Central thesis

The Max Conservation Stack is a proposed calculation architecture for general relativity in which conservation is not used only as an after-the-fact check. It is used first, at maximum independent rank, to determine the admissible response and coupling space before expensive field solves or waveform training are performed.

The program has two versions.

1. The 4D stack exhausts diffeomorphism constraints, Bianchi identities, matter and gauge Ward identities, boundary balance laws, horizon and null-infinity flux laws, first-law relations, integrability conditions, frame covariance, waveform reality, and observable-specific constraints directly in four-dimensional general relativity.
2. The 13D stack begins from a declared parent product geometry and adds information that does not arise merely from writing the same 4D equations again: internal representations, parities, topology, spectra, overlap integrals, fixed-set terms, reaction stresses, projection normalization, and a certified exclusion of forbidden internal and mixed metric directions.

The central computation is a constraint-saturation problem. Candidate coupling coefficients are collected into a vector (g). All independent conservation and Ward conditions are assembled into

The remaining dimension after four-dimensional conservation is

The internal geometry supplies a lawful coupling basis,

and the surviving internal master dimension is

The measurable information supplied by the internal dimensions is therefore

This definition is deliberately unforgiving. If , the internal geometry has supplied no additional computational information. If , the geometry has compressed the physical correction algebra. If , conservation plus the internal basis fixes the declared coupling sector uniquely.

Strong claim being pursued

The project does not claim that thirteen-dimensional arithmetic is inherently faster. The stronger and more defensible claim is:

A higher-dimensional parent object can make a four-dimensional GR computation substantially cheaper when it supplies nonredundant constraints, exact selection rules, fixed coupling ratios, or a lower-dimensional invariant master algebra before the numerical solve.

The proposed IMRI application uses a two-tensor decomposition,

where is an additive response under a frozen linearized operator and owns the complete nonlinear, boundary, reaction, and frame correction. The order in which mass parcels or parameter changes are applied is controlled by the curvature of the total response connection. In conservative sectors, exactness of a state function makes the continuation path-independent. In dissipative sectors, the failure of path independence must be owned by radiative, horizon, boundary, or reaction flux.

Current evidence

The software architecture has passed synthetic and controlled ablation tests. These tests demonstrate the algebra and expose the conditions under which large reductions are possible; they do not yet constitute physical , , or waveform results.

The strongest controlled results presently recorded are:

• additive-plus-correction reconstruction at machine precision;
• all tested mass-addition orders agreeing at approximately ;
• a 46.2% error when local frames are added without lawful transport;
• a Shape quotient reducing an ambient correction rank from 8 to 3 in a contaminated controlled model;
• a conservation and co-rotating-frame closure reducing a raw 50-mode correction family from effective rank 42 to two masters;
• a controlled training-label reduction from 25,600 labels to 16, or , under the synthetic master-family assumptions.

The physical coupling matrices and , including the fixed-set and anomaly-complete terms, remain owed. This document does not hide that debt. It defines exactly what must be calculated, how it must be tested, and which claims become lawful at each closure state.

Status classes used throughout

The document distinguishes mathematical status from implementation status and physical validation status.

Nonnegotiable claim rule

A large numerical speedup produced by a controlled family is evidence that the architecture can exploit a low-rank physical family. It is not evidence that nature supplies that family. Promotion requires an independently replayable physical matrix or waveform test with frozen tolerances.

Likewise, a relation that can be derived entirely in 4D is useful but not uniquely 13D. The higher-dimensional contribution is the part of the relation that is derivable from internal geometry and that reduces , training samples, correction rank, or certified uncertainty beyond the conservation-saturated 4D baseline.

What is being built

The Max Conservation Stack is a pre-solve constraint compiler for GR. It accepts a declared physical system, action, boundaries, observer, frames, and candidate response basis. It emits:

1. the maximal independent conservation matrix ;
2. the conservation-admissible affine space;
3. the 4D unresolved master dimension ;
4. in the 13D version, the internal basis ;
5. the conservation-compatible internal master dimension ;
6. a reconstruction and error certificate for the requested observable;
7. explicit blockers if any boundary, anomaly, reaction, or observer channel is unowned.

