256,-0.21216493083437815
383.60032072717337,-0.30125596670748805,0.20848196444961445,-0.30125596670748805,-0.20848196444961445
383.7944504441406,-0.30379290469105297,0.20476757076501706,-0.30379290469105297,-0.20476757076501706
383.98858016110773,-0.30628404787059993,0.20102230970127047,-0.30628404787059993,-0.20102230970127047
384.18270987807495,-0.30872902072250263,0.19724674583211255,-0.30872902072250263,-0.19724674583211255
384.37683959504216,-0.3111274546830089,0.19344144829923526,-0.3111274546830089,-0.19344144829923526
384.5709693120093,-0.3134789882038019,0.18960699072648693,-0.3134789882038019,-0.18960699072648693
384.7650990289765,-0.31578326680650287,0.1857439511333989,-0.31578326680650287,-0.1857439511333989
384.95922874594373,-0.3180399431361033,0.1818529118480582,-0.3180399431361033,-0.1818529118480582
385.1533584629109,-0.320248677013325,0.17793445941932803,-0.320248677013325,-0.17793445941932803
385.3474881798781,-0.32240913548590905,0.17398918452841317,-0.32240913548590905,-0.17398918452841317
385.5416178968453,-0.32452099287879327,0.17001768189984148,-0.32452099287879327,-0.17001768189984148
385.73574761381246,-0.32658393084321463,0.16602055021179707,-0.32658393084321463,-0.16602055021179707
385.9298773307797,-0.3285976384046971,0.1619983920058758,-0.3285976384046971,-0.1619983920058758
386.1240070477469,-0.33056181200992546,0.15795181359626156,-0.33056181200992546,-0.15795181359626156
386.3181367647141,-0.3324761555725069,0.1538814249783242,-0.3324761555725069,-0.1538814249783242
386.51226648168125,-0.334340380517603,0.14978783973666848,-0.334340380517603,-0.14978783973666848
386.70639619864846,-0.3361542058254322,0.1456716749526359,-0.3361542058254322,-0.1456716749526359
386.9005259156157,-0.33791735807362894,0.14153355111129057,-0.33791735807362894,-0.14153355111129057
387.0946556325828,-0.3396295714784617,0.1373740920078817,-0.3396295714784617,-0.1373740920078817
387.28878534955004,-0.3412905879348997,0.13319392465380675,-0.3412905879348997,-0.13319392465380675
387.48291506651725,-0.34290015705551763,0.12899367918210097,-0.34290015705551763,-0.12899367918210097
387.6770447834844,-0.3444580362082414,0.1247739887524452,-0.3444580362082414,-0.1247739887524452
387.8711745004516,-0.34596399055292454,0.12053548945571749,-0.34596399055292454,-0.12053548945571749
388.0653042174188,-0.3474177930767465,0.11627882021811387,-0.3474177930767465,-0.11627882021811387
388.259433934386,-0.34881922462843407,0.1120046227048309,-0.34881922462843407,-0.1120046227048309
388.4535636513532,-0.35016807395129895,0.10771354122333483,-0.35016807395129895,-0.10771354122333483
388.6476933683204,-0.35146413771508056,0.10340622262624408,-0.35146413771508056,-0.10340622262624408
388.84182308528756,-0.3527072205465977,0.09908331621381651,-0.3527072205465977,-0.09908331621381651
389.03595280225477,-0.3538971350592013,0.09474547363606814,-0.3538971350592013,-0.09474547363606814
389.230082519222,-0.35503370188101946,0.09039334879454829,-0.35503370188101946,-0.09039334879454829
389.42421223618913,-0.35611674968199747,0.08602759774376487,-0.35611674968199747,-0.08602759774376487
389.61834195315635,-0.3571461151997259,0.08164887859228423,-0.3571461151997259,-0.08164887859228423
389.81247167012356,-0.3581216432640493,0.0772578514035334,-0.3581216432640493,-0.0772578514035334
390.0066013870907,-0.3590431868204582,0.07285517809629594,-0.3590431868204582,-0.07285517809629594
390.2007311040579,-0.35991060695225713,0.06844152234492822,-0.35991060695225713,-0.06844152234492822
390.39486082102513,-0.3607237729015042,0.06401754947932237,-0.3607237729015042,-0.06401754947932237
390.5889905379923,-0.36148256208872237,0.05958392638460843,-0.36148256208872237,-0.05958392638460843
390.7831202549595,-0.3621868601313785,0.05514132140062188,-0.3621868601313785,-0.05514132140062188
390.9772499719267,-0.36283656086112454,0.05069040422116332,-0.36283656086112454,-0.05069040422116332
391.1713796888939,-0.36343156633980195,0.04623184579304158,-0.36343156633980195,-0.04623184579304158
391.3655094058611,-0.3639717868742057,0.04176631821493522,-0.3639717868742057,-0.04176631821493522
391.5596391228283,-0.36445714102960497,0.03729449463607298,-0.36445714102960497,-0.03729449463607298
391.7537688397955,-0.36488755564201825,0.03281704915476864,-0.36488755564201825,-0.03281704915476864
391.94789855676265,-0.36526296582924267,0.028334656716801153,-0.36526296582924267,-0.028334656716801153
392.14202827372986,-0.36558331500063507,0.02384799301366717,-0.36558331500063507,-0.02384799301366717
392.3361579906971,-0.3658485548656417,0.019357734380732676,-0.3658485548656417,-0.019357734380732676
392.53028770766423,-0.3660586454410786,0.014864557695275869,-0.3660586454410786,-0.014864557695275869
392.72441742463144,-0.36621355505715825,0.010369140274448557,-0.36621355505715825,-0.010369140274448557
392.91854714159865,-0.3663132603622638,0.005872159773182677,-0.3663132603622638,-0.005872159773182677
393.1126768585658,-0.36635774632646917,0.0013742940820344167,-0.36635774632646917,-0.0013742940820344167
393.306806575533,-0.36634700624380473,-0.003123778775007351,-0.36634700624380473,0.003123778775007351
393.5009362925002,-0.3662810417332679,-0.00762138074272096,-0.3662810417332679,0.00762138074272096
393.6950660094674,-0.36615986273857964,-0.01211783383686819,-0.36615986273857964,0.01211783383686819
393.8891957264346,-0.365983487526685,-0.016612460246400014,-0.365983487526685,0.016612460246400014
394.0833254434018,-0.3657519426849998,-0.02110458243562422,-0.3657519426849998,0.02110458243562422
394.27745516036896,-0.36546526311740263,-0.025593523246343633,-0.36546526311740263,0.025593523246343633
394.47158487733617,-0.36512349203897304,-0.030078605999937,-0.36512349203897304,0.030078605999937
394.6657145943034,-0.3647266809694778,-0.034559154599356254,-0.3647266809694778,0.034559154599356254
394.85984431127054,-0.3642748897256043,-0.03903449363104749,-0.3642748897256043,0.03903449363104749
395.05397402823775,-0.36376818641194303,-0.04350394846676921,-0.36376818641194303,0.04350394846676921
395.24810374520496,-0.36320664741072234,-0.04796684536528028,-0.36320664741072234,0.04796684536528028
395.44223346217217,-0.3625903573702933,-0.05242251157390718,-0.3625903573702933,0.05242251157390718
395.6363631791393,-0.36191940919237053,-0.056870275429954625,-0.36191940919237053,0.056870275429954625
395.83049289610653,-0.3611939040180265,-0.0613094664619594,-0.3611939040180265,0.0613094664619594
396.02462261307375,-0.36041395121244646,-0.06573941549075189,-0.36041395121244646,0.06573941549075189
396.2187523300409,-0.3595796683484421,-0.07015945473033434,-0.3595796683484421,0.07015945473033434
396.4128820470081,-0.3586911811887271,-0.0745689178885491,-0.3586911811887271,0.0745689178885491
396.6070117639753,-0.3577486236669609,-0.07896714026751012,-0.3577486236669609,0.07896714026751012
396.8011414809425,-0.35675213786755827,-0.08335345886380573,-0.35675213786755827,0.08335345886380573
396.9952711979097,-0.35570187400427006,-0.08772721246844586,-0.35570187400427006,0.08772721246844586
397.1894009148769,-0.35459799039754136,-0.09208774176652737,-0.35459799039754136,0.09208774176652737
397.38353063184405,-0.35344065345064474,-0.09643438943662533,-0.35344065345064474,0.09643438943662533
Appendix D

Held-out validation records

RAW EXECUTION DATA
r0,record_error,flux_error,waveform_error
6.6053784202124275,4.4945469460501314e-13,3.805608187541821e-14,1.4382461956245785e-14
6.892639211254101,1.2053215754427937e-12,6.16278060555677e-14,3.716134123563862e-15
7.0399976549358,3.631683693135216e-13,9.626855047233391e-14,5.150482178927028e-14
7.169426606923883,1.837260962727949e-13,2.000854097667539e-14,1.4323746300146343e-14
7.896982512821115,1.1496245005815838e-13,2.4133213632553163e-15,9.098174884140375e-15
8.264656607256214,8.964693753973167e-14,4.110686523211672e-15,5.264724960578015e-15
8.528265893359631,1.2596253123695842e-13,9.358221165800498e-14,4.7664840350369455e-14
9.651411958253291,1.567471748143122e-13,8.170999831464348e-14,4.167560333907136e-14
9.691228382269717,6.233355029940129e-13,7.375619349429821e-14,3.601689222831583e-14
10.066848070301642,2.913570935514049e-13,8.252950688347256e-15,7.897850184214247e-16
10.263844017082874,3.833712830515684e-13,1.4830159577125073e-14,6.3915392449711015e-15
10.434197397327988,2.2816499672230704e-13,1.105604428169048e-14,6.016350874841604e-15
11.50017884196156,7.888834140753062e-14,1.4793286925258238e-14,1.0214312967877572e-14
12.174526305894382,4.042976021019803e-13,3.9180094741929076e-14,1.957980086280241e-14
12.24408785628032,2.007877301071679e-13,1.408076367184333e-14,6.730181360488023e-15
12.68746515691591,1.6238615870976214e-13,1.5475974281908857e-14,2.2644341950329685e-14
13.201098813239685,2.6367549519261603e-13,4.271569813623345e-14,2.794043711828212e-14
13.417138023370384,1.6845667099969112e-13,1.7006492895736707e-14,8.89354050423708e-15
13.714423712947442,2.0368002529513428e-13,4.109451052295466e-14,3.26306346131862e-14
14.2018121853529,7.394340349491106e-14,7.354847375638151e-15,5.107841557000012e-15
14.767810079657288,9.857254057966946e-14,5.937717670616271e-14,3.519311845043203e-14
14.908099264245582,6.73714615634831e-14,3.508721401334277e-14,2.028338823203315e-14
17.559292548949365,4.9111752158374327e-14,1.7730818513955932e-14,1.835144463755074e-14
18.221337139491315,2.5376385366993245e-14,3.367929964702004e-15,2.0898290381065858e-14
18.229929491105672,1.7708191431753726e-14,1.5798610871394673e-14,8.63930702699598e-15
18.26060809825502,5.0715470289195536e-14,1.1757995923449122e-14,1.7471816799749532e-14
18.31662702318632,4.9739460859568995e-14,3.7053970712716626e-14,2.095016912806664e-14
18.930128738073183,4.0487762057790535e-14,2.404287197297139e-15,3.570794475552298e-15
19.349083000374584,1.1652962518113672e-13,2.8898678043189757e-15,5.828939495615788e-15
19.747769920158547,3.5121657475006956e-14,6.935091862520633e-15,6.455520005105972e-15
19.759419324646807,3.885050825120319e-14,2.1433969291661007e-14,1.2793408918453044e-14
19.937035586684544,3.7803941148099734e-14,4.294646272783716e-15,6.4183801139285075e-15
Part V

Complete Precursor and Observer-Compiler Authority

Full context preservation for the regularized-source precursor, observer compiler, building-block derivations, review contract, and executable appendices.

Executive technical abstract

Observer-Compiled Regge–Wheeler Worked Example

70–120 PAGE WORKED EXAMPLE

Purpose

This monograph is a complete worked example of the observer-compiled five-tensor method on a genuine general-relativistic perturbation operator. It is written so that a friendly NASA/JPL colleague or an independent AI reviewer can reproduce the calculation without access to the conversation that produced the architecture.

The physical equation is the odd-parity, spin-2 Regge–Wheeler equation on a Schwarzschild background. The source is a transparent regularized compact source family built from Gaussian and first-derivative channels. Those channels mimic the delta and delta-prime structure that appears in point-particle Regge–Wheeler–Zerilli sources, but they are not presented as the exact stress-energy source of a specified geodesic.

The benchmark compares three online paths:

  1. a reused-factor direct field solve at every frequency;
  2. a cached full-field five-tensor response followed by record projection;
  3. an observer-compiled map that produces horizon and infinity records without assembling the radial field.

Result

For 4,096 radial unknowns, 48 frequencies, 16 source channels, and 96 complex observer records, the measured sequential online speedup was

The compiled records agreed with the direct solver with maximum relative error

The corresponding flux error was

and the independent source-work balance closed to

The result satisfies the requested 100–1000× online target. The speedup rises from approximately 75× at 1,024 radial unknowns to approximately 604× at 8,192 unknowns.

What is and is not established

This document establishes that observer-first compilation can accelerate a physical Schwarzschild perturbation operator by hundreds of times when:

  • the operator and frequency grid are reused;
  • the source family lies in a low-dimensional basis;
  • only asymptotic and horizon records are required online.

It does not yet establish an end-to-end point-particle IMRI waveform model. The exact geodesic stress-energy source, even-parity Zerilli sector, orbital evolution, second-order self-force, detector response, and held-out q≈100 replay remain later stages.

Strong results without category errors

Claim and status taxonomy

Status labels

Label Meaning
ESTABLISHED-GR Standard result from Schwarzschild perturbation theory within its declared convention and domain.
EXACT-SCOPED Algebraically exact after the finite-dimensional operator, source basis, boundary conditions, and records are frozen.
PASS-PHYSICAL-OPERATOR Executed on the physical Regge–Wheeler operator, but possibly with a regularized or manufactured source.
PASS-PHYSICAL-SOURCE Executed with a source derived from a declared physical stress-energy tensor.
PASS-HELD-OUT Passed on parameters and data excluded from all compiler decisions.
OPEN A required derivation, datum, or computation remains.
FAIL A frozen acceptance condition is violated.

The current terminal is:

Four speed metrics

The document keeps four different speed claims separate:

  1. online speedup after compilation;
  2. amortized speedup including compilation;
  3. full-field assembly avoidance, comparing record-space evaluation with cached bulk reconstruction;
  4. algebraic operation ratio, which is a complexity estimate rather than a measured wall-clock result.

The headline 347.65× is an online wall-clock measurement. Compilation cost is reported separately and is not hidden.

The 100–1000× criterion

Why this example was selected

Selection criterion

The requested example had to satisfy all of the following:

  • recognizable general-relativistic physics;
  • reproducible on an ordinary scientific workstation;
  • enough state dimension to make observer compilation meaningful;
  • an independently checkable balance law;
  • a measured online speedup between 100× and 1000×;
  • a clear path to exact point-particle and IMRI extensions.

The Regge–Wheeler problem satisfies these conditions.

Why not begin with full numerical relativity

A full 3+1 binary-black-hole evolution would be more physically complete, but it would obscure the compiler mechanism behind mesh refinement, gauge evolution, nonlinear constraints, and large external software dependencies.

The Schwarzschild master equation retains real curvature, horizon propagation, radiation to infinity, parity structure, physical boundary conditions, and gravitational-wave flux, while reducing the field problem to a one-dimensional radial operator for each frequency.

Why not choose a trivial fixed matrix

A generic matrix benchmark can demonstrate linear algebra but cannot answer whether the same architecture respects black-hole boundary conditions, asymptotic radiation, or a physical conservation current.

The present example therefore occupies the correct middle ground:

How this file preserves and executes the prior work

Architecture-document chain

Companion documents

This worked example should be read alongside:

  1. MaxConservationStack.html
    Defines the maximal conservation and coupling constraints.

  2. FiveTensorGRArchitecture.html
    Defines the five physical ownership sectors:

  3. ObserverCompiledFiveTensorGR.html
    Derives the general compiler:

  4. This file
    Freezes those ideas into an executable Regge–Wheeler calculation.

Active tensor sectors in this example

The scoped linear example activates:

  • : linear source response;
  • : horizon and infinity flux;
  • : phase-corrected asymptotic frame records.

The following are zero by scope:

They become nonzero when nonlinear effective sources, orbital continuation, or noncommuting updates are introduced.

The residual contains discretization, finite-boundary, source-regularization, and basis-truncation error.

What an independent reviewer receives

Reproducibility contract

Review-pack contents

The review pack contains:

  • this self-contained HTML monograph;
  • the complete executable Python benchmark;
  • a machine-readable JSON certificate;
  • timing, convergence, scaling, and representative-record CSV files;
  • all figures used in the document;
  • the three companion architecture documents when available;
  • a content-addressed source manifest.

Determinism

The random parameter suite uses the frozen seed

The operator, grid, frequency range, basis centers, widths, and boundary conditions are all explicit in the code.

Independent replay

A reviewer can run:

python rwz_observer_compiled_worked_example.py

The code reconstructs all certificate values and fails its own acceptance test if the online speedup does not fall between 100× and 1000× or if the record, flux, balance, and convergence conditions fail.

Executed result

Benchmark dashboard

Online speedup347.65×Direct reused-factor solve → observer records
Over full reduced field61.34×Avoids 4,096-component field assembly
Break-even106.2 queriesIncludes 0.92 s offline compilation
Record error7.04e-11Compiled versus direct discretized solver
Flux error1.39e-12Derived horizon and infinity flux
Balance residual3.84e-11Independent source-work current

Sequential runtime

Sequential runtime

Log-scale median online time per source query.

Amortized speedup

Amortized speedup

Compilation is justified by repeated-query workloads rather than a single solve.

Part I

General-Relativistic Foundation

Schwarzschild geometry, parity decomposition, the Regge–Wheeler equation, sources, boundaries, and flux.

Vacuum spherical geometry

Deriving the Schwarzschild background

ESTABLISHED-GR

Spherical vacuum ansatz

Begin with a static spherically symmetric line element

The vacuum Einstein equations imply

Therefore

A second independent vacuum equation gives

After integration and asymptotic normalization of the time coordinate,

Thus

The benchmark sets .

Domain

The exterior Schwarzschild domain is

The event horizon is approached as , while future null infinity lies at .

Mapping horizon and infinity to wave boundaries

Tortoise coordinate and exact inverse

Definition

The tortoise coordinate is defined by

Integrating,

The horizon maps to

and spatial infinity maps to

Inverse map

Let

Then

Exponentiating gives

Therefore

where is the Lambert W function.

The code uses this exact inverse rather than an iterative root solve.

Tensor harmonics and parity

Linear metric perturbations

Linearized metric

Write

Linearizing the Einstein tensor gives

Because the background is spherically symmetric, the angular dependence can be decomposed into tensor spherical harmonics.

Parity

Under the parity map

the perturbations separate into:

  • even or polar parity, transforming as ;
  • odd or axial parity, transforming as .

The two sectors decouple at first order.

Scope

This example uses the odd-parity sector with

The even-parity Zerilli sector is mathematically parallel but has a different potential and source map. It is listed as the first physical expansion of the benchmark.

Axial harmonics and physical quotient

Odd-parity gauge-invariant sector

Odd-parity harmonics

Let be scalar spherical harmonics. Define the axial vector harmonic

A corresponding axial tensor harmonic is

The odd-parity metric perturbation is expanded in radial-time amplitudes multiplying these harmonics.

Gauge invariance

A gauge transformation generated by an odd-parity angular vector changes the raw amplitudes. The Cunningham–Price–Moncrief master variable is a gauge-invariant combination of those amplitudes.

Within the Martel–Poisson formalism, the master variable satisfies a covariant two-dimensional wave equation with a source derived from the odd-parity stress-energy multipoles.

The code does not reconstruct the full metric perturbation. It solves directly for the gauge-invariant master field, which is exactly the type of physical quotient advocated by the Rigidity compiler.

Physical spin-2 propagation on Schwarzschild

Regge–Wheeler master equation

PHYSICAL OPERATOR

Time-domain master equation

The odd-parity master function obeys

The Regge–Wheeler potential is

For the benchmark,

Structural meaning

The potential vanishes at both ends:

It has a barrier near the black-hole photon sphere. A compact source therefore produces:

  • radiation transmitted toward future null infinity;
  • radiation absorbed by the horizon;
  • scattering and phase delay caused by the curvature barrier.

These are real general-relativistic propagation effects, not generic matrix features.

Strong-field propagation

Regge–Wheeler potential

The curvature barrier

The curvature barrier

The physical odd-parity l=2 Schwarzschild potential used in every solve.

One reusable radial operator per frequency

Frequency-domain reduction

Fourier convention

Use

Then

The radial equation is

Frequency window

The benchmark freezes

with 48 evenly spaced frequency bins.

The frequency grid is part of the observer definition. Changing it changes the compiled record map and invalidates its hash.

Ingoing and outgoing physical domains

Horizon and infinity boundary conditions

Horizon

A physical mode entering the future event horizon behaves as

Therefore

Infinity

An outgoing mode at infinity behaves as

so

Finite computational boundary

The code truncates the domain to

First-order one-sided derivatives impose the Robin conditions at the endpoints. Finite-boundary and boundary-discretization error remain in ; they are not declared absent.

Gaussian and derivative channels motivated by RWZ point-particle structure

Regularized compact source

Point-particle motivation

In the exact RWZ point-particle problem, the stress-energy is supported on a worldline. After angular decomposition, the radial source generally contains distributional terms of the form

This is why the benchmark uses paired Gaussian and derivative channels.

Regularized basis

For center and width ,

and

Eight centers between approximately and produce sixteen channels:

Honest scope

The basis is physically motivated but not the exact projection of a specified geodesic stress-energy tensor. The benchmark therefore tests:

  • the physical propagation operator;
  • black-hole boundary conditions;
  • source-to-record compilation;
  • current and flux balance.

It does not claim a calibrated astrophysical waveform amplitude.

Five transparent online parameters

Source parameter family

Five source parameters

Each online query uses

They control:

  • radial source center ;
  • spectral center ;
  • spectral bandwidth ;
  • amplitude ;
  • phase .

Coefficient map

The coefficients have a separable structure:

The spectral envelope is

The radial weights interpolate among the Gaussian and derivative channels.

This parameterization is intentionally small enough for a reviewer to inspect and large enough to exercise repeated source motion across the curvature barrier.

Source representation

Regularized radial basis

Gaussian source channels

Gaussian source channels

Eight Gaussian channels are paired with eight derivative channels in the executable basis.

A conservation ledger outside the compiler equality

Independent current and source-work identity

INDEPENDENT PHYSICS CHECK

Conserved radial current

For a real potential, define

Using

one obtains

Integrating across the computational domain gives

Boundary currents

For the outgoing infinity mode,

For the horizon-directed mode,

Therefore

This identity is the independent physics ledger used in the benchmark.

From master amplitudes to relative energy flux

Infinity and horizon flux records

Gauge-invariant flux normalization

For a single odd-parity master mode, a standard RWZ energy-flux normalization has the structure

For harmonic time dependence,

The spectral flux proxy is therefore

The horizon flux is defined analogously from .

Because the regularized source has arbitrary overall normalization, the benchmark interprets these as relative, convention-consistent flux records rather than an astrophysical luminosity.

Part II

Direct Numerical Calculation

Grid, boundary rows, reused factorization, record extraction, and numerical error.

Second-order interior finite differences

Radial discretization

Radial grid

The main run uses

uniform points in tortoise coordinate:

with

Interior stencil

For interior point ,

The discrete radial equation is

This produces a complex tridiagonal matrix after the boundary rows are replaced by the Robin conditions.

Complex Robin conditions

Discrete physical boundary rows

Left row

The forward difference

gives

Right row

The backward difference

gives

Operator

For each frequency,

The matrix is factorized once. The direct baseline reuses this factorization, so it is not penalized by repeated matrix construction or repeated symbolic factorization.

The strongest fair online baseline

Reused-factor direct solver

Offline direct setup

For every frequency :

  1. evaluate the Regge–Wheeler potential;
  2. assemble ;
  3. compute a sparse LU factorization.

This cost is available to both the direct and compiled paths.

Online direct query

For every new parameter vector :

  1. generate ;

  2. form the radial source

  3. solve

  4. extract and .

Across 48 frequencies, the direct query produces and solves for all

complex field values before returning 96 complex records.

Ninety-six complex records rather than 196,608 complex field values

Observer record definition

Record space

For each frequency, define the two complex records

The phase-corrected endpoint map is

The complete record vector contains

complex numbers.

Why this record family is sufficient for the benchmark

It determines:

  • outgoing and absorbed amplitudes;
  • relative energy-flux spectra;
  • total integrated relative flux;
  • source-work balance;
  • frequency-dependent phase response.

It does not reconstruct the local radial field. A reviewer can request that diagnostic separately.

Frozen inputs and requested outputs

Representative source query

Frozen query

The representative source parameters are

The source is localized outside the potential peak and has support across the benchmark frequency band.

Outputs

For every frequency the calculation returns:

and derives

All numerical values are included in representative_records.csv.

Part III

Observer Compilation

Full response caching, direct record maps, adjoints, five-tensor ownership, and complexity.

Move the physical solves offline

Full response-basis compiler

EXACT-SCOPED

Full response basis

Let

For each frequency, solve all basis responses offline:

Then a new full field is

This removes the online sparse solve but still assembles field components per frequency.

Exact scoped identity

Because the equation is linear and the source is restricted to ,

No approximation is introduced by moving the basis solve offline. The only numerical difference is floating-point operation ordering.

Compile curvature propagation directly into records

Record-space matrix kernel

EXACT-SCOPED

Compilation

Apply the observer map offline:

The dimensions are

Online,

Across the full frequency grid, the code evaluates

Meaning of a matrix entry

The entry

is the complete precomputed response of record at frequency to source channel , including:

  • Schwarzschild curvature scattering;
  • horizon absorption;
  • outgoing propagation;
  • finite-domain boundary convention;
  • phase transport into the record frame.
Two record solves can replace sixteen source solves

Adjoint observer compiler

Adjoint derivation

Let the -th record be

With

solve the adjoint system

Then

Therefore

Why the implementation uses forward basis solves

There are 16 source channels and 2 records per frequency. Either approach is inexpensive:

  • forward compilation: 16 right-hand-side solves;
  • adjoint compilation: 2 adjoint solves.

A production implementation should prefer the adjoint formulation when the source basis is large and the record count is small.

The code stores the full response basis because it also benchmarks the cost of reconstructing the entire field. The smallest production compiler would store only .

Three active sectors and two scoped zeros

Five-tensor ownership in the RW example

Additive tensor

The complete master field in this linear example is the additive response:

Nonlinear tensor

No quadratic or higher Einstein source is admitted:

Order-curvature tensor

The frequency operators and compilation order are frozen, with no alternative continuation path:

Flux tensor

The observer records determine

Transport tensor

The phase factors

transport endpoint components into asymptotic amplitude records. They are the scoped action.

Residual

From field dimension N to record dimension q

Complexity reduction

Direct query

For a sparse tridiagonal-like solve,

after factorization, plus source formation

The implementation uses sparse LU, whose constant factors include indirect memory access and triangular solves.

Full reduced field

The cached field path costs

Observer compiled

The record path costs

where

The ideal field-assembly ratio is

The measured wall-clock ratio between full reduced-field assembly and record evaluation is smaller because highly optimized matrix operations, Python overhead, and memory behavior matter.

The 347× online result

Measured offline and online costs

PASS-PHYSICAL-OPERATOR

Offline costs

The measured compiler costs were:

Online costs

The measured median costs per query were:

Therefore

and

When compilation actually pays

Amortized total-work speedup

Total-work speedup

For queries,

The break-even point is

Measured-cost implications

Using the frozen timings:

Queries Amortized speedup
1 0.009×
10 0.094×
100 0.94×
1,000 9.19×
10,000 74.25×
100,000 254.09×
1,000,000 335.30×

The compiler exceeds 100× amortized speedup at approximately 14,900 queries.

This distinction is essential. The method is compelling for waveform libraries, parameter inference, uncertainty propagation, Monte Carlo work, and repeated design studies. It is not justified for a single isolated solve.

Not required and not used in the benchmark

Optional 13D coupling hook

Native 4D result

The entire measured benchmark is four-dimensional. No internal geometry is required to obtain the 347× online result.

Optional internal map

If a parent geometry derives

then

Define

The online dimension can fall from to .

Acceptance condition

The internal geometry earns a unique speed claim only if:

  1. is physically derived rather than fitted to the same output data;
  2. its rank is smaller after all native 4D constraints are saturated;
  3. it preserves the direct records and flux ledgers;
  4. a 4D solver supplied with the same map reproduces the same arithmetic.

This example intentionally leaves

until the physical coupling map is derived.

Part IV

Executed Worked Calculation

Potential, records, flux, balance, accuracy, runtime, scaling, and convergence.

What the potential contributes to each compiled entry

Reading the curvature barrier

Potential barrier

The Regge–Wheeler potential rises from zero near the horizon, peaks in the strong-field region, and decays toward zero at infinity.

The source can therefore excite:

  • direct outward radiation;
  • inward radiation that crosses the horizon;
  • reflected radiation from the curvature barrier;
  • frequency-dependent phase shifts.

The record kernel contains all of these effects. It is not a simple geometric projection of the source.

Frequency-dependent physical records

Asymptotic and horizon spectra

Infinity record

The infinity amplitude is largest where the source envelope and black-hole transfer response overlap.

Horizon record

The horizon amplitude measures the part of the master field transmitted inward through the curvature barrier.

Flux

Flux depends quadratically on amplitude:

Consequently, a small complex-amplitude error can have a different relative effect on a near-zero flux bin. The certificate therefore reports both record error and integrated flux-vector error.

Observer output

Complex-amplitude spectra

Infinity and horizon amplitudes

Infinity and horizon amplitudes

Phase-corrected endpoint amplitudes from the representative source query.

Derived physical record

Relative energy-flux spectra

Infinity and horizon flux

Infinity and horizon flux

Single-mode relative fluxes derived from the compiled amplitude records.

Independent closure at approximately 10^-11

Source-work balance result

PASS

Measured closure

Across all 48 frequencies in the representative query:

The median residual was

Interpretation

This agreement tests the direct field, source, boundary conditions, and observer amplitudes together. A compiler that returned the wrong asymptotic record could still match a cached matrix internally, but it would fail this independent balance ledger.

Compression error below direct discretization error

Compiled versus direct exactness

PASS

Field-basis and observer-kernel equality

For 24 independently generated source parameters, the maximum record discrepancy was

The maximum derived flux discrepancy was

Why the record error is larger than machine epsilon

The compiled response is formed by solving the basis columns and then recombining them. The direct path first combines source columns and then performs one solve. Both operations are mathematically identical, but finite-precision LU solves and cancellation occur in a different order.

The measured error remains far below the grid and finite-boundary error.

75× to 604× as radial resolution increases

Measured speedup scaling

Measured scaling

Radial unknowns Direct time/query Compiled time/query Online speedup
1,024 2,023.8 μs 27.0 μs 75.0×
2,048 4,237.1 μs 24.6 μs 172.2×
4,096 8,017.9 μs 24.7 μs 324.7×
8,192 15,009.5 μs 24.8 μs 604.3×

The main benchmark's longer timing suite produced 347.6× at .

Interpretation

The compiled cost is nearly independent of because it depends on the 16 source channels and two records per frequency. The direct solve cost grows with the radial field dimension.

This is the exact condition under which observer compilation becomes increasingly valuable:

Scaling evidence

Speedup versus field dimension

Measured scaling

Measured scaling

The compiled online cost remains almost fixed while direct field cost grows with N.

Compression is not the dominant numerical error

Grid-refinement trend

Grid-refinement trend

Using as the internal reference:

Radial unknowns Relative record difference
1,024
2,048
4,096
8,192 0

The trend is monotonic.

Important distinction

The observer compiler reproduces the discretized direct solver to approximately , while the discretization differs from the result at approximately .

Therefore the current dominant error is physical discretization and finite-boundary error—not record compilation.

This is exactly the intended architecture: compression error should sit well below the direct solver's own resolution error.

Numerical validation

Observer-record convergence

Grid refinement

Grid refinement

Relative record difference decreases monotonically toward the N=8192 internal reference.

Fifty megabytes to approximately twenty-four kilobytes

Memory compression

Stored full response

The optional full response basis has dimensions

At complex double precision, this is approximately

Stored record kernel

The production record kernel has dimensions

requiring only approximately

The storage ratio is about

The same factor that drives arithmetic savings also drives memory savings.

How the benchmark should fail

Destructive controls

Required destructive tests

Wrong horizon sign

Replace

with the outgoing sign at the horizon. The source-work balance and absorption interpretation must fail.

Wrong infinity sign

Replace

with an incoming condition. The infinity record changes and the compiled map hash must be invalidated.

Stale record kernel

Change the frequency grid, radial domain, potential, source basis, or boundary rule without recompiling . Freshness must fail.

Omitted derivative channels

Remove the columns. A source family containing derivative structure should no longer be represented exactly.

Kernel-only validation

Compare compiled and direct records without the source-work balance. This is insufficient because both could share the same wrong boundary convention.

Out-of-basis source

Inject a radial source orthogonal to the 16-channel span. The record residual must be attributed to source-basis truncation rather than hidden.

Part V

NASA/JPL Interpretation

Mission relevance, review paths, and the limits of the current result.

Repeated black-hole perturbation queries and record-space inference

Why the calculation matters for LISA work

LISA and large-mass-ratio modeling

NASA's LISA preparatory program includes waveform development, calibration, and validation for large mass-ratio systems, including work led from JPL.

The computational pattern is demanding because a single scientific analysis can require many waveform or likelihood evaluations across source parameters.

Why this benchmark is relevant

The example isolates a reusable pattern:

That pattern occurs in:

  • frequency-domain black-hole perturbation theory;
  • Green-function waveform generation;
  • source-parameter inference;
  • detector-response compilation;
  • uncertainty propagation;
  • waveform-library construction.

The benchmark is not a claim that the final LISA problem is only one Regge–Wheeler equation. It demonstrates a load-bearing acceleration mechanism inside the larger problem.

A short path and a deep path

Friendly reviewer walkthrough

Thirty-minute review path

A reviewer can:

  1. inspect the Schwarzschild and Regge–Wheeler equations;
  2. verify the tortoise-coordinate inversion;
  3. inspect the finite-difference and boundary rows;
  4. run the Python script;
  5. compare direct and compiled records;
  6. inspect the source-work balance;
  7. review timing and amortization;
  8. change , the frequency grid, or source parameters;
  9. confirm the certificate reopens when an acceptance condition fails.

Deep technical review

A deeper review should examine:

  • operator domains and finite-boundary approximations;
  • normalization of the odd-parity master function;
  • source basis adequacy;
  • sensitivity to Gaussian width;
  • convergence toward exact distributional jump conditions;
  • comparison with independent RWZ software;
  • adjoint implementation;
  • exact point-particle upgrade.
Claim tables, witnesses, and forbidden promotions

AI technical-review contract

AI review table

An AI reviewer should produce one row per claim:

Field Requirement
Claim Exact sentence under review
Scope Background, parity, mode, source, domain, records
Status Established, exact-scoped, controlled, open, or failed
Equation Governing relation
Witness Code, CSV, figure, or certificate path
Negative control Closest false-positive object
Residual Numerical or derivational debt
Reopen trigger Change that invalidates the claim

Prohibited promotion

The AI reviewer must reject these substitutions:

  • regularized source exact point particle;
  • master-function amplitude detector strain without reconstruction;
  • online speedup first-query speedup;
  • 4D result evidence for a 13D coupling advantage;
  • agreement with the same discretization continuum validation.
What may be presented today

Strongest defensible claim

PUBLIC CLAIM BOUNDARY

Defensible claim

A frequency-domain observer compiler for the physical odd-parity Schwarzschild Regge–Wheeler operator can replace repeated 4,096-component radial field solves with direct evaluation of 96 asymptotic and horizon records from 16 source channels. In the executed regularized-source benchmark, this produced 347.6× sequential online acceleration, machine-level agreement with the discretized direct solver, and an independently closing source-work balance.

Claim not yet available

The project should not yet claim:

  • a 347× exact point-particle waveform acceleration;
  • a 347× complete IMRI acceleration;
  • a 13D-derived physical speed advantage;
  • continuum-level waveform accuracy;
  • detector-likelihood acceleration.

Those claims require the upgrades explicitly listed in this document.

Part VI

Path to the Complete Physical Example

Exact point-particle sources, even parity, time evolution, nonlinear corrections, and held-out replay.

From regularized channels to geodesic stress energy

Exact point-particle source upgrade

Required source derivation

For a particle of mass on a Schwarzschild geodesic ,

Project this tensor onto odd- and even-parity harmonics to obtain the exact RWZ source coefficients.

The radial source contains distributional terms. A production solver can use:

  • analytic jump conditions;
  • discontinuous Galerkin methods;
  • effective-source regularization;
  • frequency-domain homogeneous solutions and Wronskians.

Compiler effect

The observer compilation theorem remains unchanged:

Only the physical source generator and basis change.

Validation

The upgraded example should compare:

  • circular-orbit fluxes against published RWZ values;
  • selected eccentric or plunge waveforms;
  • infinity and horizon flux;
  • independent Black Hole Perturbation Toolkit outputs where available.
Required for the dominant circular-orbit mode

Even-parity Zerilli upgrade

Even-parity sector

The Zerilli–Moncrief function obeys

With

the Zerilli potential can be written

The dominant circular-orbit waveform lives in the even sector. Adding this sector is therefore necessary before presenting the benchmark as a dominant-mode point-particle waveform model.

The observer compiler simply concatenates the even and odd record kernels or keeps separate parity ledgers.

Generic trajectories and history dependence

Time-domain and Green-function extension

Time-domain operator

The time-domain problem is

A time-domain observer compiler can use:

  • impulse-response Green functions;
  • convolution quadrature;
  • reduced temporal bases;
  • balanced truncation;
  • adjoint time integration;
  • hyperboloidal extraction.

Why frequency domain is the correct first example

The frequency-domain operator makes the reusable map explicit and separates the physics of propagation from source variation.

A time-domain extension is essential for generic nonperiodic trajectories, but it should be added after the frequency-domain compiler and its balance laws are independently verified.

Activating N and K

Second-order and operation-curvature extension

Second-order perturbations

At second order,

This activates the nonlinear tensor:

The second-order source can itself be represented in a low-rank basis and compiled into observer records.

Operation curvature

When mass ratio, orbital elements, frame, and background updates are continued along different paths, the response connection can have nonzero curvature:

This activates

The current benchmark intentionally avoids these effects so the observer compiler can be isolated and falsified cleanly.

Development, demonstration, and held-out replay

q=64 → q=100 → q=128

q=64 development

  • exact point-particle RWZ source;
  • even and odd parity;
  • frozen waveform record basis;
  • native 4D compiler;
  • conservation, flux, and frame controls;
  • rank and error selection.

q=100 demonstration

Freeze:

  • record basis;
  • source basis;
  • interpolation;
  • scale windows;
  • boundary domains;
  • tolerances;
  • runtime protocol.

Run the q=100 case without architecture changes.

q=128 held-out replay

Execute once under the frozen compiler. A failure reopens the physical claim.

The present benchmark is a precursor to this sequence. It establishes that record compilation itself is not the likely computational bottleneck.

Physics, computation, and workload must pass together

Next-stage acceptance criteria

Physical acceptance for the next-stage example

Require:

  • exact source derivation;
  • independent flux agreement;
  • waveform phase and amplitude;
  • horizon absorption;
  • grid, boundary, and source regularization convergence;
  • held-out trajectory;
  • reproducible wall-clock measurements.

Computational acceptance

Report:

  • operator factorization;
  • basis compilation;
  • adjoint/forward solve counts;
  • memory;
  • online source generation;
  • online record evaluation;
  • full-field diagnostic reconstruction;
  • break-even and amortized speedup.

Target

The current example supports retaining the target:

For end-to-end amortized speed, the query workload must be stated. In this benchmark, approximately 14,900 queries are needed to exceed 100× after including compilation.

Part VII

Technical Derivation Appendices

Discrete conservation, operation counts, distributional sources, Green functions, code, certificate, and provenance.

Appendix A

Discrete current derivation

Discrete source-work identity

For the interior stencil, multiply the discrete equation by , subtract its complex conjugate, and sum over .

The potential and terms cancel because they are real. The second-difference terms telescope into a boundary current.

The remaining source term is

The Robin rows convert the endpoint current into

up to the one-sided boundary-discretization convention.

The measured -level closure therefore also checks that the implementation's signs and phase conventions are mutually consistent.

Appendix B

Operation-count analysis

Nominal complex multiply-add counts

For one query:

Full reduced field

complex multiply-add contributions.

Observer compiled

The nominal ratio is

Why wall clock is 61× rather than 2048×

The full-field path uses optimized dense matrix-vector kernels and only reads boundary components afterward. The compiled path is so small that Python, coefficient generation, function calls, and array setup dominate.

A native compiled implementation or large batch can move the measured ratio closer to the algebraic ceiling.

Appendix C

Gaussian and derivative distributional limit

Distributional limit

A normalized Gaussian satisfies

in the distributional sense as .

Its derivative satisfies

For any smooth test function ,

and

Numerical restriction

The width cannot be taken below the grid resolution without losing convergence. The exact point-particle upgrade should therefore use analytic jump conditions or an effective-source method rather than merely shrinking the Gaussian indefinitely.

Appendix D

Green-function interpretation

Green function

Let

with ingoing horizon and outgoing infinity conditions.

Then

The infinity record is

where

The observer-compiled matrix is simply the Green-function kernel integrated against the finite source basis:

This connects the matrix compiler directly to standard black-hole Green-function theory.

Appendix E

Primary references

PRIMARY SOURCES

Primary references

  1. T. Regge and J. A. Wheeler, Stability of a Schwarzschild Singularity, Physical Review 108, 1063 (1957).
  2. F. J. Zerilli, Effective Potential for Even-Parity Regge-Wheeler Gravitational Perturbation Equations, Physical Review Letters 24, 737 (1970).
  3. K. Martel and E. Poisson, Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism, arXiv:gr-qc/0502028.
  4. K. Martel, Gravitational waveforms from a point particle orbiting a Schwarzschild black hole, arXiv:gr-qc/0311017.
  5. C. O'Toole, A. Ottewill, and B. Wardell, Characteristic formulation of the Regge-Wheeler and Zerilli Green functions, arXiv:2010.15818.
  6. A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, arXiv:2101.04592.
  7. D. Brizuela, J. M. Martín-García, and M. Tiglio, A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole, arXiv:0903.1134.
  8. L. Durkan and N. Warburton, Slow evolution of the metric perturbation due to a quasicircular inspiral into a Schwarzschild black hole, arXiv:2206.08179.
  9. NASA, LISA Preparatory Science Program, including waveform and EMRI modeling activities.

Reference-use boundary

The derivations in this monograph use standard equations and conventions from these sources. The numerical benchmark, source basis, compiler, timing, and certificate are original controlled artifacts in the review pack.

Appendix F

Complete executable Python implementation

EXECUTED

from __future__ import annotations

import csv
import hashlib
import json
import math
import platform
import time
from dataclasses import dataclass
from pathlib import Path

import matplotlib.pyplot as plt
import numpy as np
import scipy
from scipy.special import lambertw
from scipy.sparse import diags
from scipy.sparse.linalg import splu


SEED = 20260806
M = 1.0
ELL = 2
RSTAR_MIN = -60.0
RSTAR_MAX = 140.0
N_MAIN = 4096
N_FREQUENCY = 48
OMEGA_MIN = 0.12
OMEGA_MAX = 0.55
N_SOURCE_CHANNELS = 16
N_RECORDS_PER_FREQUENCY = 2


@dataclass
class Model:
    n_state: int
    rstar: np.ndarray
    r: np.ndarray
    potential: np.ndarray
    h: float
    frequencies: np.ndarray
    source_basis: np.ndarray
    source_centers_r: np.ndarray
    source_centers_rstar: np.ndarray
    lu_factors: list
    full_response_basis: np.ndarray | None
    record_kernel: np.ndarray | None
    factorization_seconds: float
    response_compile_seconds: float


def sha256(path: Path) -> str:
    h = hashlib.sha256()
    with path.open("rb") as f:
        for block in iter(lambda: f.read(1024 * 1024), b""):
            h.update(block)
    return h.hexdigest()


def schwarzschild_r_from_tortoise(rstar: np.ndarray, mass: float = M) -> np.ndarray:
    """Invert r_* = r + 2M ln(r/(2M)-1) using Lambert W."""
    argument = np.exp(np.clip(rstar / (2.0 * mass) - 1.0, -700.0, 700.0))
    return 2.0 * mass * (1.0 + lambertw(argument).real)


def regge_wheeler_potential(r: np.ndarray, ell: int = ELL, mass: float = M) -> np.ndarray:
    """Odd-parity spin-2 Schwarzschild Regge-Wheeler potential."""
    f = 1.0 - 2.0 * mass / r
    return f * (ell * (ell + 1.0) / r**2 - 6.0 * mass / r**3)


def build_source_basis(rstar: np.ndarray) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
    """Gaussian and first-derivative channels approximating delta and delta-prime source structure."""
    centers_r = np.linspace(6.2, 30.0, N_SOURCE_CHANNELS // 2)
    centers_rstar = centers_r + 2.0 * np.log(centers_r / 2.0 - 1.0)
    widths = np.linspace(0.60, 1.40, len(centers_r))
    h = float(rstar[1] - rstar[0])

    columns = []
    for center, width in zip(centers_rstar, widths):
        gaussian = np.exp(-0.5 * ((rstar - center) / width) ** 2)
        gaussian /= math.sqrt(float(np.sum(np.abs(gaussian) ** 2) * h))
        derivative = -(rstar - center) / width**2 * gaussian
        derivative /= math.sqrt(float(np.sum(np.abs(derivative) ** 2) * h))
        columns.extend([gaussian, derivative])

    basis = np.column_stack(columns).astype(np.complex128)
    basis[[0, -1], :] = 0.0
    return basis, centers_r, centers_rstar


def build_frequency_operator(
    omega: float,
    potential: np.ndarray,
    h: float,
) -> splu:
    """Second-order interior discretization with ingoing/outgoing Robin boundaries."""
    n = len(potential)
    off = np.full(n - 1, 1.0 / h**2, dtype=np.complex128)
    main = (-2.0 / h**2 + omega**2 - potential).astype(np.complex128)
    operator = diags([off, main, off], [-1, 0, 1], shape=(n, n), format="lil")

    # Horizon side: dPsi/dr_* = -i omega Psi.
    operator[0, :] = 0.0
    operator[0, 0] = -1.0 + 1j * omega * h
    operator[0, 1] = 1.0

    # Infinity side: dPsi/dr_* = +i omega Psi.
    operator[-1, :] = 0.0
    operator[-1, -2] = -1.0
    operator[-1, -1] = 1.0 - 1j * omega * h

    return splu(operator.tocsc())


def compile_model(
    n_state: int = N_MAIN,
    compile_full_response: bool = True,
) -> Model:
    rstar = np.linspace(RSTAR_MIN, RSTAR_MAX, n_state)
    h = float(rstar[1] - rstar[0])
    r = schwarzschild_r_from_tortoise(rstar)
    potential = regge_wheeler_potential(r)
    frequencies = np.linspace(OMEGA_MIN, OMEGA_MAX, N_FREQUENCY)
    source_basis, centers_r, centers_rstar = build_source_basis(rstar)

    t0 = time.perf_counter()
    lu_factors = [build_frequency_operator(w, potential, h) for w in frequencies]
    factorization_seconds = time.perf_counter() - t0

    full_response_basis = None
    record_kernel = None
    response_compile_seconds = 0.0

    if compile_full_response:
        t0 = time.perf_counter()
        response_list = []
        kernel_list = []
        for omega, lu in zip(frequencies, lu_factors):
            response = lu.solve(source_basis)
            record_map = np.vstack(
                [
                    np.exp(-1j * omega * rstar[-1]) * response[-1, :],
                    np.exp(+1j * omega * rstar[0]) * response[0, :],
                ]
            )
            response_list.append(response)
            kernel_list.append(record_map)
        full_response_basis = np.stack(response_list)
        record_kernel = np.stack(kernel_list)
        response_compile_seconds = time.perf_counter() - t0

    return Model(
        n_state=n_state,
        rstar=rstar,
        r=r,
        potential=potential,
        h=h,
        frequencies=frequencies,
        source_basis=source_basis,
        source_centers_r=centers_r,
        source_centers_rstar=centers_rstar,
        lu_factors=lu_factors,
        full_response_basis=full_response_basis,
        record_kernel=record_kernel,
        factorization_seconds=factorization_seconds,
        response_compile_seconds=response_compile_seconds,
    )


def source_coefficients(model: Model, parameters: np.ndarray) -> np.ndarray:
    """Generate frequency-dependent source coefficients.

    Parameters:
      r0       : radial center in units of M
      omega_c  : spectral center
      bandwidth: spectral width
      amplitude
      phase
    """
    r0, omega_c, bandwidth, amplitude, phase = np.asarray(parameters, dtype=float)
    rstar0 = r0 + 2.0 * np.log(r0 / 2.0 - 1.0)
    spatial_width = 2.0

    gaussian_weights = np.exp(
        -0.5 * ((model.source_centers_rstar - rstar0) / spatial_width) ** 2
    )
    gaussian_weights /= np.linalg.norm(gaussian_weights) + 1.0e-30
    derivative_weights = (
        0.18
        * np.tanh((model.source_centers_rstar - rstar0) / spatial_width)
        * gaussian_weights
    )

    spatial = np.empty(N_SOURCE_CHANNELS, dtype=float)
    spatial[0::2] = gaussian_weights
    spatial[1::2] = derivative_weights

    spectral = np.exp(
        -0.5 * ((model.frequencies - omega_c) / bandwidth) ** 2
    ) * np.exp(1j * (phase + 0.7 * model.frequencies))

    return amplitude * spectral[:, None] * spatial[None, :]


def direct_records(model: Model, parameters: np.ndarray) -> np.ndarray:
    coefficients = source_coefficients(model, parameters)
    records = np.empty(
        (len(model.frequencies), N_RECORDS_PER_FREQUENCY),
        dtype=np.complex128,
    )
    for index, (omega, lu) in enumerate(zip(model.frequencies, model.lu_factors)):
        source = model.source_basis @ coefficients[index]
        field = lu.solve(source)
        records[index, 0] = np.exp(-1j * omega * model.rstar[-1]) * field[-1]
        records[index, 1] = np.exp(+1j * omega * model.rstar[0]) * field[0]
    return records


def full_reduced_records(model: Model, parameters: np.ndarray) -> np.ndarray:
    if model.full_response_basis is None:
        raise RuntimeError("Full response basis was not compiled.")
    coefficients = source_coefficients(model, parameters)
    records = np.empty(
        (len(model.frequencies), N_RECORDS_PER_FREQUENCY),
        dtype=np.complex128,
    )
    for index, omega in enumerate(model.frequencies):
        field = model.full_response_basis[index] @ coefficients[index]
        records[index, 0] = np.exp(-1j * omega * model.rstar[-1]) * field[-1]
        records[index, 1] = np.exp(+1j * omega * model.rstar[0]) * field[0]
    return records


def observer_compiled_records(model: Model, parameters: np.ndarray) -> np.ndarray:
    if model.record_kernel is None:
        raise RuntimeError("Observer record kernel was not compiled.")
    coefficients = source_coefficients(model, parameters)
    return np.einsum("fqr,fr->fq", model.record_kernel, coefficients)


def flux_spectra(model: Model, records: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
    """Martel-Poisson-normalized single-mode spectral flux proxies.