Claims we state directly

Claim A — Conservation saturation can reduce computation before simulation.
When several apparently independent waveform corrections are tied by energy balance, angular-momentum balance, first-law identities, common-frame transport, symmetry, and boundary conditions, solving each correction independently is wasteful.

Claim B — Order independence is a curvature question.
The order of incremental source additions is physically immaterial when the response connection is flat after quotienting gauge and including all boundary flux. Nonzero curvature is not ignored; it becomes the correction ledger.

Claim C — Frames are computational structure, not presentation.
The controlled test found an untransported frame error of 46.2%, while properly transported responses agreed at machine precision.

Claim D — Internal dimensions help only by supplying information.
The current 13D numerical kernel reduces to the same 4D kernel after projection. Its demonstrated value is the projector and channel ownership. A uniquely higher-dimensional speed benefit requires internal selection rules or coupling identities that lower below the conservation-saturated 4D dimension.

Claim E — The potential payoff is large.
In the controlled invariant-master experiment, the raw family had rank 42, while the co-rotating master residual had rank 2. The synthetic expensive-label count fell from 25,600 to 16, a 1600 reduction. The physical replay remains owed.

Claims we do not make

• We do not claim that arbitrary 4D metrics superpose.
• We do not claim that conservation alone determines arbitrary radiation.
• We do not claim that a finite list of global charges determines every local spacetime observable.
• We do not claim that the current 13D compactification has been established by observation.
• We do not claim a physical waveform speedup until the public perturbative and NR datasets are executed under the frozen protocol.
• We do not call a synthetic rank result a physical rank result.

The dashboard reports controlled architecture results. The physical q=64 replay is still blocked by unavailable large binary inputs in this runtime. The document treats that distinction as a hard claim boundary.

Definition of the native 4D stack

A native 4D Max Conservation Stack is the tuple

where:

• is the four-dimensional spacetime and its declared differentiable, causal, and asymptotic structure;
• is the dynamical metric;
• contains all retained matter, gauge, auxiliary, and apparatus fields;
• is a differentiable action including required boundary and corner terms;
• and are the presymplectic potential and current;
• is the complete constraint and Ward system;
• is the boundary, horizon, null-infinity, and junction ledger;
• is the frozen observable map;
• freezes the asymptotic, co-moving, tetrad, polarization, and mode conventions.

“Maximum” does not mean that every identity that can be written is counted as independent. It means that every lawful independent constraint is generated, normalized to one convention, rank-reduced, and either used or explicitly dispositioned.

Four-dimensional conservation object

For a diffeomorphism-invariant action,

The symplectic current is

[ \omega_4

\delta_1\Theta_4 -\delta_2\Theta_4. ]

For an admitted symmetry generator , the Hamiltonian variation is a boundary integral,

[ \delta H_\xi

\int_{\partial\Sigma} \left( \delta Q_\xi-\xi\cdot\Theta_4 \right), ]

subject to the integrability and boundary conditions of the selected sector. Where symplectic current escapes through null infinity or a horizon, the correct statement is a charge-plus-flux balance rather than a globally conserved Hamiltonian.

This is the first discipline of the stack: nothing is called conserved until its support and flux channels are specified.

4D constraint registry

The following registry is the operational core of the 4D version. A project need not activate every row for every observable, but it must show that inactive rows are irrelevant rather than silently omit them.

The rows are grouped into six families:

1. variational and differential identities;
2. canonical and gauge constraints;
3. asymptotic, horizon, and regional balances;
4. binary and orbital integrability;
5. radiative and modal constraints;
6. observer, frame, numerical, and uncertainty constraints.

Each row becomes one or more normalized equations in the matrix . Nonlinear identities are linearized in the selected coupling coordinates or represented as polynomial constraints with a declared lifting. Redundant rows are removed by rank-revealing QR or SVD with rational or interval replay where exact rank matters.

4D saturation theorem

Let (V) be the chosen finite or function-space coupling coordinate domain after frame, boundary, and observable choices have been frozen. Let all independent 4D constraints be represented as

Assume the system is consistent. Then the full conservation-admissible set is the affine space

Its unresolved dimension is

This is elementary linear algebra, but it has a strong methodological consequence:

No calculation may claim that conservation has fixed more coefficients than the rank of the independent constraint matrix.

If , the remaining basis vectors are genuine unresolved masters, not license to fit each waveform mode independently. A reduced solver should learn or calculate only those masters and reconstruct the dependent coefficients through the row-reduced constraint solution.

Nonlinear and functional constraints

Many GR constraints are nonlinear or live in function spaces. The stack handles them by one of four declared methods:

1. exact symbolic elimination;
2. polynomial lifting to a higher-dimensional linear system;
3. local tangent-space rank analysis with a finite-radius residual certificate;
4. operator-nullspace analysis with domain and coercivity certificates.

A local Jacobian rank is not promoted to a global theorem unless branch connectedness, boundary preservation, and singular-set exclusions are established.

Maximum versus redundant conservation

Energy conservation, the contracted Bianchi identity, and the equations of motion may generate related statements. Counting all restatements as independent would create a false compression. The stack therefore records:

• source equation;
• derivation path;
• support and boundary;
• convention and units;
• algebraic row;
• numerical rank contribution;
• dependence certificate.

The resulting is the independent conservation stack, not a list of slogans.

Conservation does not imply sparsity by itself

A fixed total radiated energy can be distributed among one mode or arbitrarily many orthogonal modes. Therefore

does not by itself imply low rank.

The additional ingredient is a higher-order conservation, regularity, selection, or observable-insensitivity statement.

Higher-order energy and tail bounds

If a commuted-energy hierarchy controls angular derivatives of Bondi news,

then

This is a genuine conservation-to-tail theorem in the declared linear or controlled-regularity sector.

Observable-specific factorization

A stronger computational strategy is often easier: prove that the chosen observable is insensitive to the unresolved field. For a bounded linear observable,

if , then

exactly, even if the physical tail is large.

For a general detector response, the error is bounded by

This distinction is central to the stack: the physical field need not be globally low-rank for a mission observable to be certifiably low-rank.

Two invariant masters for secular phase

For a nonspinning quasi-circular adiabatic binary, define

Let the conservative binding energy and total flux be

Energy balance gives

Therefore the secular phase factors through two scalar functions. Separate phase-correction surfaces for every () mode are redundant for this observable.

After factoring the carrier,

[ h_{\ell m}

P_{\ell m} e^{-im\phi} r_{\ell m}, ]

the remaining question is whether the amplitude residual also factors through a small coupling algebra.

Controlled master-closure evidence

The manufactured physical-family test recorded:

These values validate the compression mechanism on the declared synthetic family. The physical amplitude residual rank remains an explicit replay target.

One externally verifiable project

The selected demonstration is:

Certified nonspinning quasi-circular IMRI Waveform Bridge

The project uses:

• as the visible development case;
• as the demonstration target;
• as the held-out falsification case;
• public perturbative/surrogate data;
• public numerical-relativity data on the available overlapping modes;
• a frozen specification and blind commitment;
• full versus reduced runtime and physics comparisons.

NASA’s LISA Preparatory Science Program explicitly identifies efficient, accurate waveform models across mass ratios and identifies the (1{:}100)–(1{:}1000) IMRI regime as a challenging gap. It also notes the large number of waveform realizations required by LISA analysis. This makes the benchmark directly relevant without requiring NASA to accept the 13D parent construction in advance.

Verification logic

A reviewer need not run a new numerical-relativity simulation. The proposed sequence is:

1. verify source and archive hashes;
2. run unit and synthetic controls;
3. execute the public development inputs;
4. freeze , the internal basis, retained masters, and tolerances;
5. generate the result;
6. confirm the blind commitment;
7. unblind and execute once;
8. compare phase, amplitude, mismatch, energy, angular momentum, remnant data, rank, samples, and wall time;
9. repeat the 4D/13D ablations on the same hardware.

What a pass would establish

A pass would support:

Conservation saturation and a geometry-derived coupling basis reduced the independent correction calculations for a declared vacuum binary sector while preserving specified waveform and balance-law observables against independent public references.

It would not establish generic spinning, precessing, eccentric, matter-coupled, or arbitrary strong-field closure.