    The regularized source has arbitrary overall normalization, so these are
    physically structured relative fluxes, not an astrophysical luminosity.
    """
    factorial_factor = math.factorial(ELL + 2) / math.factorial(ELL - 2)
    prefactor = factorial_factor / (64.0 * math.pi)
    infinity_flux = prefactor * model.frequencies**2 * np.abs(records[:, 0]) ** 2
    horizon_flux = prefactor * model.frequencies**2 * np.abs(records[:, 1]) ** 2
    return infinity_flux, horizon_flux


def source_work_balance(
    model: Model,
    parameters: np.ndarray,
) -> dict:
    coefficients = source_coefficients(model, parameters)
    residuals = []
    rows = []
    for index, (omega, lu) in enumerate(zip(model.frequencies, model.lu_factors)):
        source = model.source_basis @ coefficients[index]
        field = lu.solve(source)
        a_inf = np.exp(-1j * omega * model.rstar[-1]) * field[-1]
        a_hor = np.exp(+1j * omega * model.rstar[0]) * field[0]
        boundary_current = omega * (abs(a_inf) ** 2 + abs(a_hor) ** 2)
        injected_current = np.trapezoid(
            np.imag(np.conj(field) * source),
            model.rstar,
        )
        relative = abs(boundary_current - injected_current) / max(
            abs(boundary_current),
            abs(injected_current),
            1.0e-30,
        )
        residuals.append(relative)
        rows.append(
            {
                "omega": float(omega),
                "boundary_current": float(boundary_current),
                "source_work": float(injected_current),
                "relative_residual": float(relative),
            }
        )
    return {
        "maximum_relative_residual": float(max(residuals)),
        "median_relative_residual": float(np.median(residuals)),
        "rows": rows,
    }


def random_parameters(count: int, seed: int = SEED) -> list[np.ndarray]:
    rng = np.random.default_rng(seed)
    return [
        np.array(
            [
                rng.uniform(6.5, 25.0),
                rng.uniform(0.16, 0.48),
                rng.uniform(0.04, 0.12),
                rng.uniform(0.5, 1.5),
                rng.uniform(-math.pi, math.pi),
            ],
            dtype=float,
        )
        for _ in range(count)
    ]


def benchmark_method(method, parameters: list[np.ndarray], repeats: int) -> dict:
    samples = []
    checksum = 0.0
    for _ in range(repeats):
        t0 = time.perf_counter()
        value = 0.0
        for p in parameters:
            records = method(p)
            value += float(np.real(records[0, 0]))
        samples.append(time.perf_counter() - t0)
        checksum = value
    median_seconds = float(np.median(samples))
    return {
        "queries": len(parameters),
        "repeats": repeats,
        "all_seconds": [float(x) for x in samples],
        "median_seconds": median_seconds,
        "median_microseconds_per_query": float(
            1.0e6 * median_seconds / len(parameters)
        ),
        "checksum": checksum,
    }


def exactness_test(model: Model, parameters: list[np.ndarray]) -> dict:
    full_errors = []
    compiled_errors = []
    flux_errors = []
    for p in parameters:
        direct = direct_records(model, p)
        full = full_reduced_records(model, p)
        compiled = observer_compiled_records(model, p)
        denominator = max(float(np.linalg.norm(direct)), 1.0e-30)
        full_errors.append(float(np.linalg.norm(direct - full) / denominator))
        compiled_errors.append(
            float(np.linalg.norm(direct - compiled) / denominator)
        )

        direct_flux = np.concatenate(flux_spectra(model, direct))
        compiled_flux = np.concatenate(flux_spectra(model, compiled))
        flux_denominator = max(float(np.linalg.norm(direct_flux)), 1.0e-30)
        flux_errors.append(
            float(np.linalg.norm(direct_flux - compiled_flux) / flux_denominator)
        )
    return {
        "maximum_relative_record_error_full_reduced": max(full_errors),
        "maximum_relative_record_error_observer_compiled": max(compiled_errors),
        "maximum_relative_flux_error_observer_compiled": max(flux_errors),
        "median_relative_record_error_observer_compiled": float(
            np.median(compiled_errors)
        ),
    }


def scaling_test() -> list[dict]:
    rows = []
    parameters = random_parameters(24, seed=SEED + 700)
    for n_state in (1024, 2048, 4096, 8192):
        model = compile_model(n_state, compile_full_response=True)
        direct = benchmark_method(
            lambda p, m=model: direct_records(m, p),
            parameters,
            repeats=3,
        )
        compiled = benchmark_method(
            lambda p, m=model: observer_compiled_records(m, p),
            parameters,
            repeats=5,
        )
        rows.append(
            {
                "n_state": n_state,
                "factorization_seconds": model.factorization_seconds,
                "response_compile_seconds": model.response_compile_seconds,
                "direct_us_per_query": direct["median_microseconds_per_query"],
                "compiled_us_per_query": compiled["median_microseconds_per_query"],
                "online_speedup": (
                    direct["median_microseconds_per_query"]
                    / compiled["median_microseconds_per_query"]
                ),
            }
        )
        del model
    return rows


def convergence_test() -> list[dict]:
    representative = np.array([10.0, 0.32, 0.08, 1.0, 0.4])
    records_by_n = {}
    for n_state in (1024, 2048, 4096, 8192):
        model = compile_model(n_state, compile_full_response=False)
        records_by_n[n_state] = direct_records(model, representative)
        del model

    reference = records_by_n[8192]
    rows = []
    for n_state in (1024, 2048, 4096, 8192):
        denominator = max(float(np.linalg.norm(reference)), 1.0e-30)
        rows.append(
            {
                "n_state": n_state,
                "relative_record_difference_vs_8192": float(
                    np.linalg.norm(records_by_n[n_state] - reference) / denominator
                ),
            }
        )
    return rows


def make_figures(
    root: Path,
    model: Model,
    representative: np.ndarray,
    timing: dict,
    amortized: list[dict],
    scaling: list[dict],
    convergence: list[dict],
) -> None:
    records = direct_records(model, representative)
    infinity_flux, horizon_flux = flux_spectra(model, records)

    plt.figure(figsize=(9, 5))
    plt.plot(model.rstar, model.potential)
    plt.xlabel(r"Tortoise coordinate $r_*/M$")
    plt.ylabel(r"$V_{\ell=2}^{RW} M^2$")
    plt.title("Odd-parity Schwarzschild Regge-Wheeler potential")
    plt.tight_layout()
    plt.savefig(root / "regge_wheeler_potential.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    for index in range(0, N_SOURCE_CHANNELS, 2):
        plt.plot(
            model.rstar,
            np.real(model.source_basis[:, index]),
            label=f"center {model.source_centers_r[index // 2]:.1f}M",
        )
    plt.xlim(-5, 45)
    plt.xlabel(r"$r_*/M$")
    plt.ylabel("Normalized Gaussian source channel")
    plt.title("Regularized radial source basis")
    plt.legend(ncol=2)
    plt.tight_layout()
    plt.savefig(root / "source_basis.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(model.frequencies, np.abs(records[:, 0]), label="Infinity amplitude")
    plt.plot(model.frequencies, np.abs(records[:, 1]), label="Horizon amplitude")
    plt.xlabel(r"$M\omega$")
    plt.ylabel("Asymptotic master-function amplitude")
    plt.title("Representative frequency-domain observer records")
    plt.legend()
    plt.tight_layout()
    plt.savefig(root / "record_spectrum.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(model.frequencies, infinity_flux, label="Infinity flux")
    plt.plot(model.frequencies, horizon_flux, label="Horizon flux")
    plt.yscale("log")
    plt.xlabel(r"$M\omega$")
    plt.ylabel("Relative single-mode energy-flux density")
    plt.title("Derived infinity and horizon flux records")
    plt.legend()
    plt.tight_layout()
    plt.savefig(root / "flux_spectrum.svg", format="svg")
    plt.close()

    labels = ["Direct", "Full reduced field", "Observer compiled"]
    values = [
        timing["direct"]["median_microseconds_per_query"],
        timing["full_reduced"]["median_microseconds_per_query"],
        timing["observer_compiled"]["median_microseconds_per_query"],
    ]
    plt.figure(figsize=(9, 5))
    plt.bar(labels, values)
    plt.yscale("log")
    plt.ylabel("Median microseconds per query (log scale)")
    plt.title("Sequential end-to-end online timing")
    plt.tight_layout()
    plt.savefig(root / "runtime_comparison.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(
        [row["queries"] for row in amortized],
        [row["amortized_speedup"] for row in amortized],
        marker="o",
    )
    plt.xscale("log")
    plt.yscale("log")
    plt.axhline(100.0, linestyle="--")
    plt.axhline(1000.0, linestyle="--")
    plt.xlabel("Number of waveform/source queries")
    plt.ylabel("Amortized speedup including compilation")
    plt.title("Offline compilation amortization")
    plt.tight_layout()
    plt.savefig(root / "amortized_speedup.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(
        [row["n_state"] for row in scaling],
        [row["online_speedup"] for row in scaling],
        marker="o",
    )
    plt.xlabel("Radial field degrees of freedom")
    plt.ylabel("Measured online speedup")
    plt.title("Observer-compilation speedup versus field dimension")
    plt.tight_layout()
    plt.savefig(root / "scaling_speedup.svg", format="svg")
    plt.close()

    plt.figure(figsize=(9, 5))
    plt.plot(
        [row["n_state"] for row in convergence],
        [row["relative_record_difference_vs_8192"] for row in convergence],
        marker="o",
    )
    plt.xscale("log", base=2)
    plt.yscale("log")
    plt.xlabel("Radial field degrees of freedom")
    plt.ylabel("Relative record difference versus N=8192")
    plt.title("Grid-refinement trend for observer records")
    plt.tight_layout()
    plt.savefig(root / "convergence.svg", format="svg")
    plt.close()


def main() -> None:
    root = Path(__file__).resolve().parent
    model = compile_model(N_MAIN, compile_full_response=True)
    representative = np.array([10.0, 0.32, 0.08, 1.0, 0.4])

    benchmark_parameters = random_parameters(120, seed=SEED + 100)
    for method in (
        lambda p: direct_records(model, p),
        lambda p: full_reduced_records(model, p),
        lambda p: observer_compiled_records(model, p),
    ):
        for p in benchmark_parameters[:4]:
            method(p)

    timing = {
        "direct": benchmark_method(
            lambda p: direct_records(model, p),
            benchmark_parameters,
            repeats=5,
        ),
        "full_reduced": benchmark_method(
            lambda p: full_reduced_records(model, p),
            benchmark_parameters,
            repeats=5,
        ),
        "observer_compiled": benchmark_method(
            lambda p: observer_compiled_records(model, p),
            benchmark_parameters,
            repeats=9,
        ),
    }

    direct_us = timing["direct"]["median_microseconds_per_query"]
    full_us = timing["full_reduced"]["median_microseconds_per_query"]
    compiled_us = timing["observer_compiled"]["median_microseconds_per_query"]
    timing["online_speedups"] = {
        "observer_compiled_over_direct": direct_us / compiled_us,
        "observer_compiled_over_full_reduced": full_us / compiled_us,
        "full_reduced_over_direct": direct_us / full_us,
    }

    exactness = exactness_test(
        model,
        random_parameters(24, seed=SEED + 200),
    )
    balance = source_work_balance(model, representative)

    offline_seconds = model.factorization_seconds + model.response_compile_seconds
    direct_seconds_per_query = direct_us * 1.0e-6
    compiled_seconds_per_query = compiled_us * 1.0e-6
    amortized = []
    for queries in (1, 10, 100, 1000, 10000, 100000, 1000000):
        direct_total = queries * direct_seconds_per_query
        compiled_total = offline_seconds + queries * compiled_seconds_per_query
        amortized.append(
            {
                "queries": queries,
                "direct_total_seconds": direct_total,
                "compiled_total_seconds": compiled_total,
                "amortized_speedup": direct_total / compiled_total,
            }
        )

    break_even_queries = offline_seconds / max(
        direct_seconds_per_query - compiled_seconds_per_query,
        1.0e-30,
    )

    scaling = scaling_test()
    convergence = convergence_test()

    representative_records = direct_records(model, representative)
    infinity_flux, horizon_flux = flux_spectra(model, representative_records)

    with (root / "representative_records.csv").open(
        "w",
        newline="",
        encoding="utf-8",
    ) as f:
        writer = csv.writer(f)
        writer.writerow(
            [
                "omega",
                "A_infinity_real",
                "A_infinity_imag",
                "A_horizon_real",
                "A_horizon_imag",
                "infinity_flux",
                "horizon_flux",
            ]
        )
        for index, omega in enumerate(model.frequencies):
            writer.writerow(
                [
                    float(omega),
                    float(np.real(representative_records[index, 0])),
                    float(np.imag(representative_records[index, 0])),
                    float(np.real(representative_records[index, 1])),
                    float(np.imag(representative_records[index, 1])),
                    float(infinity_flux[index]),
                    float(horizon_flux[index]),
                ]
            )

    with (root / "timing_results.csv").open(
        "w",
        newline="",
        encoding="utf-8",
    ) as f:
        writer = csv.writer(f)
        writer.writerow(["method", "median_us_per_query", "speedup_over_direct"])
        writer.writerow(["direct", direct_us, 1.0])
        writer.writerow(
            [
                "full_reduced",
                full_us,
                direct_us / full_us,
            ]
        )
        writer.writerow(
            [
                "observer_compiled",
                compiled_us,
                direct_us / compiled_us,
            ]
        )

    with (root / "scaling_results.csv").open(
        "w",
        newline="",
        encoding="utf-8",
    ) as f:
        writer = csv.DictWriter(f, fieldnames=list(scaling[0].keys()))
        writer.writeheader()
        writer.writerows(scaling)

    with (root / "convergence_results.csv").open(
        "w",
        newline="",
        encoding="utf-8",
    ) as f:
        writer = csv.DictWriter(f, fieldnames=list(convergence[0].keys()))
        writer.writeheader()
        writer.writerows(convergence)

    make_figures(
        root,
        model,
        representative,
        timing,
        amortized,
        scaling,
        convergence,
    )

    certificate = {
        "certificate_id": "OBSERVER-COMPILED-RWZ-WORKED-EXAMPLE/V1",
        "date": "2026-08-06",
        "status": "PASS-PHYSICAL-OPERATOR-REGULARIZED-SOURCE-BENCHMARK",
        "environment": {
            "python": platform.python_version(),
            "numpy": np.__version__,
            "scipy": scipy.__version__,
            "platform": platform.platform(),
        },
        "physical_scope": {
            "background": "Schwarzschild spacetime, M=1",
            "sector": "linear odd-parity spin-2 Regge-Wheeler master equation",
            "ell": ELL,
            "frequency_domain": [OMEGA_MIN, OMEGA_MAX],
            "frequency_bins": N_FREQUENCY,
            "radial_domain_rstar": [RSTAR_MIN, RSTAR_MAX],
            "radial_degrees_of_freedom": N_MAIN,
            "source": (
                "Regularized compact master-equation source represented by "
                "eight Gaussian and eight derivative channels. This mimics "
                "delta/delta-prime structure but is not an exact point-particle "
                "stress-energy source."
            ),
            "records": (
                "Complex outgoing amplitude at the outer boundary and complex "
                "ingoing amplitude at the horizon-side boundary for each frequency."
            ),
        },
        "claim_boundary": [
            "The Regge-Wheeler operator and potential are physical Schwarzschild perturbation equations.",
            "The source family is regularized and parameterized, not the exact source generated by a specified geodesic stress-energy tensor.",
            "The measured speedup is end-to-end online after offline factorization and record-map compilation.",
            "Amortized speedup including compilation depends on query count and is reported separately.",
            "The 13D geometry is not used in this benchmark.",
        ],
        "dimensions": {
            "field": N_MAIN,
            "source_channels": N_SOURCE_CHANNELS,
            "complex_records": N_FREQUENCY * N_RECORDS_PER_FREQUENCY,
            "frequency_bins": N_FREQUENCY,
        },
        "offline": {
            "factorization_seconds": model.factorization_seconds,
            "response_and_record_compile_seconds": model.response_compile_seconds,
            "total_seconds": offline_seconds,
            "break_even_queries": break_even_queries,
        },
        "timing": timing,
        "amortized_speedup": amortized,
        "exactness": exactness,
        "source_work_balance": {
            "maximum_relative_residual": balance["maximum_relative_residual"],
            "median_relative_residual": balance["median_relative_residual"],
        },
        "scaling": scaling,
        "convergence": convergence,
        "representative_parameters": representative.tolist(),
        "artifacts": {},
    }

    required = [
        timing["online_speedups"]["observer_compiled_over_direct"] >= 100.0,
        timing["online_speedups"]["observer_compiled_over_direct"] <= 1000.0,
        exactness["maximum_relative_record_error_observer_compiled"] < 1.0e-9,
        exactness["maximum_relative_flux_error_observer_compiled"] < 1.0e-8,
        balance["maximum_relative_residual"] < 1.0e-8,
        convergence[0]["relative_record_difference_vs_8192"]
        > convergence[-2]["relative_record_difference_vs_8192"],
    ]
    certificate["all_required_checks_pass"] = bool(all(required))

    certificate_path = root / "rwz_worked_example_certificate.json"
    certificate_path.write_text(
        json.dumps(certificate, indent=2) + "\n",
        encoding="utf-8",
    )

    certificate["artifacts"] = {
        path.name: {
            "bytes": path.stat().st_size,
            "sha256": sha256(path),
        }
        for path in sorted(root.iterdir())
        if path.is_file() and path.name != certificate_path.name
    }
    certificate_path.write_text(
        json.dumps(certificate, indent=2) + "\n",
        encoding="utf-8",
    )

    print(
        json.dumps(
            {
                "certificate": str(certificate_path),
                "pass": certificate["all_required_checks_pass"],
                "online_speedup": timing["online_speedups"][
                    "observer_compiled_over_direct"
                ],
                "speedup_over_full_reduced": timing["online_speedups"][
                    "observer_compiled_over_full_reduced"
                ],
                "compiled_record_error": exactness[
                    "maximum_relative_record_error_observer_compiled"
                ],
                "flux_error": exactness[
                    "maximum_relative_flux_error_observer_compiled"
                ],
                "balance_residual": balance["maximum_relative_residual"],
                "offline_seconds": offline_seconds,
                "break_even_queries": break_even_queries,
            },
            indent=2,
        )
    )


if __name__ == "__main__":
    main()
Appendix G

Machine-readable execution certificate

PASS
{
  "certificate_id": "OBSERVER-COMPILED-RWZ-WORKED-EXAMPLE/V1",
  "date": "2026-08-06",
  "status": "PASS-PHYSICAL-OPERATOR-REGULARIZED-SOURCE-BENCHMARK",
  "environment": {
    "python": "3.13.5",
    "numpy": "2.3.5",
    "scipy": "1.17.0",
    "platform": "Linux-6.18.35-x86_64-with-glibc2.41"
  },
  "physical_scope": {
    "background": "Schwarzschild spacetime, M=1",
    "sector": "linear odd-parity spin-2 Regge-Wheeler master equation",
    "ell": 2,
    "frequency_domain": [
      0.12,
      0.55
    ],
    "frequency_bins": 48,
    "radial_domain_rstar": [
      -60.0,
      140.0
    ],
    "radial_degrees_of_freedom": 4096,
    "source": "Regularized compact master-equation source represented by eight Gaussian and eight derivative channels. This mimics delta/delta-prime structure but is not an exact point-particle stress-energy source.",
    "records": "Complex outgoing amplitude at the outer boundary and complex ingoing amplitude at the horizon-side boundary for each frequency."
  },
  "claim_boundary": [
    "The Regge-Wheeler operator and potential are physical Schwarzschild perturbation equations.",
    "The source family is regularized and parameterized, not the exact source generated by a specified geodesic stress-energy tensor.",
    "The measured speedup is end-to-end online after offline factorization and record-map compilation.",
    "Amortized speedup including compilation depends on query count and is reported separately.",
    "The 13D geometry is not used in this benchmark."
  ],
  "dimensions": {
    "field": 4096,
    "source_channels": 16,
    "complex_records": 96,
    "frequency_bins": 48
  },
  "offline": {
    "factorization_seconds": 0.5892536530000143,
    "response_and_record_compile_seconds": 0.33282259200001363,
    "total_seconds": 0.9220762450000279,
    "break_even_queries": 106.2203503148173
  },
  "timing": {
    "direct": {
      "queries": 120,
      "repeats": 5,
      "all_seconds": [
        1.0588391830000319,
        1.1000998229999936,
        1.006673343999978,
        0.9848617149999654,
        1.0446995199999947
      ],
      "median_seconds": 1.0446995199999947,
      "median_microseconds_per_query": 8705.82933333329,
      "checksum": -2.568498475742129
    },
    "full_reduced": {
      "queries": 120,
      "repeats": 5,
      "all_seconds": [
        0.19834815899997693,
        0.18500437500000544,
        0.1843233960000248,
        0.17670107800000778,
        0.15018286499997657
      ],
      "median_seconds": 0.1843233960000248,
      "median_microseconds_per_query": 1536.0283000002064,
      "checksum": -2.5684984757422082
    },
    "observer_compiled": {
      "queries": 120,
      "repeats": 9,
      "all_seconds": [
        0.0030672419999859812,
        0.00340185800001791,
        0.003143133999969905,
        0.002993892999995751,
        0.0029324120000069342,
        0.02235271499995406,
        0.0030050699999719654,
        0.002914035000003423,
        0.0029721109999627515
      ],
      "median_seconds": 0.0030050699999719654,
      "median_microseconds_per_query": 25.04224999976638,
      "checksum": -2.568498475742206
    },
    "online_speedups": {
      "observer_compiled_over_direct": 347.64565218438867,
      "observer_compiled_over_full_reduced": 61.33747167345331,
      "full_reduced_over_direct": 5.667753213487095
    }
  },
  "amortized_speedup": [
    {
      "queries": 1,
      "direct_total_seconds": 0.00870582933333329,
      "compiled_total_seconds": 0.9221012872500277,
      "amortized_speedup": 0.009441293981159692
    },
    {
      "queries": 10,
      "direct_total_seconds": 0.0870582933333329,
      "compiled_total_seconds": 0.9223266675000256,
      "amortized_speedup": 0.09438986901387679
    },
    {
      "queries": 100,
      "direct_total_seconds": 0.870582933333329,
      "compiled_total_seconds": 0.9245804700000045,
      "amortized_speedup": 0.94159779660209
    },
    {
      "queries": 1000,
      "direct_total_seconds": 8.70582933333329,
      "compiled_total_seconds": 0.9471184949997943,
      "amortized_speedup": 9.191911444338526
    },
    {
      "queries": 10000,
      "direct_total_seconds": 87.0582933333329,
      "compiled_total_seconds": 1.1724987449976916,
      "amortized_speedup": 74.25022304267311
    },
    {
      "queries": 100000,
      "direct_total_seconds": 870.5829333333289,
      "compiled_total_seconds": 3.4263012449766657,
      "amortized_speedup": 254.0882634326737
    },
    {
      "queries": 1000000,
      "direct_total_seconds": 8705.82933333329,
      "compiled_total_seconds": 25.964326244766404,
      "amortized_speedup": 335.299643490195
    }
  ],
  "exactness": {
    "maximum_relative_record_error_full_reduced": 7.044850167863805e-11,
    "maximum_relative_record_error_observer_compiled": 7.044850143023785e-11,
    "maximum_relative_flux_error_observer_compiled": 1.3875611315323372e-12,
    "median_relative_record_error_observer_compiled": 1.2578067209895632e-11
  },
  "source_work_balance": {
    "maximum_relative_residual": 3.8388861293463885e-11,
    "median_relative_residual": 2.3976801299648766e-12
  },
  "scaling": [
    {
      "n_state": 1024,
      "factorization_seconds": 0.10702217099998279,
      "response_compile_seconds": 0.04834984799998665,
      "direct_us_per_query": 2023.7622500000423,
      "compiled_us_per_query": 26.97199999820062,
      "online_speedup": 75.03196834254238
    },
    {
      "n_state": 2048,
      "factorization_seconds": 0.21570279899998468,
      "response_compile_seconds": 0.10779850099999067,
      "direct_us_per_query": 4237.107791666972,
      "compiled_us_per_query": 24.602250000782533,
      "online_speedup": 172.22440189544454
    },
    {
      "n_state": 4096,
      "factorization_seconds": 0.3839623319999532,
      "response_compile_seconds": 0.2917863249999755,
      "direct_us_per_query": 8017.915666667836,
      "compiled_us_per_query": 24.691583334401912,
      "online_speedup": 324.722621392074
    },
    {
      "n_state": 8192,
      "factorization_seconds": 0.8810017700000117,
      "response_compile_seconds": 0.6350319329999934,
      "direct_us_per_query": 15009.462041665718,
      "compiled_us_per_query": 24.83799999926835,
      "online_speedup": 604.2943088053728
    }
  ],
  "convergence": [
    {
      "n_state": 1024,
      "relative_record_difference_vs_8192": 0.014702727111715339
    },
    {
      "n_state": 2048,
      "relative_record_difference_vs_8192": 0.006807487090531584
    },
    {
      "n_state": 4096,
      "relative_record_difference_vs_8192": 0.002365622728526713
    },
    {
      "n_state": 8192,
      "relative_record_difference_vs_8192": 0.0
    }
  ],
  "representative_parameters": [
    10.0,
    0.32,
    0.08,
    1.0,
    0.4
  ],
  "artifacts": {
    "amortized_speedup.svg": {
      "bytes": 53292,
      "sha256": "ef0319dafae47decc1777be8988f9d020727963faa1a2a8eb8d6e97763b4d3e0"
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    "convergence.svg": {
      "bytes": 32881,
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    },
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      "bytes": 51362,
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    },
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      "bytes": 33324,
      "sha256": "d92f14aaaeaffdeb9626efbc74b808adbaa22ed88b7325da619eb44f723e19c2"
    },
    "rwz_observer_compiled_worked_example.py": {
      "bytes": 28320,
      "sha256": "c34e01a353353b5279fd2397e5dcb5dca11c50afce1c9612fbbdd7ee096f64e9"
    },
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      "bytes": 513,
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      "bytes": 51570,
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    }
  },
  "all_required_checks_pass": true
}
Appendix H

Timing result table

RAW EXECUTION DATA
method,median_us_per_query,speedup_over_direct
direct,8705.82933333329,1.0
full_reduced,1536.0283000002064,5.667753213487095
observer_compiled,25.04224999976638,347.64565218438867
Appendix I

Scaling result table

RAW EXECUTION DATA
n_state,factorization_seconds,response_compile_seconds,direct_us_per_query,compiled_us_per_query,online_speedup
1024,0.10702217099998279,0.04834984799998665,2023.7622500000423,26.97199999820062,75.03196834254238
2048,0.21570279899998468,0.10779850099999067,4237.107791666972,24.602250000782533,172.22440189544454
4096,0.3839623319999532,0.2917863249999755,8017.915666667836,24.691583334401912,324.722621392074
8192,0.8810017700000117,0.6350319329999934,15009.462041665718,24.83799999926835,604.2943088053728
Appendix J

Convergence result table

RAW EXECUTION DATA
n_state,relative_record_difference_vs_8192
1024,0.014702727111715339
2048,0.006807487090531584
4096,0.002365622728526713
8192,0.0
Appendix K

Representative record table

RAW EXECUTION DATA
omega,A_infinity_real,A_infinity_imag,A_horizon_real,A_horizon_imag,infinity_flux,horizon_flux
0.12,0.10539392726040064,0.031775377255475586,-0.003059579132275415,-0.0010556753177015897,2.082854193141854e-05,1.800601490708339e-08
0.12914893617021275,0.1623316116267809,0.044867784956404255,-0.0044420607290420085,-0.001632520597164787,5.6472992367468545e-05,4.459163897860783e-08
0.13829787234042554,0.24249009260858412,0.06709732063423274,-0.0063648719760809915,-0.0025518524120648977,0.00014452403682127646,1.07356357591393e-07
0.1474468085106383,0.35049479980287274,0.0932432078377545,-0.009031602554598159,-0.003998065676168231,0.0003413602587086679,2.531622192915851e-07
0.15659574468085108,0.500685209580339,0.13274079542346282,-0.01257536000855519,-0.006304295984351852,0.0007853646788907334,5.792307495675505e-07
0.16574468085106384,0.6878216798626721,0.18558711579123005,-0.017227404309799806,-0.009728354001630378,0.0016643012202992814,1.283537058758711e-06
0.1748936170212766,0.9350295011313726,0.2512275552753583,-0.02328797811106207,-0.01502865227407535,0.0034225685601513836,2.8047757312517932e-06
0.18404255319148938,1.2319099608980648,0.34906411876889504,-0.030673406235382165,-0.02282844942250147,0.006628503908950987,5.9110422639418705e-06
0.19319148936170213,1.5900559545876216,0.45745466764417547,-0.0397901630385442,-0.03440286642810978,0.012196024273215046,1.2326432071485568e-05
0.2023404255319149,2.0144654174175494,0.6160555897241098,-0.049302524143925885,-0.05120172919782107,0.021686747174519013,2.4691112251551653e-05
0.21148936170212768,2.4729268658270165,0.7940318702249753,-0.05937096905737636,-0.07364125868687989,0.03601599179587061,4.7772926733094306e-05
0.22063829787234043,3.003517509390171,1.0186189809180286,-0.06810686168228158,-0.10496871255495199,0.058450083553412725,9.098107338152263e-05
0.2297872340425532,3.521710150047312,1.2930259124288928,-0.07348988914692123,-0.1440473165784704,0.08870779739511478,0.0001648205707938216
0.23893617021276597,4.074982044904922,1.5776620499699574,-0.07437414069671842,-0.19642449362707287,0.13012279885803038,0.0003006232467390828
0.24808510638297873,4.579431260331039,1.9459524826328032,-0.06028901179150378,-0.2590179765269679,0.18188503403564524,0.0005195845271512988
0.2572340425531915,5.018692517986882,2.2775175411292365,-0.03472095555804665,-0.3304643984660734,0.23990838512159357,0.0008720785282842478
0.2663829787234043,5.398093133687578,2.6789908874362323,0.01814223991359966,-0.41121596412234274,0.30760766113820687,0.0014350888093454242
0.27553191489361706,5.585823635275892,3.0271690819702375,0.09099753937655769,-0.4852345152369023,0.36579022154266505,0.0022087151240155717
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Appendix L

Content-addressed artifact manifest

PROVENANCE
ArtifactBytesSHA-256
rwz_observer_compiled_worked_example.py28,320c34e01a353353b5279fd2397e5dcb5dca11c50afce1c9612fbbdd7ee096f64e9
rwz_worked_example_certificate.json8,716b346c111e1194d83813ad85679ef54c30e880d9a79d7cb32e0b70b467c29edeb
timing_results.csv18462114776095e44465558b59a9aa96688672c9e29ad3a13f73ef57244240765c1
scaling_results.csv513f43c117d8f0c1e51ed1b33ed998368e9171c00cbafd16a27f84b3a79a0525673
convergence_results.csv1352173f931c3fa67324bfe6fc8750e645515fd04078f95681431df829d78971c5c
representative_records.csv6,933df9b46485378f1f10a12424c3c9e2c6122ddfbc82b820df509267fa6990df919
Part VIII

Complete Observer-Compiler Authority

Full preservation of the general derivation, building-block provenance, AI execution contract, controlled observer benchmark, and source appendices that govern the worked Regge–Wheeler implementation.

Executive technical abstract

Observer-Compiled Five-Tensor GR

CONTEXT-PRESERVATION MONOGRAPH

Purpose

This document preserves and derives the observer-compiled five-tensor architecture so that the idea can be evaluated by a friendly human reviewer, a technical AI reviewer, or a NASA/JPL collaborator without relying on prior conversation context.

It is a companion to:

  • MaxConservationStack.html, which constructs the maximal independent conservation and coupling constraints;
  • FiveTensorGRArchitecture.html, which separates the field response into five physical ownership sectors.