Definition of the 13D parent stack

The declared parent branch is represented schematically as

with parent metric

on the admitted constrained branch. Four-dimensional gravity remains dynamical. The internal metric and mixed metric directions are either fixed, projected, or owned by geometric-admissibility reaction equations according to the frozen Shape authority.

The 13D stack is

where:

• is the frozen internal spectral and representation basis;
• is the normalized observer projection;
• denotes constraint-reaction multipliers;
• owns fixed sets, corners, interval endpoints, and localized sectors;
• owns local, global, measure, and inflow consistency.

What the internal dimensions are permitted to add

The internal dimensions may add:

• exact representation products;
• parity and domain selection rules;
• topological and cohomological labels;
• degeneracy relations;
• spectral gaps and coercive constants;
• normalized overlap integrals;
• fixed-set localized contributions;
• reaction-stress couplings;
• coupling ratios fixed by geometry;
• superselection rules and vanishing theorems.

They may not:

• override a failed conservation law;
• delete an unresolved boundary flux;
• convert a numerical near-zero into an exact selection rule;
• treat an inaccessible mode as absent without a type proof;
• repair an anomaly by silently changing the spectrum;
• claim computational benefit when the same identity is already fully derivable in the conservation-saturated 4D stack.

Conservation first, coupling algebra second

Candidate internal couplings are collected into

The complete parent and projected conservation equations are assembled into

Independently, the internal geometry supplies

Substitution gives

The surviving master dimension is

Constraint-saturation theorem

If the affine system is consistent and

then the declared internal coupling coefficients are uniquely fixed. If the rank is smaller, the nullspace basis is the complete list of unresolved coupling masters.

This theorem is exact. The physical difficulty is not the theorem; it is populating the matrices without omitted boundaries, duplicated constraints, wrong domains, or fitted post-target coefficients.

Information supplied by the internal dimensions

The 13D information contribution is not the number of additional coordinates. It is the number of independent physical coefficient directions removed or fixed:

We define four outcomes:

• Unique algebra: .
• Finite-master algebra: .
• No information gain: .
• Inconsistent parent: the conservation-compatible intersection is empty.

A fifth apparent outcome—large compression created only by projecting out directions that a well-posed native 4D formulation would never include—must be reported as a governance benefit, not a uniquely higher-dimensional physical benefit.

Coupling tensor

For an internal basis (Y_A(y)), define the complete projected coupling tensor

[ \Gamma_{ABC}

\Gamma^{\rm bulk}{ABC} +\Gamma^{\rm fixed}{ABC} +\Gamma^{\rm react}_{ABC}, ]

with

[ \Gamma^{\rm bulk}_{ABC}

\int_{X_9}d^9y\sqrt\gamma, Y_A\mathcal D. ]

The derivative operator , index contractions, bundle connections, parity domains, and normalization are frozen before integration.

Maximum-constraint algorithm

1. Freeze the physical question. Specify source class, observable, frame, boundary prescription, approximation order, and parameter domain.
2. Enumerate candidate couplings. Construct a typed coupling registry, not an unstructured coefficient vector.
3. Generate 4D identities. Diffeomorphism, gauge, canonical, asymptotic, horizon, first-law, symmetry, and observable constraints.
4. Generate parent identities. Parent Noether, internal flux, fixed-set, reaction, anomaly, domain, and projection constraints.
5. Normalize. Put all rows on one ruler, convention, and boundary support.
6. Remove dependencies. Compute exact or certified numerical rank and publish the row-dependence graph.
7. Derive internal basis. Calculate representation products, parity zeros, topological restrictions, spectra, overlaps, and localized terms.
8. Intersect. Solve .
9. Publish nullspace. Every unresolved coupling master receives a basis vector, physical interpretation, and test.
10. Map to observables. Contract the coupling masters into , fluxes, and waveform modes.
11. Run ablations. Native 4D, conservation-saturated 4D, 4D plus internal projector, and full 13D.
12. Falsify. Test wrong parity, omitted fixed set, wrong frame, duplicated constraint, altered boundary condition, and held-out waveform.

Matrix diagnostics

The machine-readable analyzer must report:

and the uncertainty in each rank decision.