The present document adds a compiler around those five tensors. Its purpose is to avoid reconstructing an entire spacetime field when the requested scientific output is a much smaller set of waveform, flux, timing, or detector records.

Central architecture

The five-tensor field response is

The compiler applies, in order:

For a frozen observer record map , a five-tensor source basis , and an effective physical operator , the compiled 4D map is

If the 13D geometry supplies a coupling map , the compiled map becomes

The online calculation is then

instead of a full nonlinear or perturbative field reconstruction followed by projection.

Controlled result

In the reproducible 900-state toy calculation embedded in this review pack:

  • the native observer-compiled 4D path achieved 40.86× sequential speedup;
  • the 13D-informed path achieved 57.37× sequential speedup;
  • the batched linear-algebra-only ceiling was 3726× and 6991×;
  • observer records agreed with the direct calculation at approximately relative error.

These are controlled architecture results, not physical Einstein-equation or IMRI waveform results. The complete realistic example is deliberately deferred until the derivation and execution contract are preserved.

How to read strong and conditional claims

Claim and status taxonomy

Four distinct claim layers

Layer What may be claimed
Algebraic identity Exact equality after operators, projectors, domains, and observer maps are frozen.
Controlled architecture test Code and calculation mechanism work on a manufactured field family.
Physical scoped result The compiled model passes against declared perturbative, numerical-relativity, or detector data.
General physical theorem A proof applies throughout a declared class of GR systems, domains, and observables.

The current project has strong algebraic identities and controlled architecture tests. The physical , , and held-out waveform replay remains open.

Terms used precisely

Observer means the complete record-producing map, not a conscious person. It includes frame, calibration, time standard, detector transfer, boundary accessibility, normalization, truncation, and coarse-graining.

Compilation means expensive operator inversions, constraint quotients, matching maps, and record contractions are performed offline and stored as a small reusable map.

13D-informed means that internal geometry supplies a coupling or selection map. It does not mean the online solver numerically evolves all thirteen dimensions.

Algebraic ceiling means coefficient generation, offline compilation, and physical data acquisition have been removed from the timed online kernel. It is an upper bound on possible acceleration, not an end-to-end claim.

Conservation, five tensors, and observer compilation

How the three documents fit together

The three-document chain

1. Max Conservation Stack

The first document answers:

Which independent conservation, boundary, gauge, frame, and internal-geometry constraints restrict the physical coupling space before calculation?

Its central objects are

and

2. Five-Tensor GR Architecture

The second document answers:

How should the remaining physical response be divided so that linear addition, conservative interaction, operation order, physical flux, and frame transport are not mixed into one correction object?

It gives

3. Observer-Compiled Five-Tensor GR

This document answers:

Once the response is lawful and correctly divided, what is the smallest object that must be evaluated online to produce the records a mission or reviewer actually requests?

The answer is usually not the entire field. It is the image of the five-tensor response under the frozen observer map.

Controlled execution evidence

Current benchmark dashboard

Sequential 4D speedup40.86×Includes Python coefficient evaluation
Sequential 13D-informed57.37×10 masters rather than 24 channels
Batched 4D ceiling3726×Algebra only; excludes master generation
Batched 13D ceiling6991×Algebra only; excludes master generation
4D record error1.39e-15Relative to direct field solve
Observer dimensions900 → 8Field to frozen record space
Sequential runtime
Speedup by layer
Part I

The Computational Opportunity

Why a complete spacetime field can be the wrong online object.

Record equivalence and information loss

Why full-field reconstruction is often unnecessary

The avoidable calculation

Suppose a discretized or basis-expanded physical field has dimension , while the requested record has dimension , with

A conventional reduced calculation may still do:

Even if is assembled from a reduced basis, the online path pays for every field component.

The observer-compiled path instead precomputes

where maps reduced coefficients to the field. Online:

The field is never assembled unless a diagnostic explicitly requests it.

Why this is not merely an implementation trick

The Observer building block requires an inventory of:

  • the map kernel;
  • the map image;
  • degeneracies;
  • inaccessible sectors;
  • indirect observables;
  • sensitivity to frame, scale, scheme, boundary access, and truncation.

Therefore record-space compilation is a physical quotient with an explicit information-loss ledger. It is not permission to delete unobserved physics from the theory.

A field direction in may remain physically real. The lawful statement is only:

Each compiler layer traced to a source obligation

Building-block provenance

Building-block origin of each compiler layer

Compiler layer Primary source obligations Contribution
Rigidity quotient RIG-C11; Rulebook R-09 Remove gauge, constrained, rigid, and domain-invalid directions before solving.
Cross-scale continuation RIG-C22; SCL-C17; SCL-C23 Replace one global basis with matched scale-window bases.
Mixed-sector Schur map INT-C06; INT-C11 Integrate out coupled sectors with generated terms and error bounds.
Factorization and gluing INT-C10; BND-C17 Prove subsystem or regional splits and restore edge data.
Observer image OBS-C06 Compute the exact kernel/image and inaccessible-sector ledger.
Observer sensitivity OBS-C07; OBS-C08 Carry frame, calibration, truncation, and covariance errors into records.
Boundary transfer BND-C03; BND-C18 Freeze lawful domains and retain horizon/null/asymptotic flux.
Interaction sparsity DYN-C10; Rulebook R-17 Generate every allowed interaction and prove forbidden edges zero.
Symmetry blocks Stage S-14, S-17; Rulebook R-22 Block-diagonalize and share responses over symmetry orbits.
Time patching TS-C06, C12, C14, C19, C22 Control holonomy, delay, moving domains, refinement, and IBVP synchronization.
Dependency cache INT-C03, INT-C13; CSE-C11, C18, C20 Recompile only descendants of a changed source object.
Anti-hang ordering CSE-C27 Run exact identity and domain tests before expensive numerical work.
Notation and mathematical object inventory

Frozen spaces, domains, and maps

Frozen spaces

Let:

  • be the candidate perturbation or response space;
  • be the gauge/redundancy image;
  • be the linearized constraint map;
  • be the physical quotient;
  • be the retained five-tensor coefficient space;
  • be the observer record space.

The physical tangent space is schematically

on the frozen operator domain.

The compiler never uses this quotient symbolically without a basis. It must publish:

  1. a domain certificate;
  2. a gauge-image basis;
  3. a constraint-nullspace basis;
  4. a rank and conditioning certificate;
  5. the projector or coordinate map used numerically.

Operators

Let

be the frozen linearized or effective operator.

Let

map five-tensor coefficients into physical source terms.

Let

be the frozen record map.

The basic compiled map is

Part II

Exact Reduction Layers

Constraint quotient, mixed-sector condensation, symmetry, boundaries, and time.

Remove nonphysical dimensions before any expensive solve

Rigidity and physical quotient compiler

Constraint-derived physical quotient

The Rigidity method explicitly requires the gauge/redundancy map and exact constraint Jacobian before stability or spectral interpretation.

Let

generate gauge directions and

be the independently justified constraint Jacobian.

A basis is chosen for a complement of in . Then

and the reduced operator is

Speed benefit

If the candidate state has dimension but the physical quotient has dimension , every subsequent factorization, response solve, basis search, and observer contraction should use .

The benefit is multiplicative with later reductions because it shrinks the object entering the Schur, symmetry, and observer compilers.

Required controls

  • A direction in but also in must not count as a physical mode.
  • A low finite-dimensional eigenvalue list must not hide essential spectrum crossing zero.
  • Alternative lawful operator domains must be dispositioned.
  • The quotient must be recomputed if Shape, Boundary, Scale, or gauge ownership changes.
Integrate out coupled sectors without deleting their influence

Schur-complement compiler

Block operator

After the physical quotient, partition the retained and eliminable sectors:

Here may represent the observer-relevant gravitational sector, while may represent heavy modes, internal modes, auxiliary fields, reaction multipliers, boundary interiors, or other coupled sectors.

If is invertible on its lawful domain,

Substitution gives

Exact and approximate cases

The Schur map is exact when the partition and inverse are exact.

When is approximated by , the generated error is

A record-level bound is

Memory and nonlocality

Integrating out a dynamical sector can produce temporal memory or spatial nonlocality. The compiler must retain those generated kernels. Replacing them by local constants is a second approximation that requires its own error certificate.

Block diagonalization, orbit sharing, and interaction sparsity

Symmetry and theorem-zero compiler

Stage-preserving symmetry

The Shape Stage freezes a block-product metric and restricts physical symmetry to transformations preserving the full Stage. Stage-changing diffeomorphisms are not treated as gauge.

Let a symmetry group act on the physical response space. Decompose

If the effective operator and boundaries preserve this action,

Orbit sharing

For parameter points related by a lawful symmetry,

A response can be computed once and transported:

The same principle applies to the five tensors, provided the ownership projectors commute with the symmetry.

Theorem-zero interaction edges

Dynamics requires a complete interaction hypergraph. Every candidate interaction is either:

  • parent-owned and computed;
  • forbidden by representation, parity, support, charge, derivative grammar, or projector orthogonality;
  • left explicitly open.

Exact zero edges are deleted before basis construction. A numerically small edge is not a theorem-zero edge.

Domains, adjoints, interfaces, horizons, and null infinity

Boundary and regional transfer compiler

Domain first

The Boundary block requires the actual domain and adjoint domain of every load-bearing operator. An adjoint compiler is invalid if is written formally while the boundary form is nonzero on the selected domains.

Green's identity has the form

The adjoint boundary conditions must make the intended boundary contribution vanish or explicitly retain it as a flux record.

Regional transfer maps

Partition a domain into regions . Interior variables can be eliminated to produce interface maps such as

The global solution is assembled by matching interface data and edge modes.

Horizons and null infinity

Flux through a horizon or null infinity is not eliminated as unobservable. It is compiled into records and balance equations. A field component may be outside a detector map but still required to close energy or angular-momentum conservation.

Matched local models instead of one oversized global basis

Scale-window and multirate compiler

Scale-window compiler

Let the parameter or frequency domain be partitioned into overlapping windows

Each window has its own:

  • effective operator ;
  • active mode shelf;
  • five-tensor ranks;
  • observer map ;
  • error certificate;
  • matching map .

The compiled maps are

Matching

On an overlap,

A single global basis is retained only if it is cheaper and its error remains controlled.

Multi-rate updates

Assign update rates according to each sector's variation:

  • slowly varying conservative masters;
  • faster local curvature corrections;
  • boundary flux at its required resolution;
  • frame transport when its interpolation tolerance is reached;
  • observer calibration on its own schedule.

Time Synchronization requires that these updates refer to consistent event cells, delays, redshifts, moving domains, and refinement classes.

Incremental rebuild, freshness, and anti-hang execution

Dependency-DAG compiler

Compiled artifact DAG

The compiler outputs are nodes in a typed dependency graph:

Optional nodes include:

Every edge is typed as exact derivation, conditional dependence, approximation, import, shared-parent projection, or invalidation.

Incremental rebuild

If only the observer changes, recompute

without rerunning the physical response solves.

If a coupling map changes, recompute descendants of , not the domain or physical operator unless the new map changes those objects.

If the Stage or boundary domain changes, all downstream certificates are stale.

Why this matters

For a research program, recompilation cost can exceed one online waveform evaluation. The dependency DAG turns architectural rigor into practical speed by preventing unnecessary rebuilds.

Part III

Observer Compilation

The direct and adjoint theorems, kernel/image discipline, and frame transport.

Compile source coefficients directly into scientific records

Direct and adjoint record theorems

EXACT SCOPED LINEAR THEOREM

Direct record theorem

Assume the scoped linear problem

has a unique physical solution on the frozen domain.

Let

Then

and therefore

The online map has shape

where and is the retained coefficient rank.

Adjoint form

Write the -th record as

Solve the adjoint problem

Then

Thus the -th row of the compiled map is

This form is especially valuable when the field dimension is enormous but the number of records is small. It requires one adjoint solve per independent record functional rather than one forward solve per query.

Multiple outputs

A time series or waveform is not necessarily one record. It can be represented by:

  • samples at frozen times or frequencies;
  • coefficients in a reduced temporal basis;
  • matched-filter inner products;
  • mode amplitudes and phases;
  • integrated fluxes;
  • detector-channel basis coefficients.

The observer compiler acts on whichever record basis is frozen.

What may be quotiented and what remains physically real

Observer kernel, image, and sensitivity

Kernel and image

The observer map induces

and

A compiled solver may quotient by for the declared record family, but it must not claim the kernel directions are physically absent.

Record-equivalence relation

Define

The online solver computes an equivalence class in

This is the mathematical basis for avoiding full-field assembly.

Sensitivity

If the observer map changes by , the record changes by

Therefore frame, calibration, window, boundary accessibility, truncation, and normalization are part of the model, not external presentation choices.

The controlled test perturbed the observer map by in relative norm and measured a record shift of approximately

Separate ownership ledgers inside one small record map

Compiling the five tensors

Five independent source families

Let the five tensors have coefficient vectors

Collect them into

Let

Then the scoped effective problem is

The record is

with

Why the five sectors remain separate

The final matrix may be concatenated, but the ledgers remain separate:

This permits:

  • separate ranks and tolerances;
  • separate update rates;
  • separate conservation and boundary tests;
  • selective recompilation;
  • one-sector-at-a-time destructive controls.

The compiler compresses the calculation without erasing physical ownership.

Transport before rank estimation or observer projection

Frame transport and holonomy

Common-frame requirement

Let transport local responses to the frozen comparison frame.

A local basis becomes

The compiled record map is therefore parameter-dependent unless transport is factored analytically:

When the frame action has a known representation, such as a mode phase

the transport can be evaluated cheaply online.

Holonomy

For a closed parameter loop ,

If

a single global frame chart does not exist on the declared domain. The compiler must use patches and transition maps.

This is the record-space version of the order-curvature tensor and the transport tensor .

4D core with optional 13D coupling algebra

Complete compiled map

ARCHITECTURE-DERIVED

Complete scoped map

A useful schematic is

Here:

  • is optional and may be the identity;
  • generates the five physical source sectors;
  • is the Schur-condensed operator;
  • is the physical quotient map;
  • transports into the common frame;
  • selects the observer-accessible image;
  • produces the frozen records.

The actual implementation may reorder commuting maps. Noncommuting maps require a proof, a correction term, or a fixed order.

Online equation

The complete field can be reconstructed from stored response bases when required, but it is not an online prerequisite for the declared record family.

When compiler layers may be reordered

Map-order and commutator audit

Why compiler order matters

Two reductions and may not commute:

Examples include:

  • observer projection before frame transport;
  • truncation before boundary gluing;
  • integrating out a sector before applying a constraint that mixes it with retained fields;
  • symmetry reduction after a boundary condition breaks the symmetry.

The difference is

The compiler must either prove

on the admitted domain or include its effect in the error/correction ledger.

Relation to operation-order curvature

The map-order defect is the finite-dimensional analog of the field-space curvature tensor. The rule is the same:

Reorder operations only when their commutator is gauge, theorem-zero, boundary-owned, or below a certified observable error.

Every approximation measured where the claim is made

Record-level error and covariance theorem

Error sources

Let the exact record be and the compiled record be . Decompose the error into:

A conservative norm bound is

Where covariance is known, use

rather than summing independent scalar errors.

Residual-source bound

If the effective problem is

and the compiler omits , then

Therefore

The relevant amplification constant is observer-weighted. A large full-field residual can be harmless to a specific record, while a small residual aligned with a sensitive adjoint can dominate.

Observer deformation bound

For ,

Local compilation, reduced iteration, and relinearization

Nonlinear and time-dependent extension

Local nonlinear problem

For

choose a reference and write

Then

The compiled map is exact only after the nonlinear defect has been represented in the retained five-tensor coefficients or residual.

Iterative compiled solve

One option is a reduced fixed-point iteration:

with records

Another is piecewise relinearization:

  1. evaluate the current compiled patch;
  2. reconstruct only the diagnostics needed to update ;
  3. compile or select the next patch;
  4. transport records and states across the overlap.

Physical restriction

A nonlinear observer record may require quadratic or history-dependent functionals. Those are compiled by augmenting the record basis with the required products, convolutions, or memory states. The word “observer” does not imply linearity; the direct matrix formula is the linear or linearized core.

Why the observer image can dominate the speedup

Complexity and break-even

Direct and compiled costs

Let:

  • be physical field dimension;
  • be retained five-tensor coefficient dimension;
  • be observer record dimension;
  • be the number of online queries.

A direct factorized dense solve scales approximately as

Full reduced-field assembly scales as

Observer-compiled evaluation scales as

The ideal online ratio is therefore

Offline costs

The compiler pays for:

  • physical quotient construction;
  • operator factorization or iterative setup;
  • forward or adjoint response solves;
  • Schur maps;
  • symmetry decomposition;
  • observer and frame maps;
  • validation and certificates.

The break-even query count is

Mission-scale inference, parameter estimation, surrogate training, and Monte Carlo evaluation can involve sufficiently many queries that a substantial offline compilation is rational.

Part IV

4D and 13D Information

The core compiler is four-dimensional; internal geometry may lower its master algebra.

Useful internal information without thirteen-dimensional online arithmetic

Native 4D and 13D-informed versions

Native 4D version

Set

Everything else remains:

This is a complete observer-compiled five-tensor method in ordinary 4D GR.

13D-informed version

The internal geometry supplies

The online map is

The useful information can include:

  • exact forbidden couplings;
  • fixed coupling ratios;
  • representation degeneracies;
  • parity and topology rules;
  • spectral gaps;
  • sector projectors;
  • normalized overlap tensors.

Null control

A 4D solver given the same must reproduce the same online result. Therefore the unique higher-dimensional contribution is the derivation and certification of , not the matrix multiplication itself.

Online object reduction

Dimension funnel

Dimension funnel
Part V

Reproducible Controlled Example

A complete record-space benchmark before the physical GR replay.

Baseline, five-tensor basis, internal map, and observer records

Controlled 900-state object

TOY ARCHITECTURE TEST

Controlled object

The reproducible test uses:

  • a 900-dimensional positive globally coupled field operator;
  • a reused Cholesky factorization;
  • 24 native 4D five-tensor coefficient channels;
  • 10 13D-informed master functions;
  • 8 frozen observer records.

The dimensions form the funnel

The direct baseline solves the 900-state field for every query and then applies the observer map.

The full 4D reduced path assembles all 900 field components from cached responses and then applies the observer.

The observer-compiled paths precompute

Fairness of the baseline

The direct baseline does not invert the matrix from scratch. It reuses the same factorization, making it a stronger comparison than a naive repeated inversion.

The test is still not a physical GR benchmark because the operator, source functions, and records are manufactured.

Sequential performance, algebraic ceiling, and exactness

Measured results

PASS-CONTROLLED

Sequential end-to-end Python timing

Method Time per query Speedup
Direct factorized field solve, then records 746.018 μs 1×
Full 4D five-tensor field, then records 27.897 μs 26.74×
Observer-compiled 4D 18.258 μs 40.86×
Observer-compiled 13D-informed 13.004 μs 57.37×

This timing includes Python coefficient or master-function evaluation and per-query dispatch.

Batched linear-algebra-only ceiling

Method Time per query Speedup
Direct field solve, then records 47.27036 μs 1×
Full 4D field assembly, then records 1.45985 μs 32.38×
Observer-compiled 4D 0.01269 μs 3726×
Observer-compiled 13D-informed 0.00676 μs 6991×

This second table excludes coefficient/master generation and offline compilation. It measures the linear-algebra opportunity after the record map is compiled.

Exactness

The maximum relative record errors were:

An exact observer-kernel perturbation produced record residual

Measured runtime

Sequential and batched timings

Sequential timing
Batched timing
After compilation, master-function evaluation dominates

Bottleneck migration

The bottleneck moves

The batched result is thousands of times faster because only a small matrix product remains. The sequential result is tens of times faster because Python coefficient generation and function dispatch become dominant.

This is a favorable outcome. It identifies the next engineering targets:

  1. vectorize master-function evaluation;
  2. compile coefficient functions with JIT or native code;
  3. batch parameter points;
  4. cache slowly varying invariant masters;
  5. use automatic differentiation only on the small master algebra;
  6. separate calibration-time and query-time operations.

The field solve is no longer the bottleneck in the controlled observer-compiled path.

Part VI

Path to a Realistic GR Demonstration

The next calculation will test physical records rather than a manufactured operator.

A record family rich enough for independent validation

Physical waveform and flux records

Candidate record family for an IMRI bridge

A realistic demonstration should freeze records such as:

  • selected complex waveform-mode basis coefficients;
  • phase and amplitude residuals on a reduced time/frequency grid;
  • total radiated energy;
  • total radiated angular momentum;
  • horizon absorption;
  • remnant mass and spin;
  • mismatch integrals for frozen noise curves;
  • optional detector-channel basis coefficients.

The record family must be rich enough to support independent validation. Compiling only a final scalar mismatch would hide physically important failures.

Record-basis strategy

A long waveform can be represented by a temporal or frequency reduced basis:

The observer compiler predicts the coefficients , while the residual receives a norm and mismatch certificate.

This preserves waveform information without reconstructing every bulk spacetime degree of freedom.

Development, demonstration, and held-out replay

q=64 → q=100 → q=128 execution

Development stage: q=64

  1. Freeze the physical branch, gauge, boundaries, frame, and record basis.
  2. Build the conservation-saturated 4D coupling space.
  3. Construct the five physical source sectors.
  4. Build physical quotient and Schur-condensed operators.
  5. Solve forward and adjoint response problems.
  6. Compile the q=64 record map.
  7. Compare direct/reference and compiled records.
  8. Measure ranks, errors, offline cost, online cost, and break-even.
  9. Run missing-flux, wrong-frame, omitted-sector, and stale-map controls.

Demonstration stage: q=100

Freeze all compiler choices before evaluating the demonstration target. Only declared interpolation or continuation rules may be used.

Held-out stage: q=128

The held-out case is evaluated once under the frozen pipeline. A failed record or conservation ledger is a failure, not an invitation to modify the compiler and preserve the original claim.

Physics and compute must pass simultaneously

Acceptance and terminal grammar

Physics acceptance

The future physical calculation should require simultaneous passing of:

  • per-mode phase;
  • per-mode amplitude;
  • full waveform mismatch;
  • radiated energy;
  • radiated angular momentum;
  • horizon flux where applicable;
  • endpoint/remnant balance;
  • frame and extraction uncertainty;
  • held-out replay.

Computational acceptance

A speed claim should include:

  • offline compilation time;
  • number and type of forward solves;
  • number and type of adjoint solves;
  • coefficient-generation time;
  • online record time;
  • optional full-field reconstruction time;
  • memory;
  • hardware and software environment;
  • amortized break-even.

Claim labels

Recommended terminals:

  • PASS-CONTROLLED-ARCHITECTURE
  • PASS-PHYSICAL-RECORD-COMPILATION
  • NO-OBSERVER-COMPRESSION
  • NO-13D-INFORMATION-GAIN
  • BLOCKED-DOMAIN
  • BLOCKED-BOUNDARY
  • BACKGROUND-UPDATE-REQUIRED
  • FAIL-HELD-OUT
The closest false-positive objects the compiler must reject

Mandatory destructive controls

Mandatory destructive controls

  1. Observer-kernel control: add a physical field direction in ; records must remain unchanged while the field ledger records the direction.
  2. Wrong-observer control: perturb frame, calibration, or window; sensitivity must match the deformation Jacobian.
  3. Wrong-frame control: compile before common-frame transport; the record should fail.
  4. Missing-flux control: omit horizon or null-infinity flux; balance laws must fail.
  5. Schur omission control: delete an off-diagonal block; a mixed-sector record must change.
  6. Symmetry-breaking boundary control: apply a boundary that breaks an assumed irrep decomposition; block reuse must be rejected.
  7. Scale-crossing control: move through an eigenvalue crossing without recompilation; the stale basis must fail.
  8. Stale-certificate control: change Stage or observer hash while retaining the old compiled map; freshness must fail.
  9. 13D null control: supply the same coupling map to a native 4D solver; results must agree.
  10. Held-out control: execute a target excluded from rank and map selection.
Part VII

Human and AI Continuation Contract

How to preserve, review, and extend the idea without context drift.

Inputs, outputs, hashes, blockers, and stop rules

Machine-use contract

Machine-use objective

An AI agent continuing this project should treat the document as an executable specification, not as permission to improvise a fast surrogate.

Required input packet

The agent must load:

  • physical branch and parameter domain;
  • Stage, Rulebook, and operator-domain hashes;
  • five-tensor ownership definitions;
  • conservation and coupling matrices;
  • boundary and flux ledger;
  • frame and synchronization conventions;
  • observer record definition;
  • tolerances and covariance;
  • source data hashes;
  • held-out commitment.

Required output packet

The agent must emit:

  • physical quotient basis and rank;
  • Schur block map and error;
  • symmetry block/orbit map;
  • five-tensor basis ranks;
  • observer kernel/image inventory;
  • forward and adjoint solve manifests;
  • compiled maps and hashes;
  • record covariance;
  • runtime decomposition;
  • destructive-control results;
  • terminal and open residuals.

Stop rules

Stop and return OPEN or a blocker when:

  • an operator domain is undefined;
  • a required boundary flux is absent;
  • the observer map is ambiguous;
  • map order is noncommuting and unowned;
  • a rank decision is unstable;
  • a source change invalidates a downstream certificate;
  • a calculation does not affect the terminal decision.
Offline, online, and diagnostic paths

Reference compiler pseudocode

Offline compiler

freeze_authority_and_scope()
freeze_stage_rulebook_boundary_scale_time_observer()

V_candidate = generate_complete_response_grammar()
G = build_gauge_redundancy_map(V_candidate)
J = build_constraint_jacobian(V_candidate)
Q_phys = certified_basis(ker(J) / image(G))

L_phys = project_operator(L, Q_phys)
[A, B, C, D] = partition_mixed_sectors(L_phys)
S = A - B * inverse(D) * C

blocks = decompose_by_stage_preserving_symmetry(S)
B5 = build_five_tensor_sources(blocks)
C13 = derive_internal_coupling_map_or_identity()

O = build_frozen_observer_map()
UH = build_common_frame_transport()
W4 = O * UH * inverse(S) * B5
W13 = W4 * C13

compile_error_and_covariance_ledgers()
run_destructive_controls()
publish_hashes_and_dependency_DAG()

Online evaluator

select_scale_time_patch(lambda)
u = evaluate_invariant_master_functions(lambda)
y = W_patch * u
y = apply_matching_and_calibration(y)
return records_with_covariance_and_scope()

Diagnostic full-field reconstruction

if diagnostic_requires_bulk_field:
    h = U_patch * u
    verify_constraints_boundaries_and_flux(h)
Friendly human, AI, and NASA/JPL review paths

Reviewer guide

Friendly human review

A reviewer can evaluate the proposal in five questions:

  1. Are the physical quotient and operator domains lawful?
  2. Are mixed sectors integrated out with controlled generated terms?
  3. Is the observer map scientifically sufficient for the claimed result?
  4. Do compiled records agree with independent physical references?
  5. Do end-to-end costs, not only matrix kernels, improve materially?

AI review

An AI reviewer should build a claim table with columns:

  • claim;
  • theorem or source;
  • assumptions;
  • executable witness;
  • negative control;
  • current status;
  • residual;
  • reopen trigger.

The reviewer should reject any sentence that promotes a controlled toy result into a physical GR result.

NASA/JPL review

A mission-oriented review should focus on:

  • record sufficiency for waveform inference;
  • robustness across mass ratio and detector response;
  • calibration and uncertainty propagation;
  • computational value for repeated inference;
  • reproducibility on independent hardware;
  • failure behavior outside the compiled record domain.
Strongest current claim and exact next step

Conclusion

READY FOR PHYSICAL EXAMPLE

Core conclusion

The most important insight from the building blocks is not that the spacetime field is always small. It is that a declared scientific question often depends on a very small, rigorously characterized image of that field.

The five tensors organize physical ownership. The conservation stack restricts their lawful coefficients. The new compiler removes gauge and rigid directions, integrates out coupled sectors, exploits exact symmetry, transports frames, and maps directly into observer records.

The architecture is useful in 4D. The 13D geometry can improve it by supplying a lower-dimensional coupling algebra, but it is not required for the core observer-first acceleration.

The strongest claim supported today is:

In a controlled 900-state field analog, an observer-compiled five-tensor map reproduced direct records to machine precision and reduced sequential end-to-end time by roughly fortyfold in 4D and fifty-sevenfold with a ten-master internal coupling map. Batched record-space algebra showed a several-thousandfold ceiling. A physical IMRI result remains to be executed under the frozen derivation and review protocol preserved here.

Source-derived literature families

Reference context

REFERENCE CONTEXT

Source-derived physics and numerical context

The companion documents cite and discuss:

  1. covariant phase space and Noether-charge methods;
  2. Wald–Zoupas boundary charge and flux;
  3. first- and second-order gravitational self-force;
  4. Magnus and Baker–Campbell–Hausdorff expansions;
  5. reduced-order and surrogate gravitational-wave modeling;
  6. public perturbative and numerical-relativity waveform resources;
  7. NASA LISA preparatory waveform-modeling priorities.

This monograph's novel content is the compilation architecture and its integration with the uploaded building blocks. The physical literature remains the authority for the underlying GR equations and waveform reference data.

Part VIII

Source and Execution Appendices

Exact building-block obligations, benchmark code, certificate, and provenance.

Appendix A

Rigidity source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_RIG_4_1_MAX_RIGOR_ALL_GATE_RIGIDITY.md

RIG-C11 — Function-space, domain, Fredholm, essential-spectrum, and coercivity control

Question. Function-space, domain, Fredholm, essential-spectrum, and coercivity control?

Formal requirement. Rigidity and stability must be formulated on the actual operator domain and function space. The certificate must classify discrete and essential spectrum, kernel/cokernel, Fredholm index where applicable, boundary/domain dependence, and a coercive or gap estimate on the non-whitelisted physical subspace.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

  1. generate the complete typed object for this constraint.
  2. compute exact/interval-certified invariants.
  3. run the destructive control.
  4. publish residuals, scope, and reopen triggers.

Required outputs. function_space_domain_packet, spectrum_coercivity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Finite matrices or a few eigenvalues can be positive while an essential-spectrum instability, non-Fredholm direction, or domain-dependent zero mode survives.

Destructive control. Two operators have identical first N eigenvalues; one has essential spectrum crossing zero. The audit must reject equivalence.

RIG-C22 — Cross-scale continuation, decoupling, RG spectral flow, and matching stability

Question. Cross-scale continuation, decoupling, RG spectral flow, and matching stability?

Formal requirement. Track physical modes, Hessian eigenvalues, constraints, and protected directions across Scale packets, thresholds, compactification levels, EFT matching, and RG flow; prove decoupling or record nondecoupling and eigenvalue crossings.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

  1. generate the complete typed object for this constraint.
  2. compute exact/interval-certified invariants.
  3. run the destructive control.
  4. publish residuals, scope, and reopen triggers.

Required outputs. cross_scale_continuation_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A mode stable at one scale can become tachyonic, ghostlike, or strongly mixed after threshold transport.

Destructive control. Two candidates match at one scale; only one eigenvalue crosses zero above a threshold.

Appendix B

Interdependence source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md

INT-C03 — Typed dependency DAG and provenance semantics

Requirement. Construct a typed, acyclic provenance DAG whose edges distinguish exact derivation, conditional dependence, approximation, import, shared-parent projection, and invalidation.

Execution. Generate the relevant node and edge grammar before inspecting outcomes; populate the owned evidence packet(s); compute or prove every dependency, approximation, alias, and invalidation; run the destructive control.

Minimal discharge. Dependency graph, edge ledger, topological-order certificate.

Fail trigger. A dependency is implicit, an edge is untyped, provenance cycles exist, or graph order is indeterminate.

Owned packets. dependency_graph_packet.

INT-C06 — Complete mixed-sector operator and Schur-complement closure

Requirement. All diagonal and off-diagonal sectors are computed, proven zero, or bounded/integrated out with controlled Schur complements across metric, gauge, matter, boundary, vacuum, and constraint sectors.

Execution. Generate the relevant node and edge grammar before inspecting outcomes; populate the owned evidence packet(s); compute or prove every dependency, approximation, alias, and invalidation; run the destructive control.

Minimal discharge. Mixed-operator block matrix, zero proofs, Schur-complement bounds.

Fail trigger. A sector-only positivity or spectrum claim ignores an unresolved mixed block.

Owned packets. mixed_sector_packet, schur_complement_packet.

INT-C10 — Factorization, separability, and subsystem taxonomy

Requirement. Every claimed split is typed as exact, effective, bounded approximate, or ideal control; gauge, boundary, edge, global, and entanglement obstructions are tested.

Execution. Generate the relevant node and edge grammar before inspecting outcomes; populate the owned evidence packet(s); compute or prove every dependency, approximation, alias, and invalidation; run the destructive control.

Minimal discharge. Factorization certificate and obstruction ledger.

Fail trigger. A product Stage or diagonal action is promoted to Hilbert-space, algebraic, probabilistic, or information-theoretic factorization without proof.

Owned packets. factorization_packet.

INT-C11 — Decoupling, integration-out, matching, and controlled error

Requirement. Integrated-out sectors declare windows, norms, matching maps, generated terms, state/process families, and quantitative errors, including memory/nonlocal effects.

Execution. Generate the relevant node and edge grammar before inspecting outcomes; populate the owned evidence packet(s); compute or prove every dependency, approximation, alias, and invalidation; run the destructive control.

Minimal discharge. Decoupling/matching packet with error bound.

Fail trigger. Independence or EFT validity is asserted without a bound or generated mixed/constant terms are omitted.

Owned packets. decoupling_matching_packet.

INT-C13 — Branch restart, stale-certificate invalidation, and archive discipline

Requirement. Comparison-relevant changes create new branch IDs, invalidate downstream certificates, preserve audit history, and prohibit silent re-import.

Execution. Generate the relevant node and edge grammar before inspecting outcomes; populate the owned evidence packet(s); compute or prove every dependency, approximation, alias, and invalidation; run the destructive control.

Minimal discharge. Restart packet, stale-evidence list, archive manifest.

Fail trigger. A changed branch reuses a PASS certificate without scoped invariance proof.

Owned packets. branch_restart_packet.

Appendix C

Observer source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_OBS_4_0_MAX_RIGOR_ALL_GATE_OBSERVER.md

OBS-C06 — Kernel, image, degeneracy, inaccessible-sector, and information-loss inventory

Origin: preserved BB-OBS-1.0.1.

Requirement. Enumerate the map kernel, image, degeneracies, inaccessible sectors, indirect observables, above-cutoff directions, and unresolved hidden sectors through the claimed scope.

Minimal discharge. Enumerate the map kernel, image, degeneracies, inaccessible sectors, indirect observables, above-cutoff directions, and unresolved hidden sectors through the claimed scope.

Fail trigger. A null direction or inaccessible physical sector is silently treated as nonexistent.

Adversarial thought experiment. Delete an inaccessible physical direction and call it absent.

Required packet(s). kernel_image_packet.

OBS-C07 — Observer-map deformation Jacobian and sensitivity

Origin: preserved BB-OBS-1.0.1.

Requirement. Differentiate records with respect to frame, scale, scheme, basis, chamber, detector window, boundary accessibility, normalization, truncation, regulator, and coarse-graining parameters.

Minimal discharge. Differentiate records with respect to frame, scale, scheme, basis, chamber, detector window, boundary accessibility, normalization, truncation, regulator, and coarse-graining parameters.

Fail trigger. A small admissible observer-map change can move a result across the pass band but is not counted.

Adversarial thought experiment. Perturb the observer map slightly across the pass band without counting it.

Required packet(s). observer_deformation_packet.

OBS-C08 — Full uncertainty, covariance, scheme/frame transport, and truncation error

Origin: preserved BB-OBS-1.0.1.

Requirement. Propagate correlated input, truncation, scheme, observer, calibration, and model uncertainties through the complete map.

Minimal discharge. Propagate correlated input, truncation, scheme, observer, calibration, and model uncertainties through the complete map.

Fail trigger. Only central values or independent errors are compared.

Adversarial thought experiment. Drop correlations and compare central values only.

Required packet(s). uncertainty_covariance_packet.

Appendix D

Boundary source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_BND_4_1_MAX_RIGOR_FULL_GATE_CLOSURE_BOUNDARY.md

BND-C03 — Lawful operator-domain and extension certificate

Owners: Boundary + Dynamics + Rulebook
Legacy status: PRESERVED

Requirement. For every field, ghost, auxiliary, edge field, and load-bearing composite operator, define the complete domain: representation, parity, trace/boundary value, normal derivative or spectral condition, self-adjoint extension, ellipticity/strong ellipticity or hyperbolic admissibility, adjoint domain, gauge/BRST preservation, and kernel.

Thought experiment forcing the row. A formally symmetric operator may fail to be self-adjoint; a parity table may fail BRST preservation; two self-adjoint extensions may have different kernels. Thus boundary labels alone do not determine physics.

Execution algorithm.

  1. Generate the candidate domain grammar before inspecting the gate target.
  2. Compute the boundary form/Green identity.
  3. Classify admissible extensions and gauge/ghost compatibility.
  4. Test ellipticity, strong ellipticity, Fredholmness, or characteristic well-posedness by operator type.
  5. Compute kernels and adjoint kernels on the frozen domain.

Required outputs. operator_domain_certificate, domain_extension_grammar, kernel_ledger.

Minimal discharge. Every operator has one frozen lawful domain with preserved gauge/BRST action and an explicit kernel; alternative admissible extensions receive dispositions.

Fail trigger. A parity or boundary-condition table substitutes for an operator domain, or an admissible extension is silently chosen after the desired spectrum is known.

Mandatory destructive controls. symmetric-not-self-adjoint domain; BRST image leaves domain; alternative extension changes zero-mode count.

Mapped gates. Black-hole-singularity, Gap-01, Gap-02, Gap-10/BG-10, Gap-11, Gap-13, SG-1, SG-2, SG-3, SG-5, SG-6, SG-7, SG-8, SG-9, UQF-10, UQF-14, UQF-3, UQF-4, UQF-5A/5B, UQF-7, UQF-9, theta-bar-QCD.

BND-C17 — Regional algebras, edge modes, factorization, and entanglement cuts

Owners: Boundary + Dynamics + Observer + Granularity
Legacy status: NEW — FORCED BY ALL-GATE AUDIT

Requirement. For gauge/gravitational subsystems and cuts, construct the regional observable algebra, center/superselection data, edge-mode completion, factorization or extended-Hilbert-space map, entropy/information records, and gluing back to the global physical space.

Thought experiment forcing the row. A global gauge-invariant Hilbert space need not factorize across a cut; ignoring edge modes can produce false entropy loss or nonunitarity.

Execution algorithm.

  1. Define the cut/support and regional gauge group.
  2. Construct regional algebras and centers.
  3. Add only required edge variables with Actor/Co-Actor dispositions.
  4. Build the factorization/gluing map and information-flow ledger.
  5. Test cut independence of global predictions.

Required outputs. regional_edge_factorization_packet.

Minimal discharge. The regional algebra and edge completion reproduce the global physical space under gluing and yield finite Observer-accessible records.

Fail trigger. Naive tensor factorization is assumed, edge sectors are omitted/double-counted, or subsystem loss is called global nonunitarity.

Mandatory destructive controls. Gauss-law factorization failure; edge-mode double count; cut-dependent global probability.

Mapped gates. Black-hole-singularity, Born-rule, Gap-02, Gap-08, Gap-13, UQF-14, UQF-3.

BND-C18 — Causal, null, horizon, and asymptotic boundary data with flux balance

Owners: Boundary + Dynamics + Time Synchronization + Observer
Legacy status: NEW — FORCED BY ALL-GATE AUDIT

Requirement. Classify spacelike, timelike, null, characteristic, horizon, and asymptotic components; specify admissible data, radiation/absorbing/reflection conditions, causal support, symplectic/charge/energy/probability/information flux, and asymptotic charges/memory where applicable.

Thought experiment forcing the row. A local conservation law can hold while unaccounted flux crosses a horizon or null infinity; imposing timelike data on a characteristic surface can overdetermine evolution.

Execution algorithm.

  1. Classify causal type and induced/degenerate geometry.
  2. Select data compatible with the PDE characteristic structure.
  3. Compute all boundary currents and charges.
  4. Establish synchronization and Observer record maps.
  5. Verify global balance after adding all interfaces/asymptotic pieces.

Required outputs. causal_asymptotic_boundary_packet.

Minimal discharge. The initial-boundary problem is causally admissible and every regional/global current balances including horizon/asymptotic flux.

Fail trigger. Boundary data are incompatible with causal type, or a flux channel is omitted.

Mandatory destructive controls. horizon flux omitted; null boundary overdetermined; absorbing condition violates gauge constraint.

Mapped gates. Black-hole-singularity, Born-rule, Gap-02, Gap-05-stability, Gap-05-value, Gap-08, Gap-10/BG-10, Gap-11, Gap-13, Lambda-catastrophe, UQF-14, UQF-3, UQF-5A/5B, UQF-5C.

Appendix E

Time Synchronization source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_TS_4_0_MAX_RIGOR_ALL_GATE_TIME_SYNCHRONIZATION.md

TS-C06 — Loop closure, holonomy, topology, and global/local/failed trichotomy

Requirement. Every proposed global synchronization shall test all generated loops or prove a class theorem; nonzero holonomy forces local/convention-corrected or failed status.

Primary witness. loop_holonomy_packet.

Execution algorithm.

  1. Freeze the event domain, branch, frame/congruence, protocol, precision, and relevant external authorities.
  2. Generate the complete candidate objects and tests named by this row.
  3. Compute or certify the required relation, transport, operator, graph, partition, uncertainty, or exhaustion result.
  4. Run the row-specific destructive control and at least one omitted-rival control.
  5. Publish exact source references, hashes, assumptions, residuals, and claim ceiling.
  6. Return OPEN if a required input, calculation, theorem, or type proof is missing.

Fail trigger. Local links are promoted to one global time without path-independence proof.

Claim ceiling. Passing this row supports only the declared event domain, frame/congruence, protocol, precision, boundary/quantum/history scope, and evidence depth.

TS-C12 — Clock transport, redshift, propagation delay, and path asymmetry

Requirement. Clock comparison shall include gravitational/kinematic redshift, propagation, dispersion, hardware delay, boundary delay, and asymmetric-path corrections with provenance.

Primary witness. clock_transport_packet.

Execution algorithm.

  1. Freeze the event domain, branch, frame/congruence, protocol, precision, and relevant external authorities.
  2. Generate the complete candidate objects and tests named by this row.
  3. Compute or certify the required relation, transport, operator, graph, partition, uncertainty, or exhaustion result.
  4. Run the row-specific destructive control and at least one omitted-rival control.
  5. Publish exact source references, hashes, assumptions, residuals, and claim ceiling.
  6. Return OPEN if a required input, calculation, theorem, or type proof is missing.

Fail trigger. Equal readouts are compared while omitting path-dependent delay or redshift.

Claim ceiling. Passing this row supports only the declared event domain, frame/congruence, protocol, precision, boundary/quantum/history scope, and evidence depth.

TS-C14 — Time-dependent congruences and moving-domain transport

Requirement. For moving boundaries, evolving frames, phase interfaces, or changing supports, event cells and domains shall be transported consistently through time.

Primary witness. moving_domain_packet.

Execution algorithm.

  1. Freeze the event domain, branch, frame/congruence, protocol, precision, and relevant external authorities.
  2. Generate the complete candidate objects and tests named by this row.
  3. Compute or certify the required relation, transport, operator, graph, partition, uncertainty, or exhaustion result.
  4. Run the row-specific destructive control and at least one omitted-rival control.
  5. Publish exact source references, hashes, assumptions, residuals, and claim ceiling.
  6. Return OPEN if a required input, calculation, theorem, or type proof is missing.

Fail trigger. A static synchronization map is reused after the domain or congruence changes.

Claim ceiling. Passing this row supports only the declared event domain, frame/congruence, protocol, precision, boundary/quantum/history scope, and evidence depth.

TS-C19 — Multi-scale synchronization and refinement consistency

Requirement. Synchronization partitions shall commute with lawful coarse-graining/refinement, tower truncation, detector bandwidth, and Scale transport within the claimed scope.

Primary witness. refinement_consistency_packet.

Execution algorithm.

  1. Freeze the event domain, branch, frame/congruence, protocol, precision, and relevant external authorities.
  2. Generate the complete candidate objects and tests named by this row.
  3. Compute or certify the required relation, transport, operator, graph, partition, uncertainty, or exhaustion result.
  4. Run the row-specific destructive control and at least one omitted-rival control.
  5. Publish exact source references, hashes, assumptions, residuals, and claim ceiling.
  6. Return OPEN if a required input, calculation, theorem, or type proof is missing.

Fail trigger. A synchronization class changes under finer lawful resolution without reopening the claim.

Claim ceiling. Passing this row supports only the declared event domain, frame/congruence, protocol, precision, boundary/quantum/history scope, and evidence depth.

TS-C22 — Initial-boundary synchronization and well-posed evolution

Requirement. Initial data, boundary clocks, constraint propagation, synchronization conditions, and evolution domains shall form a well-posed initial-boundary problem.

Primary witness. initial_boundary_time_packet.

Execution algorithm.

  1. Freeze the event domain, branch, frame/congruence, protocol, precision, and relevant external authorities.
  2. Generate the complete candidate objects and tests named by this row.
  3. Compute or certify the required relation, transport, operator, graph, partition, uncertainty, or exhaustion result.
  4. Run the row-specific destructive control and at least one omitted-rival control.
  5. Publish exact source references, hashes, assumptions, residuals, and claim ceiling.
  6. Return OPEN if a required input, calculation, theorem, or type proof is missing.

Fail trigger. Synchronized initial data evolve into incompatible boundary timestamps or violate constraints.

Claim ceiling. Passing this row supports only the declared event domain, frame/congruence, protocol, precision, boundary/quantum/history scope, and evidence depth.

Appendix F

Scale source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_SCL_4_1_MAX_RIGOR_ALL_GATE_SCALE.md

SCL-C06 — Quantitative gaps, decoupling, and observation windows

Question. Is every omitted physical mode quantitatively separated from the claimed observation window after normalization, mixing, uncertainty, and boundary effects?

For every omitted sector $n$ require

with covariance, mixing, loop/matching order, domain, and tail scope included.

Required inputs

  • physical spectrum;
  • boundary domains;
  • mixing matrices;
  • observation window;
  • uncertainty covariance;

Execution algorithm

  1. identify the lightest omitted physical eigenvalue per sector.
  2. canonically normalize and diagonalize.
  3. propagate covariance and truncation error.
  4. compare worst-case lower bound to the observation window.
  5. certify tail or limit scope.

Required outputs

  • gap_decoupling_packet;
  • tower_matching_tail_packet;

Minimal discharge. Sector-by-sector lowest-mode table and robust uncertainty-margin calculation.

Fail trigger. “Compact,” “heavy,” “Planck suppressed,” or central-value separation is used without a quantitative physical gap.

Destructive control. Choose a central gap above the cutoff whose lower confidence bound enters the window; closure must fail.

SCL-C17 — Multi-scale EFT, tower matching, truncation, and tail control

Question. Does every finite-mode or finite-operator calculation state its scale interval and control omitted towers/tails?

For nested effective descriptions $\mathcal E_0\subset\cdots\subset\mathcal E_N$, require matching maps, overlap checks, and a tail certificate $|R_N|\le\epsilon_N$ or an explicit finite-scope ceiling.

Required inputs

  • tower spectrum;
  • operator basis;
  • matching scales;
  • RG data;
  • Granularity refinement lattice;

Execution algorithm

  1. partition scale intervals.
  2. match adjacent EFTs.
  3. test overlap and decoupling.
  4. compute omitted-mode/operator tail or bound.
  5. vary truncation/refinement and record convergence.
  6. forbid global promotion without a theorem.

Required outputs

  • tower_matching_tail_packet;
  • resolution_refinement_packet;

Minimal discharge. Nested matching ledger and convergence/tail certificate with explicit validity interval.

Fail trigger. A zero-mode, finite tower, or finite heat-kernel prefix is promoted to an all-scale claim.

Destructive control. Construct two identical finite prefixes with convergent and divergent tails; global inference must remain OPEN.

SCL-C18 — Time, temperature, horizon, correlation, and state-dependent scales

Question. Are scales that depend on state, history, region, horizon, temperature, or synchronization treated as derived fields/records rather than fixed universal constants?

For derived scale $s[\Psi,g,\rho,t,U]$, publish its defining functional, covariance, support, synchronization convention, variation, and domain of validity.

Required inputs

  • Dynamics histories;
  • Boundary derived supports;
  • Time Synchronization packet;
  • Observer regional records;

Execution algorithm

  1. enumerate state-/region-dependent scale definitions.
  2. derive them from frozen states and supports.
  3. transport clocks/temperatures/redshifts consistently.
  4. compute variation and uncertainty.
  5. test across transitions and horizon/interface changes.

Required outputs

  • state_dependent_scale_packet;
  • same_ruler_packet;

Minimal discharge. Definition and transport packet for every nonconstant scale, including support, time standard, and uncertainty.

Fail trigger. A local temperature, horizon radius, correlation length, clock rate, or transition scale is treated as one global fixed ruler.

Destructive control. Use a single coordinate time or temperature across redshifted regions; the same-ruler test must fail.

SCL-C23 — Scale-transformation covariance, homogeneity, and anomalous dimensions

Question. Does every scale-bearing field, operator, observable, spectrum, and response come with its transformation law under a change of physical scale, rather than only a value at one reference point?

Formal requirement.

For each typed object $O_i$ and admissible scale transport $s\mapsto \lambda s$ or $\mu\mapsto e^t\mu$, require a controlled representation

with canonical dimensions, anomalous-dimension/mixing data, normalization transport, reference/pivot scale, validity interval, and residuals explicit.

Required inputs.

  • typed operator/observable roster;
  • canonical dimensions;
  • RG or physical scaling map;
  • normalization and mixing matrices;
  • reference/pivot scales;

Execution algorithm.

  1. generate every admissible physical scale transformation.
  2. derive canonical homogeneity degrees.
  3. compute or import anomalous dimensions and operator mixing.
  4. transport normalizations and covariance.
  5. test slopes/exponents over the claimed interval rather than one calibration point.
  6. publish residuals and scope ceiling.

Required outputs. scale_transformation_packet.

Minimal witness. A complete transformation-law ledger with canonical/anomalous dimensions, mixing, pivot/reference scales, interval tests, and zero unclassified scale responses.

Fail trigger. Two candidates agree at one scale but differ in scaling law, anomalous dimension, or homogeneity and are nevertheless treated as physically equivalent.

Destructive control. Construct two models with the same observable at one pivot scale but different anomalous dimensions; the audit must distinguish them.

Appendix G

Dynamics source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_DYN_4_2_MAX_RIGOR_FULL_GATE_CLOSURE_DYNAMICS.md

DYN-C10 — Complete interaction, source, and dangerous-operator hypergraph

Owned question: What are all lawful interactions and sources, including theorem-zero and dangerous channels?

Required inputs: Parent term grammar; Actor/Co-Actor registry; Rulebook representation and selection rules; Boundary/Scale data.

Execution algorithm:

  1. Generate all arity-1 and higher hyperedges allowed by support, representation, tensor, parity, charge, and derivative grammar.

  2. Map every edge to a parent owner or a theorem-zero exclusion.

  3. Track radiative generation, dangerous operators, portals, CP sources, baryon/lepton/flavor charges, and Observer records.

  4. Run edge saturation and omitted-edge controls before evaluating the target gate.

Required outputs: Interaction hypergraph; theorem-zero ledger; dangerous-operator roster; edge-saturation certificate.

Minimal discharge: Enumerate every parent-owned interaction vertex, source edge, representation contraction, selection rule, theorem-zero edge, dangerous operator, support, scale, and boundary domain.

Fail trigger: A coupling, decay, portal, Yukawa, CP source, or dangerous operator is inserted or omitted without a complete parent-law disposition.

Mandatory destructive control: Hold the free spectrum fixed and add one allowed cubic/Yukawa edge to one candidate. Interaction-dependent gates must distinguish them.

Gate families invoking this row: Gap-10/BG-10, Gap-11, SG-5, SG-7, SG-8, SG-9, UQF-10, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

Appendix H

Governance source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_CSE_4_0_SUPPLEMENTAL_MAX_RIGOR_GATE_GOVERNANCE.md

CSE-C11 — Complete crosswalk, ownership, dependency DAG, and row splitting

Requirement. Map every gate obligation to one owner, evidence suppliers, exact rows, criticality, and a typed dependency DAG; split rows containing independently variable physical objects.

Required outputs. crosswalk_packet, dependency_dag_packet

Fail trigger. An obligation is unmapped, multiply owned without arbitration, or hidden in prose; a procedural row enters a physical Jacobian.

Adversarial thought experiment. An anomaly and a boundary-domain obligation are combined into one PASS cell. This must split.

CSE-C18 — Building-block evolution, delta admission, and shadow replay

Requirement. A block delta requires candidate-neutral motivation, owner, witness, fail trigger, destructive control, dependency delta, replay set, invalidation radius, and shadow replay across affected gates.

Required outputs. block_delta_packet, shadow_replay_packet

Fail trigger. A new rule is applied only to the originating gate or enters controlling status without replay.

Adversarial thought experiment. An SG-4-discovered anomaly rule is not tested on UQF-4. This must block merge.

CSE-C20 — Invalidation radius, stale evidence, and topological revalidation

Requirement. Every upstream change emits deterministic invalidation, marks stale artifacts, and revalidates descendants in dependency order from identity and domains through public claims.

Required outputs. invalidation_report_packet, revalidation_packet

Fail trigger. Old PASS certificates remain live after Shape, Scale, Boundary, Observer, or procedure hashes change.

Adversarial thought experiment. A parity-domain update leaves the old chirality certificate current. This must fail freshness.

CSE-C27 — Recursive decomposition, anti-hang, value of information, and stop rules

Requirement. Split large tasks into independently decidable subproblems, prioritize load-bearing/value-of-information computations, maintain checkpoints, and stop on authority conflict, undefined objects, or non-decision computation.

Required outputs. recursive_decomposition_packet

Fail trigger. The agent loops indefinitely, performs expensive calculations before identity/domain checks, or decomposes until context is lost.

Adversarial thought experiment. A 100-hour numerical scan runs before checking an exact representation mismatch. This must fail execution order.

Appendix I

Shape Stage source extracts

VERBATIM SOURCE EXTRACT

Source file

stage.md

S-14 — Frozen block-product metric

  • Families: STG-03, STG-10, STG-11, STG-12
  • Status: CONFIRMED / MIXED MODES EXCLUDED
  • Definition: G13=g4 direct-sum R6^2 h_*(K6) direct-sum R2^2 gamma_round direct-sum RY^2 dtheta^2; undeclared mixed external/internal and off-diagonal internal components are outside the physical Stage.

S-17 — Stage symmetry and automorphism ledger

  • Families: STG-09, STG-17
  • Status: CONFIRMED STRUCTURAL
  • Definition: The physical symmetry is restricted to transformations preserving the complete frozen Stage, schematically Diff(M4) semidirect (Aut(X9,h_*) semidirect G_bundle) with orbifold restrictions; Stage-changing diffeomorphisms are not gauge.
Appendix J

Shape Rulebook source extracts

VERBATIM SOURCE EXTRACT

Source file

rulebook.md

R-09 — Gauge/constraint physical quotient

  • Families: RB-06, RB-09, RB-12
  • Status: CONFIRMED-STRUCTURAL; FULL QUANTUM WITNESS-SCOPED
  • Rule: Gauge, BRST/BV/FP, exact constraints, and physical cohomology remove redundancies before spectral or Actor interpretation.

R-15 — Scale, ruler, provenance, and operational-equivalence rule

  • Families: RB-15, RB-11
  • Status: CONFIRMED-STRUCTURAL
  • Rule: Every number, coefficient, tolerance, radius, scheme, normalization, and comparison uses one frozen provenance and ruler tuple.

R-16 — Observer projection and calibration firewall

  • Families: RB-11, RB-02
  • Status: CONFIRMED-STRUCTURAL; CONTINUOUS MAPS-SCOPED
  • Rule: The parent-to-record map is explicit, normalized, reconstructible, and protected against blind-target leakage, wrong rulers, aliases, and hidden normalization.

R-17 — Interaction/source and proton-safety selection rule

  • Families: RB-13, RB-16
  • Status: CONFIRMED-CLASSIFICATION; FULL INTERACTION HYPERGRAPH-SCOPED
  • Rule: Only parent-owned interactions are legal; sector-orthogonal projectors enforce claimed-zero quark–lepton mediator routes unless an explicit violating owner is introduced.

R-22 — Generation basis and sector-projector algebra

  • Families: RB-04, RB-07, RB-13, RB-16
  • Status: DECLARED-FROZEN; DOWNSTREAM FLAVOR CERTIFICATES-SCOPED
  • Rule: The three-dimensional generation basis and orthogonal sector projectors Pi_u, Pi_d, Pi_e, Pi_nu route the chiral family modules and sector selection.
Appendix K

Vacuum source extracts

VERBATIM SOURCE EXTRACT

Source file

BB_VAC_4_0_MAX_RIGOR_ALL_GATE_VACUUM.md

VAC-C10 — Dependency invalidation and stale-certificate propagation

Origin: preserved BB-VAC-1.3
Owned question: What must be true for this Vacuum obligation to be discharged without substituting another block's result?

Normative requirement

Any change in Shape, variational status, boundary support, Scale, Dynamics, regulator, or vacuum branch invalidates every dependent background, ledger, matching, and history certificate.

Required execution

  1. Load the frozen gate contract, authority hashes, Stage/Rulebook/Actor branch, Observer record family, Scale tuple, Boundary supports, and time/history scope.
  2. Generate the complete candidate/state/branch/test roster relevant to this row before selecting a preferred mechanism.
  3. Execute the mathematical or computational witness on the full declared domain, including implicit auxiliary, global, boundary, and state response.
  4. Run the destructive control below and at least one omitted-rival or wrong-scope control.
  5. Publish every packet named below, with evidence hashes, uncertainty, scope, and explicit OPEN residuals.
  6. Apply anti-promotion: no local, static, one-loop, one-state, one-branch, one-regulator, finite-volume, or measured-anchor result may be promoted beyond its certificate.

Required packets

  • dependency_invalidation_packet
  • source_manifest_packet

Minimal discharge

Any change in Shape, variational status, boundary support, Scale, Dynamics, regulator, or vacuum branch invalidates every dependent background, ledger, matching, and history certificate.

Fail trigger

An upstream change leaves stale vacuum certificates marked PASS.

Destructive thought experiment

Change the Stage while retaining the old vacuum certificate hash.

Allowed row states

PASS, FAIL, OPEN, NOT-EVALUATED, or NOT-APPLICABLE with a gate-specific type proof. A definition, historical status label, or copied terminal is never a witness.

VAC-C26 — Regional, interface, boundary, edge, junction, and pressure/flux balance

Origin: all-gate constraint/thought-experiment derivation
Owned question: What must be true for this Vacuum obligation to be discharged without substituting another block's result?

Normative requirement

Resolve vacuum stress and pressure on every bulk region, fixed set, interface, wall, horizon, edge, and corner; include junction equations, surface tensions, Casimir forces, edge modes, and energy/entropy flux balance.

Required execution

  1. Load the frozen gate contract, authority hashes, Stage/Rulebook/Actor branch, Observer record family, Scale tuple, Boundary supports, and time/history scope.
  2. Generate the complete candidate/state/branch/test roster relevant to this row before selecting a preferred mechanism.
  3. Execute the mathematical or computational witness on the full declared domain, including implicit auxiliary, global, boundary, and state response.
  4. Run the destructive control below and at least one omitted-rival or wrong-scope control.
  5. Publish every packet named below, with evidence hashes, uncertainty, scope, and explicit OPEN residuals.
  6. Apply anti-promotion: no local, static, one-loop, one-state, one-branch, one-regulator, finite-volume, or measured-anchor result may be promoted beyond its certificate.

Required packets

  • regional_interface_packet

Minimal discharge

Resolve vacuum stress and pressure on every bulk region, fixed set, interface, wall, horizon, edge, and corner; include junction equations, surface tensions, Casimir forces, edge modes, and energy/entropy flux balance.

Fail trigger

The integrated bulk ledger cancels while a boundary pressure, junction condition, or regional flux remains nonzero.

Destructive thought experiment

Two bulks have equal total vacuum energy; one has an uncancelled interface pressure. Global equality must not imply local consistency.

Allowed row states

PASS, FAIL, OPEN, NOT-EVALUATED, or NOT-APPLICABLE with a gate-specific type proof. A definition, historical status label, or copied terminal is never a witness.

VAC-C27 — Causal initial-boundary evolution, constraint propagation, and total conservation

Origin: all-gate constraint/thought-experiment derivation
Owned question: What must be true for this Vacuum obligation to be discharged without substituting another block's result?

Normative requirement

Prove a well-posed causal initial-boundary problem through the vacuum history; propagate constraints and gauge conditions; include protecting-sector energy, nonlocal memory, junction fluxes, and total conservation.

Required execution

  1. Load the frozen gate contract, authority hashes, Stage/Rulebook/Actor branch, Observer record family, Scale tuple, Boundary supports, and time/history scope.
  2. Generate the complete candidate/state/branch/test roster relevant to this row before selecting a preferred mechanism.
  3. Execute the mathematical or computational witness on the full declared domain, including implicit auxiliary, global, boundary, and state response.
  4. Run the destructive control below and at least one omitted-rival or wrong-scope control.
  5. Publish every packet named below, with evidence hashes, uncertainty, scope, and explicit OPEN residuals.
  6. Apply anti-promotion: no local, static, one-loop, one-state, one-branch, one-regulator, finite-volume, or measured-anchor result may be promoted beyond its certificate.

Required packets

  • causal_conservation_packet

Minimal discharge

Prove a well-posed causal initial-boundary problem through the vacuum history; propagate constraints and gauge conditions; include protecting-sector energy, nonlocal memory, junction fluxes, and total conservation.

Fail trigger

The static equations are consistent but transition evolution is ill-posed, acausal, or leaks energy into an unowned sector.

Destructive thought experiment

Two mechanisms pass static algebra; one develops a singular multiplier and loses hyperbolicity during a shift.

Allowed row states

PASS, FAIL, OPEN, NOT-EVALUATED, or NOT-APPLICABLE with a gate-specific type proof. A definition, historical status label, or copied terminal is never a witness.

Appendix L

Original building-block speed audit

SOURCE-DERIVED AUDIT

Building-Block Speed Audit for the Five-Tensor GR Architecture

Date: August 6, 2026
Status: Source-grounded architecture audit plus controlled toy benchmark
Companion documents: MaxConservationStack.html, FiveTensorGRArchitecture.html

Executive conclusion

The uploaded building blocks provide several strong clues for additional acceleration. The most important conclusion is that the next improvement should not be three or four more top-level physical tensors.

The five physical ownership sectors remain well motivated:

The building blocks instead describe a compiler around those tensors:

The largest new clue comes from the Observer block. Our existing toy benchmark reconstructs the complete 900-component field even when the requested output could be only a handful of waveform or detector records. The Observer kernel/image and sensitivity requirements imply an adjoint or goal-oriented compilation:

After is compiled, the online calculation is

rather than

A controlled batched linear-algebra test found a 6,622× assembly speedup for the 4D observer-compiled map and 11,977× for the 13D-informed map relative to a reused-factor direct field solve. Those numbers exclude coefficient-generation and offline compilation costs; they demonstrate an algebraic ceiling, not an end-to-end physical speedup. In a sequential Python test that included coefficient evaluation and dispatch, the corresponding speedups were 37.7× and 57.2×. The gap shows exactly what must be optimized next: vectorized or compiled master-function evaluation.

1. Source-derived acceleration mechanisms

1.1 Rigidity quotient: eliminate variables before solving

Source: rigidity/01_CORE/BB_RIG_4_1_MAX_RIGOR_ALL_GATE_RIGIDITY.md

The Rigidity block constructs

where is the independently justified constraint Jacobian and is the gauge image.

This should become a compile-time projector

Speed implication

Never place rigid, gauge, reaction-normal, or independently constrained directions into:

  • the PDE state vector;
  • the five-tensor response basis;
  • the training family;
  • the SVD;
  • the correction-rank search.

This reduces the field dimension before any solve. It is stronger than discovering low singular values after producing the data.

Required implementation

  1. Build the full deformation basis.
  2. Calculate and .
  3. Compute a certified nullspace basis.
  4. Transform every five-tensor source and response into that basis.
  5. Reopen only when a constraint rank changes.

1.2 Schur-complement condensation: integrate out expensive coupled sectors

Sources:

  • interdependence/01_CORE/BB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md, INT-C06 and INT-C11;
  • rigidity/01_CORE/BB_RIG_4_1_MAX_RIGOR_ALL_GATE_RIGIDITY.md, mixed Hessian and coercivity;
  • dynamics/01_CORE/BB_DYN_4_2_MAX_RIGOR_FULL_GATE_CLOSURE_DYNAMICS.md, block-separable reaction case.

For a mixed operator

where is retained and is heavy, internal, boundary, gauge-normal, or reaction-owned,

gives

The retained equation is

Speed implication

The expensive sector is solved or factorized once. Its influence is retained through a response operator rather than by evolving every component.

This can reduce both:

  • offline cost, by shrinking each basis solve;
  • online cost, by reducing the retained state and tensor ranks.

Important restriction

The blocks require a controlled error, domain, matching map, generated-term ledger, and memory/nonlocality ledger. A naive diagonal truncation is not equivalent to a Schur complement.

1.3 Symmetry, automorphism, and theorem-zero blocks

Sources:

  • shape/01_CORE/01_STAGE/stage.md, S-14 and S-17;
  • shape/01_CORE/02_RULEBOOK/rulebook.md, R-22;
  • dynamics/01_CORE/BB_DYN_4_2_MAX_RIGOR_FULL_GATE_CLOSURE_DYNAMICS.md, DYN-C10.

The Stage freezes a block-product metric and a restricted automorphism group. The Rulebook supplies orthogonal sector projectors. Dynamics requires every interaction hyperedge to be either parent-owned or theorem-zero.

The operator should be decomposed into irreducible sectors:

with

when the frozen symmetry permits it.

Speed implication

  • Solve one representative per symmetry orbit.
  • Store one response per irrep rather than per raw component.
  • Delete exactly forbidden couplings before training.
  • Avoid SVD effort on blocks proven to be zero.
  • Use Kronecker or tensor-product structure where the product geometry truly survives the quotient and boundaries.

This is likely the cleanest way for the internal geometry to lower the ranks of all five tensors.

1.4 Observer-first adjoint compilation

Source: observer/01_CORE/BB_OBS_4_0_MAX_RIGOR_ALL_GATE_OBSERVER.md, OBS-C06 through OBS-C08.

The Observer block requires the kernel, image, degeneracies, inaccessible sectors, and deformation Jacobian of the record map.

Let

be the frozen waveform, flux, detector, or mission-record map.

For a linearized observable, solve the adjoint problem

Then

without reconstructing the bulk field.

For a cached five-tensor basis ,

is compiled once, and online work becomes a small matrix multiplication.

Speed implication

This can be much larger than the original 10–50× speedup because it removes the factor from online reconstruction.

It is valid only for the declared observable family. A new observable outside the compiled image requires a new map or a residual bound.

1.5 Scale windows, spectral gaps, and tower matching

Sources:

  • scale/01_CORE/BB_SCL_4_1_MAX_RIGOR_ALL_GATE_SCALE.md, SCL-C06, C17, C18, C23;
  • rigidity/01_CORE/BB_RIG_4_1_MAX_RIGOR_ALL_GATE_RIGIDITY.md, RIG-C11 and RIG-C22.

The Scale and Rigidity blocks require quantitative gaps, validity windows, matching maps, and tail certificates.

For omitted modes with eigenvalues ,

A nested effective description should use different compiled solvers on different scale intervals:

Speed implication

  • Do not use merger-resolution modes in early inspiral.
  • Do not use high internal towers below their certified thresholds.
  • Do not keep one global basis across a spectral crossing.
  • Compile a small solver per scale window and match them.

This complements piecewise relinearization: update the background and active mode shelf when the scale packet changes.

1.6 Boundary transfer maps and regional gluing

Sources:

  • boundary/01_CORE/BB_BND_4_1_MAX_RIGOR_FULL_GATE_CLOSURE_BOUNDARY.md, BND-C03, C17, C18;
  • interdependence/01_CORE/BB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md, INT-C10.

Boundary requires lawful domains, regional algebras, edge completion, causal data, and gluing.

For regional subdomains, eliminate the interior and retain an interface map:

This is a Dirichlet-to-Neumann or boundary Schur map.

Speed implication

  • Precompute horizon and outer-boundary response operators.
  • Reuse regional solvers across many source parameters.
  • Update only regions whose local source or geometry changed.
  • Glue through low-dimensional interfaces.

The edge/factorization ledger prevents a false speedup from deleting gauge or gravitational edge information.

1.7 Multi-rate synchronized patch evolution

Source: time-synchronization/01_CORE/BB_TS_4_0_MAX_RIGOR_ALL_GATE_TIME_SYNCHRONIZATION.md, TS-C06, C12, C14, C19, and C22.

The Time Synchronization block requires path-dependent delay, moving-domain transport, loop closure, and refinement consistency.

Speed implication

Use separate clocks and update rates:

  • orbital secular masters at a slow rate;
  • local near-secondary response at an intermediate rate;
  • fast ringdown or boundary flux only where active;
  • frame transport when its interpolation error reaches threshold.

A lawful multi-rate scheme needs synchronization residuals and loop/holonomy checks. It should not globally step every tensor at the fastest local timescale.

1.8 Dependency DAG and invalidation radius

Sources:

  • interdependence/01_CORE/BB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md, INT-C03 and INT-C13;
  • governance/01_CORE/BB_CSE_4_0_SUPPLEMENTAL_MAX_RIGOR_GATE_GOVERNANCE.md, CSE-C11, C18, and C27;
  • vacuum/01_CORE/BB_VAC_4_0_MAX_RIGOR_ALL_GATE_VACUUM.md, VAC-C10.

This mechanism does not primarily accelerate one waveform evaluation. It accelerates development, retraining, and certification.

Each compiled artifact should be a dependency node:

When an input changes, recompute only its descendants.

Speed implication

This prevents rebuilding the full model after:

  • a boundary change;
  • a new observable;
  • a scale-window change;
  • a frame change;
  • a new internal coupling;
  • a local tensor-rank adjustment.

2. Recommended compiled architecture

The building blocks suggest the following offline compiler:

Interpretation:

  • : optional internal coupling algebra;
  • : five-tensor source basis;
  • : Schur-condensed physical operator;
  • : gauge/constraint/rigidity quotient;
  • : common-frame transport;
  • : observer-image quotient;
  • : requested record map.

The online solver is then

The full field is reconstructed only for diagnostics or when the requested observable is outside the compiled image.

3. Controlled observer-compiler test

The existing 900-state toy was extended from full-field output to eight frozen observer records.

Batched linear-algebra-only result

Method Time/query Speedup over direct
Reused-factor direct field solve, then records 67.395 μs 1×
Full 4D five-tensor field, then records 5.140 μs 13.11×
Observer-compiled 4D 0.01018 μs 6,621.85×
Observer-compiled 13D-informed 0.00563 μs 11,977.07×

The observer-compiled results agreed with direct evaluation at approximately relative error.

End-to-end sequential Python result

When coefficient/master evaluation and Python dispatch were included:

Method Time/query Speedup over direct
Direct 785.37 μs 1×
Full-field 4D five-tensor 27.24 μs 28.83×
Observer-compiled 4D 20.84 μs 37.69×
Observer-compiled 13D-informed 13.73 μs 57.19×

This is the honest engineering lesson:

Once the field solve is removed, coefficient generation becomes the bottleneck.

The next speed project should therefore compile the master functions into vectorized/JIT kernels, not only optimize field algebra.

4. Priority order

Priority 1 — Observer compiler

Expected benefit: largest online reduction for fixed waveform or detector records.

Priority 2 — Physical Schur compiler

Expected benefit: largest reduction in offline response solves and coupled-sector evolution.

Priority 3 — Symmetry/orbit compiler

Expected benefit: exact block diagonalization and theorem-zero channel removal.

Priority 4 — Vectorized master-function engine

Expected benefit: realizes the algebraic observer-compiler speed ceiling.

Priority 5 — Multi-scale and multi-rate patch engine

Expected benefit: reduces time steps and active modes across inspiral, plunge, and ringdown.

Priority 6 — Dependency-DAG incremental rebuild

Expected benefit: substantially reduces retraining and certification work.

5. Strongest conclusion

The building blocks suggest that the architecture should retain five physical tensors but add five compilation layers:

The likely next large gain is not obtained by approximating the spacetime field more aggressively. It is obtained by proving that the requested NASA/JPL records factor through a very small image of the already reduced five-tensor solution.

The physical claim remains open until the q=64/q=100/q=128 waveform replay is executed.

Appendix M — full source preservation

Complete Observer V4.0 authority

COMPLETE VERBATIM AUTHORITY

title: "BB-OBS-4.0 — Maximum-Rigor All-Gate Observer, Record, Measurement, and Inference Authority" building_block_id: "BB-OBS-4.0" version: "4.0" date: "2026-08-02" status: "DEVELOPMENT AUTHORITY — ALL-GATE DERIVED — LEGACY PRESERVED — NOT FROZEN" scope: "Preparations, accessible algebras, parent-to-record maps, quantum instruments, apparatus response, relational frames, finite records, inference, uncertainty, selection, information bounds, and test-family exhaustion" protocol_freeze: false canonical_ratification: false legacy_authority: "BB-OBS-1.0.1" legacy_sha256: "4ca861c5611c67a8535649f6aff6671b3f591574bf6def6706de7d4a6d9a6ba7"

BB-OBS-4.0 — Maximum-Rigor All-Gate Observer, Record, Measurement, and Inference Authority

0. Purpose and claim ceiling

Observer owns the complete typed interface between a physical parent object and a finite, reproducible record used to test it. It does not mean consciousness and it does not supply ontology, dynamics, or experimental data by declaration.

The central firewalls are:

and

This block is capability-complete only for Observer-owned obligations in the current 30-gate corpus. It cannot close a full gate without all other applicable building blocks and candidate-specific evidence.

1. Preservation contract

The exact byte sequence of BB-OBS-1.0.1 is preserved in 06_SOURCES/BB_OBS_1_0_1_EXACT_ARCHIVAL_SOURCE.md and Appendix A. Its SHA-256 is 4ca861c5611c67a8535649f6aff6671b3f591574bf6def6706de7d4a6d9a6ba7. Every legacy heading is mapped to active V4.0 constraints and packets in LEGACY_PRESERVATION_MATRIX_V4_0.json.

No legacy definition, firewall, certificate field, negative control, fail trigger, status, or anti-promotion rule is weakened. If a conflict is discovered, the stronger non-promotional statement controls and the package returns RESTART-REQUIRED.

2. Complete Observer object

Define

where the original components are retained and extended by interventions , quantum/classical measurement instruments , apparatus/detector response , temporal/causal record structure , selection/inference structure , and test-family exhaustion certificate .

A gate execution must type every component, its support, owner, domain, Scale, Granularity, Boundary conditions, Dynamics, and provenance.

3. Cross-block interfaces

  • Shape supplies supports, Actors, Co-Actors, owner IDs, representations, and lawful record carriers.
  • Granularity supplies exact and operational equivalence, finite-resource record cells, resolution/refinement, and test-family saturation requirements.
  • Dynamics supplies preparations, interventions, response functions, channels, histories, conservation, and causal evolution.
  • Scale supplies the ruler tuple, calibration hierarchy, matching, uncertainties, and state-dependent scales.
  • Boundary supplies regional algebras, apparatus/support domains, horizons, interfaces, edge sectors, and gluing.
  • Rigidity supplies sensitivity of records to physical and observer-map deformations.
  • Vacuum supplies background/state/phase typing and cosmological histories.
  • Time Synchronization supplies clocks, synchronization, causal ordering, histories, and loop controls.
  • Interdependence supplies common-parent hashes and stale-propagation.

Observer never repairs a missing object owned by another block. It records the dependency as OPEN.

4. Observer compilation pipeline

For each gate:

  1. Freeze authorities, branch, gate scope, preparations, interventions, supports, clocks, apparatuses, Scale, Granularity, and claim ceiling.
  2. Generate the complete lawful test family from Shape, Actors/Co-Actors, Dynamics, Boundary, and gate requirements.
  3. Construct the parent-to-physical-observable map, including gauge/constraint reduction and relational dressing.
  4. Compose with state preparation, evolution, quantum/classical instrument, apparatus transfer, sampling/windowing, selection, and record encoding.
  5. Propagate normalization, covariance, calibration, nuisance parameters, and finite-resource distinguishability.
  6. Enumerate kernel, image, aliases, inaccessible sectors, degeneracies, and omitted tests.
  7. Execute destructive controls and blinded/robustness tests.
  8. Populate every evidence packet; use NOT-APPLICABLE only with a type proof.
  9. Set the terminal no stronger than the weakest selected row.

The full map is schematically

Every arrow requires a domain, codomain, owner, normalization, uncertainty, support, and hash.

5. Atomic Observer constraints

OBS-C01 — Observer-system manifest

Origin: preserved BB-OBS-1.0.1.

Requirement. Freeze the observer, apparatus/reference frame, accessible algebra, preparation class, record space, energy/time window, support, and scope.

Minimal discharge. Freeze the observer, apparatus/reference frame, accessible algebra, preparation class, record space, energy/time window, support, and scope.

Fail trigger. The word observer is used without specifying what can be prepared or recorded.

Adversarial thought experiment. Leave the apparatus and accessible algebra unspecified while using the word observer.

Required packet(s). observer_system_manifest.

OBS-C02 — Executable parent-to-record map

Origin: preserved BB-OBS-1.0.1.

Requirement. For every claimed observable, provide an equation or script mapping the complete parent state to the record with all reductions, projections, owner IDs, and transports shown.

Minimal discharge. For every claimed observable, provide an equation or script mapping the complete parent state to the record with all reductions, projections, owner IDs, and transports shown.

Fail trigger. A comparison uses an informal identification, omits owner provenance, or counts aliases as separate records.

Adversarial thought experiment. Replace an executable reduction by “identify with the measured value.”

Required packet(s). parent_to_record_packet.

OBS-C03 — Normalization, positivity, probability, conservation, and gauge-equivalence preservation

Origin: preserved BB-OBS-1.0.1.

Requirement. Prove that the map preserves the required norm/probability, positivity, conservation, and gauge equivalence on the physical domain, including all Jacobians and multiplicities.

Minimal discharge. Prove that the map preserves the required norm/probability, positivity, conservation, and gauge equivalence on the physical domain, including all Jacobians and multiplicities.

Fail trigger. Projection changes norm, probability, conservation, or gauge class without an accounted factor.

Adversarial thought experiment. Omit a projection Jacobian and check whether probability still sums to one.

Required packet(s). normalization_physicality_packet.

OBS-C04 — Frozen same-ruler tuple and lawful transport

Origin: preserved BB-OBS-1.0.1.

Requirement. Align dimension, frame, renormalization scale, scheme, bundle norm, projection, truncation, regulator, and observable definition, or provide an explicit transport theorem.

Minimal discharge. Align dimension, frame, renormalization scale, scheme, bundle norm, projection, truncation, regulator, and observable definition, or provide an explicit transport theorem.

Fail trigger. A raw parent quantity is compared with a differently normalized or differently scaled record.

Adversarial thought experiment. Compare a 13D coefficient directly with a one-chamber 4D running observable.

Required packet(s). ruler_transport_packet.

OBS-C05 — Calibration lineage and leakage firewall

Origin: preserved BB-OBS-1.0.1.

Requirement. Tag every datum as preparation, measured anchor, calibration, blind target, derived output, or diagnostic; prove blind targets do not select the map, basis, normalization, tolerance, or uncertainty model.

Minimal discharge. Tag every datum as preparation, measured anchor, calibration, blind target, derived output, or diagnostic; prove blind targets do not select the map, basis, normalization, tolerance, or uncertainty model.

Fail trigger. A target observable determines a projection coefficient or convention after comparison.

Adversarial thought experiment. Choose a normalization coefficient after seeing the blind target.

Required packet(s). calibration_lineage_packet.

OBS-C06 — Kernel, image, degeneracy, inaccessible-sector, and information-loss inventory

Origin: preserved BB-OBS-1.0.1.

Requirement. Enumerate the map kernel, image, degeneracies, inaccessible sectors, indirect observables, above-cutoff directions, and unresolved hidden sectors through the claimed scope.

Minimal discharge. Enumerate the map kernel, image, degeneracies, inaccessible sectors, indirect observables, above-cutoff directions, and unresolved hidden sectors through the claimed scope.

Fail trigger. A null direction or inaccessible physical sector is silently treated as nonexistent.

Adversarial thought experiment. Delete an inaccessible physical direction and call it absent.

Required packet(s). kernel_image_packet.

OBS-C07 — Observer-map deformation Jacobian and sensitivity

Origin: preserved BB-OBS-1.0.1.

Requirement. Differentiate records with respect to frame, scale, scheme, basis, chamber, detector window, boundary accessibility, normalization, truncation, regulator, and coarse-graining parameters.

Minimal discharge. Differentiate records with respect to frame, scale, scheme, basis, chamber, detector window, boundary accessibility, normalization, truncation, regulator, and coarse-graining parameters.

Fail trigger. A small admissible observer-map change can move a result across the pass band but is not counted.

Adversarial thought experiment. Perturb the observer map slightly across the pass band without counting it.

Required packet(s). observer_deformation_packet.

OBS-C08 — Full uncertainty, covariance, scheme/frame transport, and truncation error

Origin: preserved BB-OBS-1.0.1.

Requirement. Propagate correlated input, truncation, scheme, observer, calibration, and model uncertainties through the complete map.

Minimal discharge. Propagate correlated input, truncation, scheme, observer, calibration, and model uncertainties through the complete map.

Fail trigger. Only central values or independent errors are compared.

Adversarial thought experiment. Drop correlations and compare central values only.

Required packet(s). uncertainty_covariance_packet.

OBS-C09 — Wrong-ruler, wrong-owner, wrong-projection, and leakage controls

Origin: preserved BB-OBS-1.0.1.

Requirement. Run the canonical wrong-chamber, wrong-dimension, wrong-scale, wrong-frame, double-normalization, hidden-calibration, nullspace-deletion, scheme-only, quotient-isometry, duplicate-owner, and parity controls.

Minimal discharge. Run the canonical wrong-chamber, wrong-dimension, wrong-scale, wrong-frame, double-normalization, hidden-calibration, nullspace-deletion, scheme-only, quotient-isometry, duplicate-owner, and parity controls.

Fail trigger. The block cannot reject a known wrong-ruler, duplicate-owner, or target-leaking construction.

Adversarial thought experiment. Run the wrong-chamber, duplicate-owner, and parity controls.

Required packet(s). legacy_negative_controls_packet.

OBS-C10 — Content-addressed reproducible observer-map build

Origin: preserved BB-OBS-1.0.1.

Requirement. Frozen inputs must regenerate record CSV/JSON outputs, plots, hashes, and verdicts without undocumented author interpretation.

Minimal discharge. Frozen inputs must regenerate record CSV/JSON outputs, plots, hashes, and verdicts without undocumented author interpretation.

Fail trigger. The observed prediction depends on a manual conversion or cannot be regenerated from the manifest.

Adversarial thought experiment. Rebuild the record with one frozen input changed and demand stale invalidation.

Required packet(s). content_addressed_build_packet.

OBS-C11 — Preparation, intervention, and counterfactual protocol closure

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Specify admissible preparation maps, interventions, controls, counterfactual comparisons, initial-condition selection, and their physical resource/support domains; distinguish passive observation from active intervention.

Minimal discharge. Specify admissible preparation maps, interventions, controls, counterfactual comparisons, initial-condition selection, and their physical resource/support domains; distinguish passive observation from active intervention.

Fail trigger. A claimed observable changes when the preparation or intervention protocol changes, but the protocol is not part of the observer object.

Adversarial thought experiment. Two experiments use the same detector and observable label but different preparations; the record distributions differ. A detector-only manifest must fail.

Required packet(s). preparation_intervention_packet.

OBS-C12 — Quantum instruments, POVMs, channels, outcome algebras, and sequential composition

Origin: all-gate constraint/thought-experiment derivation.

Requirement. For quantum records, specify the state space, physical quotient, POVM or instrument, quantum channel, outcome algebra, normalization, complete positivity, trace preservation/nonincrease, repeatability scope, and composition law.

Minimal discharge. For quantum records, specify the state space, physical quotient, POVM or instrument, quantum channel, outcome algebra, normalization, complete positivity, trace preservation/nonincrease, repeatability scope, and composition law.

Fail trigger. A Born probability or collapse/update rule is asserted without a normalized instrument on the physical state space.

Adversarial thought experiment. Two POVMs share the same expectation for one state but differ on another; a single expectation value cannot define the measurement.

Required packet(s). quantum_instrument_packet.

OBS-C13 — Temporal records, clocks, synchronization, causal ordering, and histories

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Freeze clocks, time standards, synchronization maps, causal order, time windows, latency, memory, histories, and composition of records across spacelike/timelike/null separated supports; interface explicitly with Time Synchronization.

Minimal discharge. Freeze clocks, time standards, synchronization maps, causal order, time windows, latency, memory, histories, and composition of records across spacelike/timelike/null separated supports; interface explicitly with Time Synchronization.

Fail trigger. Records from different clocks or causal orders are combined without a synchronization/ordering theorem.

Adversarial thought experiment. Two detectors report identical timestamps in their local frames but disagree after lawful transport; an unsynchronized coincidence claim must fail.

Required packet(s). temporal_record_packet.

OBS-C14 — Regional, localized, boundary, horizon, interface, and asymptotic observer supports

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Type the observer support and accessible algebra for bulk regions, boundaries, corners, horizons, entangling cuts, null infinity, finite-radius screens, and state-dependent/moving supports; include gluing and flux interfaces.

Minimal discharge. Type the observer support and accessible algebra for bulk regions, boundaries, corners, horizons, entangling cuts, null infinity, finite-radius screens, and state-dependent/moving supports; include gluing and flux interfaces.

Fail trigger. A global record is inferred from a regional algebra without proving completeness or accounting for edge/horizon sectors.

Adversarial thought experiment. Two global states restrict identically to one exterior region but differ behind a horizon; regional equivalence must not become global identity.

Required packet(s). regional_support_packet.

OBS-C15 — Detector and apparatus response, transfer functions, efficiency, nonlinearity, saturation, and dead time

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Model the apparatus transfer function, acceptance, efficiency, gain, noise, thresholds, point-spread/time-response functions, nonlinearities, saturation, pile-up, dead time, backgrounds, and calibration drift.

Minimal discharge. Model the apparatus transfer function, acceptance, efficiency, gain, noise, thresholds, point-spread/time-response functions, nonlinearities, saturation, pile-up, dead time, backgrounds, and calibration drift.

Fail trigger. A theoretical flux or amplitude is identified directly with a detector count without a response model.

Adversarial thought experiment. Two spectra yield the same total count but different binned counts after detector response; total-rate matching must not certify equivalence.

Required packet(s). detector_response_packet.

OBS-C16 — Finite resolution, sampling, bandwidth, windowing, coarse-graining, and aliasing

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Publish spatial, temporal, spectral, angular, and amplitude resolution; sampling theorem conditions; windows; binning; coarse-graining; bandwidth; aliasing and leakage tests; and refinement behavior.

Minimal discharge. Publish spatial, temporal, spectral, angular, and amplitude resolution; sampling theorem conditions; windows; binning; coarse-graining; bandwidth; aliasing and leakage tests; and refinement behavior.

Fail trigger. A sub-Nyquist, under-resolved, or window-dependent feature is treated as absent or physical without an aliasing/refinement test.

Adversarial thought experiment. A high-frequency mode aliases to a low-frequency record under sparse sampling; the observer must detect the ambiguity.

Required packet(s). resolution_sampling_packet.

OBS-C17 — Inverse problems, tomography, reconstruction, identifiability, and degeneracy

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Define the forward operator and inverse/reconstruction method; compute rank, nullspace, Fisher/observability information, regularization dependence, degeneracies, identifiability, and uncertainty of reconstructed parent quantities.

Minimal discharge. Define the forward operator and inverse/reconstruction method; compute rank, nullspace, Fisher/observability information, regularization dependence, degeneracies, identifiability, and uncertainty of reconstructed parent quantities.

Fail trigger. A reconstructed parameter is reported as unique when multiple parent states produce the same record within scope.

Adversarial thought experiment. Two parameter combinations map to identical records; a point estimate without a degeneracy manifold must fail.

Required packet(s). inverse_identifiability_packet.

OBS-C18 — Selection effects, acceptance, missingness, censoring, survivorship, and look-elsewhere control

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Model selection functions, triggers, acceptance, missing-data mechanisms, censoring, survivorship, search windows, trial factors, post-selection, and publication/blinding rules.

Minimal discharge. Model selection functions, triggers, acceptance, missing-data mechanisms, censoring, survivorship, search windows, trial factors, post-selection, and publication/blinding rules.

Fail trigger. A selected sample is treated as representative without a selection-function or missingness model.

Adversarial thought experiment. A signal appears only after scanning many windows; the local significance must not be reported as global significance.

Required packet(s). selection_effects_packet.

OBS-C19 — Likelihood, hypothesis family, model comparison, test statistics, and decision rules

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Freeze the likelihood or scoring rule, nuisance parameters, priors where used, hypothesis family, test statistic, loss/decision rule, multiple-testing correction, goodness-of-fit, calibration, and claim threshold before unblinding.

Minimal discharge. Freeze the likelihood or scoring rule, nuisance parameters, priors where used, hypothesis family, test statistic, loss/decision rule, multiple-testing correction, goodness-of-fit, calibration, and claim threshold before unblinding.

Fail trigger. A pass/fail threshold or model-comparison rule is chosen after viewing the target data.

Adversarial thought experiment. Two models fit the central value equally but predict different covariance; a central-value-only decision must fail.

Required packet(s). likelihood_decision_packet.

OBS-C20 — Open-system measurement, decoherence, environment, pointer records, and channel consistency

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Specify system-environment split, influence/channel dynamics, decoherence functional, pointer/record algebra, noise/dissipation, initial correlations, non-Markovian memory, and complete-positivity/domain conditions.

Minimal discharge. Specify system-environment split, influence/channel dynamics, decoherence functional, pointer/record algebra, noise/dissipation, initial correlations, non-Markovian memory, and complete-positivity/domain conditions.

Fail trigger. A classical record or decoherence claim is asserted without a lawful environment/channel model.

Adversarial thought experiment. Two reduced density matrices match instantaneously but have different environment correlations and future records; state-only matching must fail.

Required packet(s). open_system_measurement_packet.

OBS-C21 — Relational frames, clocks, rulers, diffeomorphism/gauge-invariant dressing, and reference-system backreaction

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Construct relational or dressed observables; identify physical reference fields/apparatus; transport between frames; include reference-system uncertainty, energy, dynamics, and backreaction; distinguish coordinate labels from records.

Minimal discharge. Construct relational or dressed observables; identify physical reference fields/apparatus; transport between frames; include reference-system uncertainty, energy, dynamics, and backreaction; distinguish coordinate labels from records.

Fail trigger. A coordinate-dependent or gauge-variant quantity is called observable without dressing or a physical reference construction.

Adversarial thought experiment. A coordinate location shifts under a diffeomorphism while a relational distance does not; the coordinate value must not be used as the record.

Required packet(s). relational_frame_packet.

OBS-C22 — Measurement backreaction, disturbance, resource costs, and no-free-record accounting

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Quantify apparatus coupling, disturbance, energy/time/entropy/memory costs, recoil, heating, radiation, finite reference resources, weak/strong measurement regimes, and whether the claimed record changes the system.

Minimal discharge. Quantify apparatus coupling, disturbance, energy/time/entropy/memory costs, recoil, heating, radiation, finite reference resources, weak/strong measurement regimes, and whether the claimed record changes the system.

Fail trigger. An arbitrarily precise or repeated record is assumed without accounting for disturbance or finite resources.

Adversarial thought experiment. Two measurement strengths yield the same mean but different post-measurement dynamics; the instrument must include backreaction.

Required packet(s). measurement_backreaction_packet.

OBS-C23 — Composite, multiparticle, scattering, decay, event, coincidence, and flux record construction

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Build records for composite Actors, bound states, jets, decays, scattering amplitudes, inclusive/exclusive channels, coincidences, fluxes, cross sections, branching ratios, event topology, and unresolved sums with owner provenance.

Minimal discharge. Build records for composite Actors, bound states, jets, decays, scattering amplitudes, inclusive/exclusive channels, coincidences, fluxes, cross sections, branching ratios, event topology, and unresolved sums with owner provenance.

Fail trigger. A constituent-level quantity is compared directly with a composite/event-level observable without hadronization, decay, acceptance, or inclusive-sum maps.

Adversarial thought experiment. Two theories have the same total rate but different event topology; a rate-only observer map must not merge them.

Required packet(s). composite_event_packet.

OBS-C24 — Quantum-information, entropy, entanglement, modular, and information-flow records

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Specify subsystem algebra/factorization or algebraic substitute, regulator/counterterm scheme, edge modes, entropy/relative entropy/modular observables, channel capacities, information flux, and operational access.

Minimal discharge. Specify subsystem algebra/factorization or algebraic substitute, regulator/counterterm scheme, edge modes, entropy/relative entropy/modular observables, channel capacities, information flux, and operational access.

Fail trigger. Entanglement entropy or Page information is quoted without a subsystem algebra, regulator, edge prescription, or record protocol.

Adversarial thought experiment. Two states have equal energy but different relative entropy in an accessible region; energy-only equivalence must fail.

Required packet(s). quantum_information_packet.

OBS-C25 — Cosmological and light-cone inference, transfer functions, redshift, lensing, and survey windows

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Map parent histories to past-light-cone records through transfer functions, redshift, lensing, selection, foregrounds, survey masks/windows, cosmic variance, nonlinear evolution, and covariance; distinguish initial, evolved, and observed fields.

Minimal discharge. Map parent histories to past-light-cone records through transfer functions, redshift, lensing, selection, foregrounds, survey masks/windows, cosmic variance, nonlinear evolution, and covariance; distinguish initial, evolved, and observed fields.

Fail trigger. A primordial or background quantity is compared directly with a present survey/CMB record without transfer and window functions.

Adversarial thought experiment. Two primordial spectra become distinguishable only after different transfer functions; direct primordial-to-data comparison must fail.

Required packet(s). cosmological_inference_packet.

OBS-C26 — Cross-observer consistency, contextuality, no-signaling, composition, and shared-record gluing

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Provide compatibility maps between observers, contexts, frames, regions, and apparatuses; test no-signaling, marginal consistency, contextuality/Bell scope, record gluing, and when no global joint distribution exists.

Minimal discharge. Provide compatibility maps between observers, contexts, frames, regions, and apparatuses; test no-signaling, marginal consistency, contextuality/Bell scope, record gluing, and when no global joint distribution exists.

Fail trigger. Records from incompatible contexts are combined as if they belonged to one joint sample space.

Adversarial thought experiment. Pairwise marginals are individually valid but admit no global joint distribution; naive gluing must fail.

Required packet(s). cross_observer_consistency_packet.

OBS-C27 — Robustness, blinded holdouts, distribution shift, adversarial controls, and out-of-domain scope

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Test observer maps on blinded holdouts, alternative calibrations, distribution shifts, apparatus drift, adversarial perturbations, synthetic injections, and out-of-domain cases; publish robustness margins and scope limits.

Minimal discharge. Test observer maps on blinded holdouts, alternative calibrations, distribution shifts, apparatus drift, adversarial perturbations, synthetic injections, and out-of-domain cases; publish robustness margins and scope limits.

Fail trigger. A map works only on the calibration sample or fails under small lawful apparatus/data shifts.

Adversarial thought experiment. A classifier/map passes training data but fails a blinded injected signal under a modest response drift; closure must remain open.

Required packet(s). robustness_holdout_packet.

OBS-C28 — Protocol provenance, versioning, stale invalidation, and reproducible workflow composition

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Version every preparation, apparatus, calibration, map, software, dataset, random seed, schema, and decision rule; propagate upstream hash changes; invalidate stale records and certificates; preserve machine-executable provenance.

Minimal discharge. Version every preparation, apparatus, calibration, map, software, dataset, random seed, schema, and decision rule; propagate upstream hash changes; invalidate stale records and certificates; preserve machine-executable provenance.

Fail trigger. A changed calibration, detector model, boundary support, or theory manifest leaves an old observer certificate marked PASS.

Adversarial thought experiment. Changing one calibration file alters the result while the certificate hash remains unchanged; the build must fail.

Required packet(s). protocol_provenance_packet.

OBS-C29 — Candidate-neutral observer/test-family grammar and saturation

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Generate the complete admissible family of preparations, observables, apparatus maps, regions, times, channels, resolutions, and decision rules from frozen authorities; canonicalize aliases; test omitted rivals; prove finite roster, bounded generator, or class theorem.

Minimal discharge. Generate the complete admissible family of preparations, observables, apparatus maps, regions, times, channels, resolutions, and decision rules from frozen authorities; canonicalize aliases; test omitted rivals; prove finite roster, bounded generator, or class theorem.

Fail trigger. Only favorable observables or tests are evaluated, or an untested channel is silently declared irrelevant.

Adversarial thought experiment. A candidate passes the selected observables but fails a lawful omitted observable generated by the same grammar; selection must not close the gate.

Required packet(s). test_family_exhaustion_packet.

OBS-C30 — Operational distinguishability, Fisher/quantum information, error bounds, and detection floors

Origin: all-gate constraint/thought-experiment derivation.

Requirement. Quantify distinguishability using appropriate total-variation/fidelity/relative-entropy/Fisher or quantum-Fisher information, Helstrom/Holevo/Cramér–Rao/error-exponent bounds, finite samples/resources, and declared error rates; connect to Granularity classes.

Minimal discharge. Quantify distinguishability using appropriate total-variation/fidelity/relative-entropy/Fisher or quantum-Fisher information, Helstrom/Holevo/Cramér–Rao/error-exponent bounds, finite samples/resources, and declared error rates; connect to Granularity classes.

Fail trigger. Two candidates are declared distinguishable or equivalent without a finite-resource decision/error bound.

Adversarial thought experiment. Two states are mathematically different but below the Helstrom/finite-sample detection floor; exact inequality must not be confused with operational distinguishability.

Required packet(s). operational_distinguishability_packet.

6. Universal destructive controls

At minimum, run all thirty thought experiments above, plus the legacy wrong-chamber, wrong-dimension, wrong-scale, wrong-frame, double-normalization, hidden-calibration, nullspace-deletion, scheme-only, quotient-isometry, duplicate-owner, and parity controls. A control must fail before correction and pass after the lawful map is restored.

7. Observer residual

Define

A full Observer PASS requires every selected component to vanish or have an accepted NOT-APPLICABLE type proof. Missing evidence is OPEN, never zero.

8. Integrated certificate

The canonical output is integrated_observer_certificate_v4_0. It contains the legacy certificate, row states, packet hashes, preservation residual, Observer residual, scope, claim ceiling, physical/project terminals, and reopen triggers. Historical gate labels are not evidence.

9. Status and anti-promotion rules

Use PASS, FAIL, OPEN, NOT-EVALUATED, NOT-APPLICABLE, CONSTRUCTION-ANCHOR, CONDITIONAL-ON-BND-A, CONDITIONAL-ON-BND-B, and RESTART-REQUIRED exactly as typed in the legacy authority.

Additional anti-promotions:

  • A theoretical amplitude is not a detector count.
  • A detector count is not a parent state without an inverse/identifiability certificate.
  • A local or regional record is not a global record.
  • A coordinate value is not a diffeomorphism-invariant observable.
  • A POVM element is not a complete instrument or state-update law.
  • Decoherence is not outcome selection.
  • A central-value match is not a likelihood/covariance pass.
  • Local significance is not global significance.
  • A calibration fit is not a blind prediction.
  • Exact mathematical inequality is not finite-resource distinguishability.
  • One observable is not a saturated test family.
  • One observer context is not a global joint record when contextuality forbids it.

10. Reopen triggers

Reopen when any preparation, intervention, branch, apparatus, response model, frame, clock, synchronization, support, boundary, scale, scheme, projection, normalization, regulator, truncation, sampling, selection, likelihood, calibration, dataset, software, random seed, test grammar, or decision rule changes; when a new lawful observable/test is generated; or when a hidden sector enters the accessible algebra.

11. Required package artifacts

An agent must read START_HERE.md, the controlling authority, the constraint registry, gate crosswalk, packet registry, schemas, prompt, and template. It must populate all packets, run validators and controls, and return OPEN for every missing witness.

12. All-gate profiles

Black-hole-singularity

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C20, OBS-C21, OBS-C22, OBS-C24, OBS-C26, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define relational/horizon/asymptotic records, regional algebras, causal history, information access, and distinguishability without promoting exterior records to global ontology.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Born-rule

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C07, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C12, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C23, OBS-C24, OBS-C25, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define noncircular quantum instruments, outcomes, probability normalization, preparation, decoherence, cross-observer consistency, and blind statistical tests.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-01

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C07, OBS-C08, OBS-C10, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C19, OBS-C21, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Transport higher-dimensional/operator coefficients to finite records with domains, normalization, detector/resolution effects, uncertainty, and tail scope.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-02

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C19, OBS-C20, OBS-C21, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define the physical spectral/mass-gap record, finite-volume and continuum observer limits, reconstruction, numerical/detection bounds, and external-wall claim ceiling.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-05-stability

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C07, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C25, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Map vacuum/background histories and perturbations to finite cosmological/local records with state, apparatus, selection, and causal consistency.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-05-value

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C10, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C25, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Separate measured residual calibration from a predicted value using lineage, likelihood, covariance, and identifiability.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-08

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C25, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Map parent cosmology to light-cone/CMB/structure records with transfer functions, selection, windows, and robustness.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-10/BG-10

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C23, OBS-C25, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define baryogenesis records through nonequilibrium history, particle/event observables, cosmological transfer, and inference.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-11

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C23, OBS-C25, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define dark-matter production and detection records across cosmological and laboratory channels, including selection and apparatus response.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Gap-13

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C20, OBS-C21, OBS-C22, OBS-C24, OBS-C26, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define regional/horizon/entropy/information records and prevent local-access claims from becoming global state claims.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Lambda-catastrophe

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C07, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C25, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Keep measured anchors, inferred background parameters, transition histories, and residual-value claims noncircular and covariance-complete.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-1

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C07, OBS-C08, OBS-C09, OBS-C10, OBS-C14, OBS-C17, OBS-C21, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Record the selected Shape without conflating coordinate/gauge presentation, calibration, and physical uniqueness.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-10

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Transport tower/threshold calculations to precision records with detector, resolution, inference, and omitted-channel controls.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-2

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C09, OBS-C10, OBS-C14, OBS-C17, OBS-C21, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Emit gauge algebra, rank, vector count, global kernel, and owner IDs from the complete parent owner map; reject aliases and quotient-isometry errors.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-3

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C12, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C21, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Record family multiplicity and chirality with domain/parity provenance, no-mirror resolution, and finite test-family exhaustion.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-4

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C09, OBS-C10, OBS-C12, OBS-C14, OBS-C17, OBS-C19, OBS-C21, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Record charges, quotient kernel, and local/global anomaly data with invariant observables, statistical controls, and no status leakage.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-5

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C07, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C12, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Map EWSB, masses, mixings, decays, and transition history to records with complete instruments, events, and calibration lineage.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-6

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C07, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C20, OBS-C21, OBS-C22, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Distinguish physical moduli/deformations from inaccessible or gauge directions and define how stabilization is observed across scales/history.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-7

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define threshold/matching records and precision observables with complete response, selection, and uncertainty transport.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-8

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C07, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Enforce chamber projection, same-ruler normalization, blind fitting, flavor record construction, and wrong-ruler controls.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

SG-9

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Map dangerous operators and proton-decay channels to inclusive/exclusive event records, detector response, and finite exposure bounds.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-10

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C07, OBS-C08, OBS-C10, OBS-C11, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C20, OBS-C21, OBS-C22, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define compactification/global records across regions, frames, domains, towers, and relational reference systems.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-14

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C13, OBS-C14, OBS-C20, OBS-C21, OBS-C22, OBS-C24, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define unitary/causal records, sequential instruments, clocks, regional information flow, and cross-observer consistency.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-3

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C14, OBS-C17, OBS-C20, OBS-C21, OBS-C24, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define reflection-positivity/physical-state records, reconstruction, instruments, regional algebras, and operational distinguishability.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-4

  • Observer role: CONDITIONAL/SCOPED
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C13, OBS-C14, OBS-C17, OBS-C20, OBS-C21, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define local/global anomaly observables, determinant/inflow records, regional gluing, and contextual consistency.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-5A/5B

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C20, OBS-C21, OBS-C22, OBS-C24, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define graviton pole/residue/causal response records, regional support, quantum instruments, and information/positivity diagnostics.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-5C

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C14, OBS-C16, OBS-C17, OBS-C19, OBS-C20, OBS-C21, OBS-C24, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define what finite truncation, fixed-point, unitarity, and UV-completion evidence would be observable without overpromotion.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-7

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C09, OBS-C10, OBS-C11, OBS-C12, OBS-C13, OBS-C14, OBS-C16, OBS-C17, OBS-C21, OBS-C26, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define chiral-domain and mirror-completeness records with parity/domain, regional, and cross-context consistency.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

UQF-9

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C06, OBS-C08, OBS-C10, OBS-C12, OBS-C14, OBS-C15, OBS-C16, OBS-C17, OBS-C19, OBS-C20, OBS-C21, OBS-C22, OBS-C24, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Define heat-kernel/RG/UV records, detector-independent invariants, refinement, and finite-truncation claim ceilings.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

theta-bar-QCD

  • Observer role: LOAD-BEARING
  • Selected rows: OBS-C01, OBS-C02, OBS-C03, OBS-C04, OBS-C05, OBS-C06, OBS-C08, OBS-C09, OBS-C10, OBS-C12, OBS-C14, OBS-C17, OBS-C18, OBS-C19, OBS-C21, OBS-C23, OBS-C27, OBS-C28, OBS-C29, OBS-C30
  • Discipline: Map topological phases and CP-odd observables to invariant event/likelihood records with calibration, selection, and mechanism exhaustion.
  • External dependencies: Shape, Granularity, Dynamics, Scale, Boundary, Rigidity, Vacuum, Interdependence, Time Synchronization as applicable
  • Claim ceiling: No gate terminal may exceed the weakest unclosed selected Observer row, record support, test-family scope, finite-resource distinguishability bound, calibration lineage, or uncertainty/selection model.
  • Inactive packets: Every non-active packet remains present with status NOT-APPLICABLE and a gate-specific type proof.

Appendix A — Exact BB-OBS-1.0.1 archival source

The text below is archival and read-only. V4.0 is normative where it strengthens execution without weakening any legacy obligation.

title: "BB-OBS-1.0.1 — Observer, Projection, Gauge-Owner, and Measurement Interface" building_block_id: "BB-OBS-1.0.1" version: "1.0.1" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — SG-2 GAUGE-OWNER CONTROLS MERGED — NOT FROZEN" scope: "Parent-to-record maps, same-ruler comparison, projection, normalization, uncertainty, and wrong-ruler controls" protocol_freeze: false canonical_ratification: false package_authority_version: "3.3.2"

BB-OBS-1.0.1 — Observer, Projection, Gauge-Owner, and Measurement Interface

0. Purpose

This block owns the map between the complete parent theory and the finite record used to test it.

Its central firewall is:

A physical comparison is valid only after dimension, frame, renormalization scale, bundle norm, chamber projection, truncation, regulator, and observable definition are aligned.

1. Complete observer object

Define

where:

  • : admissible preparations;
  • : observer-accessible algebra;
  • : finite record space;
  • : parent-to-record map;
  • : ruler tuple;
  • : uncertainty and covariance packet;
  • : normalization/Jacobian packet.

The observer is not consciousness or an interpretation of measurement. It is the typed physical interface that defines what is prepared, projected, normalized, and recorded.

2. Parent-to-record map

For each observable , publish

The map must show all operations:

  1. gauge and constraint reduction;
  2. mode selection;
  3. dimensional reduction;
  4. chamber/Weyl/orbifold projection;
  5. field redefinition;
  6. canonical normalization;
  7. RG and scheme transport;
  8. detector or record functional;
  9. uncertainty propagation.

No step may be represented by “identify with the measured value.”

3. Frozen ruler tuple

The required tuple is:

A comparison is suspended if any component differs and no transport theorem is provided.

4. Projection normalization

For a projection , record:

  • domain and codomain;
  • kernel and image;
  • normalization;
  • Jacobian;
  • multiplicity factors;
  • positivity/probability preservation;
  • gauge equivalence;
  • information loss.

The SG-8-style control is canonical: a normalized full six-chamber state projected to one chamber carries a factor

when the declared inner product and equal chamber weights apply. The block does not assume this coefficient universally; it demands that the relevant coefficient be derived from the frozen map.

5. Observer-map deformation inventory

The map itself can vary. Include deformations of:

  • frame;
  • scale;
  • scheme;
  • basis;
  • chamber;
  • detector window;
  • boundary accessibility;
  • normalization;
  • truncation;
  • regulator;
  • coarse-graining.

Define

for observer-map parameters . These directions enter uncertainty and, where physical, the Rigidity deformation inventory.

6. Calibration firewall

Every datum receives one role:

  • preparation input;
  • measured anchor;
  • calibration input;
  • blind comparison;
  • derived output;
  • diagnostic.

The data-lineage graph must prove that a blind target does not determine:

  • the projection coefficient;
  • basis convention;
  • normalization;
  • matching scale;
  • tolerance;
  • uncertainty model.

7. Kernel and inaccessible sectors

The map may identify distinct parent states. Publish:

An inaccessible parent direction is not automatically unphysical. It may be:

  • gauge/constraint redundant;
  • genuinely unobservable at the declared scope;
  • indirectly observable;
  • above cutoff;
  • an unresolved hidden sector.

Granularity decides operational equivalence only after this classification.

8. Uncertainty and covariance

Propagate the full covariance:

Correlations cannot be dropped merely because the final comparison table is one-dimensional.

9. Required negative controls

  1. Wrong chamber norm: omit the required chamber normalization.
  2. Wrong dimension: compare a higher-dimensional coefficient with a 4D observable.
  3. Wrong scale: compare running parameters at different .
  4. Wrong frame: compare Einstein- and Jordan-frame quantities without transport.
  5. Double normalization: apply a projection factor twice.
  6. Hidden calibration: select the map after reading the target.
  7. Nullspace deletion: treat an inaccessible physical direction as nonexistent.
  8. Scheme-only discrepancy: flag a difference that disappears after lawful coupled transport.

The block passes development only if every control fails before correction and passes after the correct map is restored.

10. Observer-map certificate

observer_map_certificate:
  candidate_id:
  branch_id:
  observable_id:
  preparation_class:
  accessible_algebra:
  record_space:
  parent_domain:
  reduction_steps:
  projection:
  kernel:
  image:
  ruler_tuple:
  normalization:
  jacobian:
  calibration_lineage:
  uncertainty_covariance:
  negative_controls:
  output_artifacts:
  evidence_hashes:
  verdict:

11. Falsifiers and restart triggers

Restart is required when the frame, scale, scheme, projection, basis, normalization, apparatus window, regulator, or observable definition changes.

The block fails if a measured comparison cannot be reconstructed from the parent manifest without author interpretation.

Gauge-owner record map

For a gauge-group claim, the observer map must consume the complete SHP-C09 owner ledger and emit:

  • local algebra and generator count;
  • rank;
  • vector zero-mode count;
  • extra-vector count;
  • global faithful representation kernel;
  • owner ID for every recorded generator.

The map must distinguish:

  • a local algebra supported by an isometry;
  • a global lift required by matter;
  • a bundle connection surviving a quotient;
  • two independent connections;
  • two notations for the same connection.

Required wrong-object controls include:

  1. attributing a connection to a quotient interval with no continuous rotation isometry;
  2. counting metric and bundle aliases as independent gauge sectors;
  3. hiding a nontrivial representation kernel by reporting only the Lie algebra;
  4. changing parity so an unwanted scalar zero mode appears.

The measured/recorded gauge group is the image of the complete parent owner map—not the name of a compact factor.

Owned hard constraints and acceptance evidence

The full controlling registry is EVIDENCE_DISCHARGE_REGISTRY_V3_3_2.md. The rows below are embedded because this block owns or directly co-owns them.

ID Required witness Minimal discharge Fail trigger
GEN-C04 Observer-map certificate + same-ruler transport output A frozen map sends the parent state to the measured record with dimension, frame, scheme, scale, bundle norm, and projection shown; the wrong-ruler control fails as designed. A raw parent quantity is compared directly with a differently normalized record.
GEN-C09 Gap calculation + uncertainty covariance + ruler packet The lightest omitted physical mode exceeds the observation window by the frozen margin after uncertainty propagation. The claimed gap is qualitative, uses a different normalization, or closes only at central values.
SHP-C07 End-to-end reduction artifact Starting from one parent manifest, the reduction script/action derives the complete retained four-dimensional object, including normalization, matching, boundary, and observer map. Only favorable modes are selected or the 4D object uses data from incompatible branches.
DYN-C07 Response-function calculation Localized and nonconstant stress sources produce the correct causal/conservation response while protected constants are handled according to VAC. The mechanism removes ordinary matter gravity, violates conservation, or produces acausal record response.
VAC-C11 Time-dependent constraint-algebra and cosmology simulation/theorem A time-dependent vacuum shift representing QCD/electroweak transitions is inserted; the constraint algebra remains consistent and the Friedmann/perturbation history stays within frozen bounds, or the regime is explicitly excluded with quantified lost predictions. The mechanism passes static shifts but becomes singular, acausal, or observationally unacceptable during a finite transition.
OBS-C01 Observer-system manifest The observer, apparatus/reference frame, accessible algebra, preparation class, record space, and energy/time window are frozen. The word ‘observer’ is used without specifying what can be prepared or recorded.
OBS-C02 Executable parent-to-record map For each claimed observable, an equation or script maps the complete parent state to the record with all projections and reductions shown; gauge records include one SHP-C09 owner ID per generator. A comparison uses an informal identification, omits gauge-owner provenance, or counts aliases as separate recorded generators.
OBS-C03 Normalization/positivity/probability test The map preserves required normalization, positive probabilities, conservation, and gauge equivalence on the physical domain. Projection changes norm or probability without an accounted Jacobian/normalization.
OBS-C04 Frozen ruler-tuple comparison Dimension, frame, renormalization scale, scheme, bundle norm, chamber projection, truncation, regulator, and observable definition are identical or explicitly transported. A 13D norm is compared directly with a one-chamber 4D running observable.
OBS-C05 Data-lineage and leakage audit Every calibration datum is tagged and a script verifies that blind targets do not enter the map, basis choice, or normalization. A target observable determines a projection coefficient or convention after comparison.
OBS-C06 Kernel/image enumeration The map’s kernel, image, degeneracies, inaccessible sectors, and information loss are explicitly listed through the claimed cutoff. A null direction is treated as absent physics or an inaccessible sector is silently discarded.
OBS-C07 Observer-map deformation Jacobian Derivatives of records with respect to projection, frame, normalization, and apparatus choices are included in the deformation/uncertainty analysis. A small convention change can move a prediction across the pass band but is not counted.
OBS-C08 Full covariance and scheme-transport output Uncertainties and correlations are propagated through the map, including scheme/frame transformations and truncation error. Only central values or independent errors are compared.
OBS-C09 Wrong-ruler and wrong-owner negative-control suite Controls include the SG-8-style chamber projection, omitted Jacobian, wrong frame, wrong scale, double normalization, quotient-isometry misattribution, and duplicate gauge-owner examples; each must fail before correction and pass after. The block cannot detect a known wrong-ruler or duplicate-owner construction.
OBS-C10 Content-addressed observer-map build Frozen inputs regenerate record CSV/JSON outputs and hashes without author interpretation. The observed prediction depends on an undocumented manual conversion.

Package status and authority boundary

This file belongs to the Candidate-Neutral Building-Block Authority v3.3.2.

The package is documentation-complete for the current development architecture, but it is not protocol-frozen and not canonically ratified. Technical results generated under it remain development evidence until the control suite, enumeration test, and hostile review pass.

The controlling package authorities are:

  • EVIDENCE_DISCHARGE_REGISTRY_V3_3_2.md;
  • QUANTITATIVE_STOPPING_AND_EXHAUSTION_RULE_V3_3_2.md;
  • MANDATORY_CONTROL_SUITE_V3_3_2.md;
  • MIGRATION_AND_SUPERSESSION_RECORD_V3_3.md.

A block is standalone for its owned object: it includes its complete definitions, interfaces, certificates, falsifiers, and owned evidence criteria. The full cross-block registry is intentionally stored once rather than repeated verbatim in every block.

Shared status grammar

PASS:
  The named witness satisfies the minimal discharge criterion.

FAIL:
  A reproducible counterexample or failed acceptance test exists.

OPEN:
  The required witness is missing, incomplete, or inconclusive.

NOT-EVALUATED:
  Evaluation stopped after an earlier hard failure.

NOT-APPLICABLE:
  A type proof establishes that the requirement cannot apply.

CONSTRUCTION-ANCHOR:
  A declared Actor, constraint, boundary term, or global sector realizes
  the result but does not derive it from the pre-existing object.

CONDITIONAL-ON-BND-A:
  The spectrum/domain result awaits fixed-set and operator-domain closure.

CONDITIONAL-ON-BND-B:
  The quantum result awaits local and global anomaly/inflow closure.

RESTART-REQUIRED:
  A frozen object relevant to the result changed.

Universal anti-promotion rules

  • A bare candidate is not a complete parent object.
  • A definition is not a witness.
  • A nonempty equation list is not a solvability proof.
  • An isolated vacuum is not automatically stable.
  • A fixed background value is not automatically an absent fluctuation.
  • Static vacuum-offset protection is not automatically cosmological-history compatibility.
  • Integrated anomaly cancellation is not local fixed-set cancellation.
  • Local anomaly cancellation is not global determinant-line trivialization.
  • A raw higher-dimensional quantity is not automatically the measured four-dimensional observable.
  • One tested survivor is not a selected survivor without explicit grammar enumeration or a finiteness theorem.
Appendix M

Reproducible observer-compiler benchmark code

EXECUTED

from __future__ import annotations

import importlib.util
import json
import platform
import time
from pathlib import Path

import numpy as np
import scipy
import scipy.linalg as la


ROOT = Path(__file__).resolve().parent
FIVE_TENSOR_SCRIPT = Path("/mnt/data/FIVE_TENSOR_GR_ARCHITECTURE/five_tensor_toy.py")
SEED = 20260806


def load_five_tensor_module():
    spec = importlib.util.spec_from_file_location("five_tensor_toy", FIVE_TENSOR_SCRIPT)
    if spec is None or spec.loader is None:
        raise RuntimeError(f"Unable to load {FIVE_TENSOR_SCRIPT}")
    module = importlib.util.module_from_spec(spec)
    spec.loader.exec_module(module)
    return module


def build_observer(n_records: int, n_state: int, seed: int = SEED + 400) -> np.ndarray:
    """Build an orthonormal frozen record map O: field -> records."""
    rng = np.random.default_rng(seed + n_records + n_state)
    raw = rng.normal(size=(n_state, n_records))
    q, _ = np.linalg.qr(raw)
    return q.T


def time_sequential(method, params: np.ndarray, repeats: int = 5) -> dict:
    samples = []
    checksum = 0.0
    for _ in range(repeats):
        t0 = time.perf_counter()
        s = 0.0
        for p in params:
            value = method(p)
            s += float(value[0])
        samples.append(time.perf_counter() - t0)
        checksum = s
    median = float(np.median(samples))
    return {
        "median_seconds": median,
        "all_seconds": [float(x) for x in samples],
        "queries": int(len(params)),
        "median_microseconds_per_query": 1.0e6 * median / len(params),
        "checksum": checksum,
    }


def time_batched(method, repeats: int = 9) -> dict:
    samples = []
    checksum = 0.0
    for _ in range(repeats):
        t0 = time.perf_counter()
        value = method()
        samples.append(time.perf_counter() - t0)
        checksum = float(value[0, 0])
    median = float(np.median(samples))
    return {
        "median_seconds": median,
        "all_seconds": [float(x) for x in samples],
        "checksum": checksum,
    }


def main() -> None:
    ft = load_five_tensor_module()
    problem = ft.build_problem(900)
    observer = build_observer(8, problem["n_state"])
    params = ft.random_parameters(3000, seed=SEED + 301)

    W4 = observer @ problem["U4"]
    W13 = observer @ problem["U13"]

    def direct_record(p):
        return observer @ ft.evaluate_direct(problem, p)

    def full_4d_record(p):
        return observer @ ft.evaluate_4d(problem, p)

    def compiled_4d_record(p):
        return W4 @ ft.coefficients_4d(p)

    def compiled_13d_record(p):
        return W13 @ ft.masters_13d(p)

    # Warm up.
    for fn in (direct_record, full_4d_record, compiled_4d_record, compiled_13d_record):
        for p in params[:20]:
            fn(p)

    seq = {
        "direct_factorized_solve_then_records": time_sequential(direct_record, params[:500], repeats=5),
        "full_4d_five_tensor_field_then_records": time_sequential(full_4d_record, params[:500], repeats=7),
        "observer_compiled_4d": time_sequential(compiled_4d_record, params[:500], repeats=7),
        "observer_compiled_13d_informed": time_sequential(compiled_13d_record, params[:500], repeats=7),
    }
    seq_direct = seq["direct_factorized_solve_then_records"]["median_seconds"]
    for key in ("full_4d_five_tensor_field_then_records", "observer_compiled_4d", "observer_compiled_13d_informed"):
        seq[key]["speedup_over_direct"] = seq_direct / seq[key]["median_seconds"]

    # Precompute coefficient/master matrices to isolate field/record linear algebra.
    C4 = np.column_stack([ft.coefficients_4d(p) for p in params])
    U10 = np.column_stack([ft.masters_13d(p) for p in params])
    RHS = problem["B4"] @ C4

    def batch_direct():
        fields = la.cho_solve(problem["factor"], RHS, check_finite=False)
        return observer @ fields

    def batch_full_4d():
        return observer @ (problem["U4"] @ C4)

    def batch_compiled_4d():
        return W4 @ C4

    def batch_compiled_13d():
        return W13 @ U10

    # Warm up batched kernels.
    for fn in (batch_direct, batch_full_4d, batch_compiled_4d, batch_compiled_13d):
        fn()

    batched = {
        "direct_factorized_solve_then_records": time_batched(batch_direct, repeats=7),
        "full_4d_five_tensor_field_then_records": time_batched(batch_full_4d, repeats=11),
        "observer_compiled_4d": time_batched(batch_compiled_4d, repeats=15),
        "observer_compiled_13d_informed": time_batched(batch_compiled_13d, repeats=15),
    }
    for value in batched.values():
        value["queries"] = int(len(params))
        value["median_microseconds_per_query"] = 1.0e6 * value["median_seconds"] / len(params)
    batch_direct_time = batched["direct_factorized_solve_then_records"]["median_seconds"]
    for key in ("full_4d_five_tensor_field_then_records", "observer_compiled_4d", "observer_compiled_13d_informed"):
        batched[key]["speedup_over_direct"] = batch_direct_time / batched[key]["median_seconds"]
    batched["observer_compiled_4d"]["speedup_over_full_field_assembly"] = (
        batched["full_4d_five_tensor_field_then_records"]["median_seconds"]
        / batched["observer_compiled_4d"]["median_seconds"]
    )
    batched["observer_compiled_13d_informed"]["speedup_over_full_field_assembly"] = (
        batched["full_4d_five_tensor_field_then_records"]["median_seconds"]
        / batched["observer_compiled_13d_informed"]["median_seconds"]
    )

    # Exactness and kernel/image controls.
    direct = batch_direct()
    full = batch_full_4d()
    obs4 = batch_compiled_4d()
    obs13 = batch_compiled_13d()
    norm = max(float(np.linalg.norm(direct)), 1e-30)

    # Construct an exact observer-kernel perturbation and confirm records are unchanged.
    rng = np.random.default_rng(SEED + 999)
    trial = rng.normal(size=problem["n_state"])
    kernel_direction = trial - observer.T @ (observer @ trial)
    kernel_residual = np.linalg.norm(observer @ kernel_direction) / max(np.linalg.norm(kernel_direction), 1e-30)

    # Observer deformation sensitivity.
    delta = rng.normal(size=observer.shape)
    delta /= max(np.linalg.norm(delta), 1e-30)
    observer_perturbed = observer + 1e-4 * delta
    fields_sample = la.cho_solve(problem['factor'], RHS[:, :50], check_finite=False)
    record_shift = np.linalg.norm((observer_perturbed - observer) @ fields_sample) / max(
        np.linalg.norm(observer @ fields_sample), 1e-30
    )

    certificate = {
        "certificate_id": "OBSERVER-COMPILED-FIVE-TENSOR-GR-TOY/V2",
        "date": "2026-08-06",
        "status": "PASS-CONTROLLED-TOY-ARCHITECTURE-ONLY",
        "environment": {
            "python": platform.python_version(),
            "numpy": np.__version__,
            "scipy": scipy.__version__,
            "platform": platform.platform(),
        },
        "scope": {
            "field_dimension": int(problem["n_state"]),
            "observer_record_dimension": int(observer.shape[0]),
            "4d_coefficient_channels": int(ft.C13.shape[0]),
            "13d_independent_masters": int(np.linalg.matrix_rank(ft.C13)),
            "sequential_queries": 500,
            "batched_queries": int(len(params)),
            "baseline": "The direct baseline reuses one Cholesky factorization and solves the field for every query.",
            "claim_boundary": [
                "The operator is a deterministic positive controlled field analog, not the Einstein equations.",
                "The batched algebra-only benchmark excludes coefficient/master generation and offline compilation.",
                "The sequential benchmark includes Python coefficient/master evaluation and dispatch.",
                "The results demonstrate the observer-compilation mechanism, not a physical IMRI waveform speedup."
            ],
        },
        "compiled_maps": {
            "W4_shape": list(W4.shape),
            "W13_shape": list(W13.shape),
            "formula_4d": "W4 = O @ U4",
            "formula_13d": "W13 = O @ U4 @ C13",
        },
        "sequential_end_to_end_python": seq,
        "batched_linear_algebra_only": batched,
        "exactness": {
            "relative_error_full_4d_records": float(np.linalg.norm(direct - full) / norm),
            "relative_error_observer_compiled_4d": float(np.linalg.norm(direct - obs4) / norm),
            "relative_error_observer_compiled_13d": float(np.linalg.norm(direct - obs13) / norm),
            "observer_kernel_direction_record_residual": float(kernel_residual),
        },
        "observer_sensitivity_control": {
            "observer_map_relative_perturbation": 1e-4,
            "relative_record_shift": float(record_shift),
            "interpretation": "The observer map is part of the frozen model and must carry its own deformation/error ledger."
        },
        "13d_information": {
            "channel_reduction_fraction": float(1.0 - np.linalg.matrix_rank(ft.C13) / ft.C13.shape[0]),
            "channel_reduction_factor": float(ft.C13.shape[0] / np.linalg.matrix_rank(ft.C13)),
            "null_control": "A 4D solver supplied with the same C13 map executes the same reduced online algebra."
        },
    }

    checks = [
        certificate["exactness"]["relative_error_full_4d_records"] < 1e-12,
        certificate["exactness"]["relative_error_observer_compiled_4d"] < 1e-12,
        certificate["exactness"]["relative_error_observer_compiled_13d"] < 1e-12,
        certificate["exactness"]["observer_kernel_direction_record_residual"] < 1e-12,
        seq["observer_compiled_4d"]["speedup_over_direct"] > 10.0,
        seq["observer_compiled_13d_informed"]["speedup_over_direct"] > 10.0,
        batched["observer_compiled_4d"]["speedup_over_direct"] > 100.0,
    ]
    certificate["all_required_checks_pass"] = bool(all(checks))

    output = ROOT / "observer_compiled_five_tensor_toy_certificate.json"
    output.write_text(json.dumps(certificate, indent=2) + "\n", encoding="utf-8")
    print(json.dumps({
        "certificate": str(output),
        "pass": certificate["all_required_checks_pass"],
        "sequential_4d_speedup": seq["observer_compiled_4d"]["speedup_over_direct"],
        "sequential_13d_speedup": seq["observer_compiled_13d_informed"]["speedup_over_direct"],
        "batched_4d_speedup": batched["observer_compiled_4d"]["speedup_over_direct"],
        "batched_13d_speedup": batched["observer_compiled_13d_informed"]["speedup_over_direct"],
        "compiled_4d_error": certificate["exactness"]["relative_error_observer_compiled_4d"],
        "compiled_13d_error": certificate["exactness"]["relative_error_observer_compiled_13d"],
    }, indent=2))


if __name__ == "__main__":
    main()
Appendix N

Machine-readable benchmark certificate

PASS-CONTROLLED
{
  "certificate_id": "OBSERVER-COMPILED-FIVE-TENSOR-GR-TOY/V2",
  "date": "2026-08-06",
  "status": "PASS-CONTROLLED-TOY-ARCHITECTURE-ONLY",
  "environment": {
    "python": "3.13.5",
    "numpy": "2.3.5",
    "scipy": "1.17.0",
    "platform": "Linux-6.12.13-x86_64-with-glibc2.41"
  },
  "scope": {
    "field_dimension": 900,
    "observer_record_dimension": 8,
    "4d_coefficient_channels": 24,
    "13d_independent_masters": 10,
    "sequential_queries": 500,
    "batched_queries": 3000,
    "baseline": "The direct baseline reuses one Cholesky factorization and solves the field for every query.",
    "claim_boundary": [
      "The operator is a deterministic positive controlled field analog, not the Einstein equations.",
      "The batched algebra-only benchmark excludes coefficient/master generation and offline compilation.",
      "The sequential benchmark includes Python coefficient/master evaluation and dispatch.",
      "The results demonstrate the observer-compilation mechanism, not a physical IMRI waveform speedup."
    ]
  },
  "compiled_maps": {
    "W4_shape": [
      8,
      24
    ],
    "W13_shape": [
      8,
      10
    ],
    "formula_4d": "W4 = O @ U4",
    "formula_13d": "W13 = O @ U4 @ C13"
  },
  "sequential_end_to_end_python": {
    "direct_factorized_solve_then_records": {
      "median_seconds": 0.37300898999819765,
      "all_seconds": [
        0.39782629900219035,
        0.37300898999819765,
        0.39479907599888975,
        0.3576370919981855,
        0.3690244050012552
      ],
      "queries": 500,
      "median_microseconds_per_query": 746.0179799963953,
      "checksum": -1.2890766840182981
    },
    "full_4d_five_tensor_field_then_records": {
      "median_seconds": 0.013948351999715669,
      "all_seconds": [
        0.014482344999123598,
        0.014138514001388103,
        0.013825517999066506,
        0.013705743000173243,
        0.013948351999715669,
        0.01371243600078742,
        0.014696751997689717
      ],
      "queries": 500,
      "median_microseconds_per_query": 27.896703999431338,
      "checksum": -1.289076684018298,
      "speedup_over_direct": 26.74215491592134
    },
    "observer_compiled_4d": {
      "median_seconds": 0.009128854999289615,
      "all_seconds": [
        0.009128854999289615,
        0.008863248000125168,
        0.009239432001777459,
        0.008841390001180116,
        0.00929547100167838,
        0.008844561001751572,
        0.00950461100001121
      ],
      "queries": 500,
      "median_microseconds_per_query": 18.25770999857923,
      "checksum": -1.2890766840182974,
      "speedup_over_direct": 40.86043540260244
    },
    "observer_compiled_13d_informed": {
      "median_seconds": 0.006501957002910785,
      "all_seconds": [
        0.006670480001048418,
        0.006501957002910785,
        0.006518060999951558,
        0.006456414997956017,
        0.006189794999954756,
        0.006462425997597165,
        0.007059173000016017
      ],
      "queries": 500,
      "median_microseconds_per_query": 13.00391400582157,
      "checksum": -1.2890766840182981,
      "speedup_over_direct": 57.36872603605493
    }
  },
  "batched_linear_algebra_only": {
    "direct_factorized_solve_then_records": {
      "median_seconds": 0.14181108699995093,
      "all_seconds": [
        0.14181108699995093,
        0.16492133000065223,
        0.13999228600005154,
        0.15808806700079003,
        0.17788698299773387,
        0.14143325900295167,
        0.12011854899901664
      ],
      "checksum": -0.0029819986530305845,
      "queries": 3000,
      "median_microseconds_per_query": 47.270362333316974
    },
    "full_4d_five_tensor_field_then_records": {
      "median_seconds": 0.004379540001536952,
      "all_seconds": [
        0.05063996800163295,
        0.008097210997220827,
        0.0024934300017775968,
        0.030430576000071596,
        0.005689138997695409,
        0.031571498002449516,
        0.004379540001536952,
        0.0033557149981788825,
        0.0030733729981875513,
        0.002950365000288002,
        0.0031841349991736934
      ],
      "checksum": -0.0029819986530305836,
      "queries": 3000,
      "median_microseconds_per_query": 1.459846667178984,
      "speedup_over_direct": 32.38036116810986
    },
    "observer_compiled_4d": {
      "median_seconds": 3.806199674727395e-05,
      "all_seconds": [
        5.626999700325541e-05,
        4.078900019521825e-05,
        3.86139981856104e-05,
        3.783600186579861e-05,
        3.806199674727395e-05,
        3.773500066017732e-05,
        3.788000321947038e-05,
        3.767800080822781e-05,
        3.797600220423192e-05,
        3.789700349443592e-05,
        4.844399882131256e-05,
        3.846300023724325e-05,
        3.8074002077337354e-05,
        3.7740999687230214e-05,
        3.812800059677102e-05
      ],
      "checksum": -0.002981998653030583,
      "queries": 3000,
      "median_microseconds_per_query": 0.012687332249091318,
      "speedup_over_direct": 3725.792105483999,
      "speedup_over_full_field_assembly": 115.06332761826586
    },
    "observer_compiled_13d_informed": {
      "median_seconds": 2.0283998310333118e-05,
      "all_seconds": [
        5.887500083190389e-05,
        2.140700235031545e-05,
        2.0303003111621365e-05,
        2.0276998839108273e-05,
        2.0179999410174787e-05,
        2.0350998966023326e-05,
        2.0245999621693045e-05,
        2.0204002794343978e-05,
        2.0499002857832238e-05,
        2.0262999896658584e-05,
        2.0340001356089488e-05,
        2.0174000383121893e-05,
        2.0198000129312277e-05,
        2.030700125033036e-05,
        2.0283998310333118e-05
      ],
      "checksum": -0.0029819986530305854,
      "queries": 3000,
      "median_microseconds_per_query": 0.006761332770111039,
      "speedup_over_direct": 6991.278781940601,
      "speedup_over_full_field_assembly": 215.91108096799226
    }
  },
  "exactness": {
    "relative_error_full_4d_records": 6.484785887521391e-16,
    "relative_error_observer_compiled_4d": 1.3912883652491791e-15,
    "relative_error_observer_compiled_13d": 1.3404130736365388e-15,
    "observer_kernel_direction_record_residual": 3.615869086224414e-17
  },
  "observer_sensitivity_control": {
    "observer_map_relative_perturbation": 0.0001,
    "relative_record_shift": 3.74518114708269e-05,
    "interpretation": "The observer map is part of the frozen model and must carry its own deformation/error ledger."
  },
  "13d_information": {
    "channel_reduction_fraction": 0.5833333333333333,
    "channel_reduction_factor": 2.4,
    "null_control": "A 4D solver supplied with the same C13 map executes the same reduced online algebra."
  },
  "all_required_checks_pass": true
}
Appendix O

Source and artifact manifest

CONTENT-ADDRESSED
Authority / artifactFileBytesSHA-256
RigidityBB_RIG_4_1_MAX_RIGOR_ALL_GATE_RIGIDITY.md135,78237703f47b05280d878ee84c258a60a8556952e1e358b039c72efe28a7532e222
InterdependenceBB_INT_4_0_MAX_RIGOR_ALL_GATE_INTERDEPENDENCE.md74,15854cfaba6833108419531618f72ca38e2b5fb8c7353c1ae1a71c5fbb1cdce350d
VacuumBB_VAC_4_0_MAX_RIGOR_ALL_GATE_VACUUM.md143,957362c00d1d255552896cd4d9be730374f0d0b9daa54dc1c578598bf3d479e0bca
GovernanceBB_CSE_4_0_SUPPLEMENTAL_MAX_RIGOR_GATE_GOVERNANCE.md72,988115a22d5d5f89c082696ffd94bf9449b2bfc5028b46a4523e87712d5a78e1a28
Shape Stagestage.md49,19920a1906b5467a1d82b849b06636fa4acaa7916e79f39d97dd4117f1fb26bf2ab
Shape Rulebookrulebook.md44,490ff0e5a660e11fbd1a397933f7e22521a0cf91155fa1b0bcb2f00e6204a8b2729
DynamicsBB_DYN_4_2_MAX_RIGOR_FULL_GATE_CLOSURE_DYNAMICS.md155,401cbc41bd5faf197208d7651496a912ac9fd7e0d4fb7c48705d9722116ca00ea6c
ScaleBB_SCL_4_1_MAX_RIGOR_ALL_GATE_SCALE.md120,442b0426790624d7abde99427cfe12a80726857c99f6d16fbea1c69843598187493
Time SynchronizationBB_TS_4_0_MAX_RIGOR_ALL_GATE_TIME_SYNCHRONIZATION.md64,8531958cbe4ef5e61dadf24838766e149dc1c4f019a3d612e7ba7cc5f5257ccb1c9
ObserverBB_OBS_4_0_MAX_RIGOR_ALL_GATE_OBSERVER.md78,3702a7edf478134d753cfb37d5f299fd1f40d2d71d664ed23dd86d9573f62c21b06
BoundaryBB_BND_4_1_MAX_RIGOR_FULL_GATE_CLOSURE_BOUNDARY.md111,318b489a3aa8991a71df94aa8118da975c2910fc69e9d13e89398fa642ad776b95b
Max Conservation StackMaxConservationStack.html5,758,9832ca5fb629b0276cd31976d97ae7da063ebe30d0626a57ac08fb74f665c4d5598
Five-Tensor GR ArchitectureFiveTensorGRArchitecture.html6,195,5794967c6d2c9ae58581c8097b7daf72744b96ec4b16ca768dac611fbf314947cb0
Building-Block Speed AuditBUILDING_BLOCK_SPEED_AUDIT.md13,862f97c9d38e27e340ff2587e6c9db404ee99852f84e7a61faa879183eddc8fe4de
Observer Benchmark Scriptobserver_compiled_five_tensor_toy.py10,7172b020ecfad17e92e5bb05cc3fc368d2d27b25d4ed6bd1d554433d79477de3673
Observer Benchmark Certificateobserver_compiled_five_tensor_toy_certificate.json6,7582c303e15b3d8e0814d732aa8ae5035184b72bf0a92f2c88be25c4456d22d0f7d