Near-singular values are not rounded away without a scale and uncertainty certificate. Exact selection-rule zeros should be represented symbolically or with interval bounds that exclude nonzero values.

Native 4D versus 13D-projected arithmetic

The direct 4D-versus-13D numerical ablation produced the same field, rank, singular values, reconstruction error, conservation closure, and order-independence result after projection. The demonstrated uniquely 13D online cycle saving was therefore approximately zero in that test.

This is the expected result when the internal geometry is fixed and only the projected 4D metric is evolved. The numerical kernel is 4D.

Shape quotient ablation

A three-way controlled test compared:

1. a naive ambient 4D learner;
2. a 4D learner supplied with the Shape projector;
3. the 13D-native Shape implementation.

The Shape-constrained versions were identical. In the contaminated ambient family, the projector reduced effective rank from 8 to 3 and minimum training samples from 10 to 4. The null control, with no forbidden-channel contamination, gave rank 3 in both approaches.

The conclusion is precise:

The geometry supplied useful information by deriving the quotient. Once the quotient is known, a 4D solver can execute it at the same cost.

Test for uniquely internal information

A uniquely internal advantage requires at least one of:

where the 4D baseline already receives the full maximum conservation stack.

Any gain that disappears when the same projector or identity is supplied to the 4D solver is an information-governance gain, not evidence that higher-dimensional arithmetic is faster.

Additive response tensor

Choose a frozen background and gauge/domain prescription. Let be the linearized parent field operator. For an incremental conserved source,

[ \mathsf A[\delta\mathcal T]

\mathcal L_\star^{-1}. ]

Because the inverse is applied to the frozen linear operator,

[ \mathsf A[\delta\mathcal T_1+\delta\mathcal T_2]

\mathsf A[\delta\mathcal T_1] + \mathsf A[\delta\mathcal T_2]. ]

This is the lawful superposition object. It is not the full metric.

Correction tensor

Define

It obeys the exact defect equation

[ \mathcal L_\star\mathsf C

-\mathcal N_\star[\mathsf A+\mathsf C] +\kappa,\Delta\mathcal T_{\rm int/bnd/react}, ]

where every nonlinear, interaction, boundary, horizon, fixed-set, reaction, and frame correction is owned.

The exact reconstruction is

Why this can be cheap

The additive response can be cached, transported, and superposed. The correction is the only state-dependent nonlinear object. If

then online cost is controlled by (r), not by the bulk field dimension.

The Max Conservation Stack attempts to reduce (r) before learning:

1. quotient forbidden and gauge directions;
2. transport every response into one frame;
3. factor known representation carriers;
4. solve conservation identities;
5. use internal coupling relations;
6. learn only the residual nullspace.

Response connection

Let label operations or physical parameters: mass addition, frequency change, spin change, eccentricity, source displacement, or frame transport. Define the total response one-form

Its field-space curvature is

[ \mathsf F_{ab}

D_a\mathsf R_b-D_b\mathsf R_a+[\mathsf R_a,\mathsf R_b]. ]

For an observable projection , order independence holds in a simply connected admissible region when

after quotienting gauge and including all boundary flux.

Conservative exactness

If a sector is generated by a state function (E),

then

and the endpoint energy is independent of the path. This makes “add the secondary mass in pieces” lawful for the conservative endpoint quantity, provided every step remains in the same physical branch and the state function is regular.

For circular binary mechanics,

Thus mass continuation and frequency continuation commute at the level of the conservative state function wherever the mixed partials exist.

Dissipative curvature

Radiation prevents a blanket path-independence statement. For dissipative evolution,

[ \Delta_{\gamma_1}-\Delta_{\gamma_2}

\int_{\mathcal A}d\alpha ]

is owned by symplectic or physical flux through null infinity, horizons, fixed sets, reaction channels, or other declared boundaries.

The computational proposal is therefore:

[ \boxed{ \text{hard path}

\text{easy path} + \text{curvature/flux correction}. } ]

If the curvature correction is low-rank or determined by a small set of flux masters, a nonlinear evolution can be replaced by cached additive transport plus a small correction solve.

Why one frame is mandatory

A tensor response calculated near one source cannot be added directly to a response expressed in another tetrad, orbital phase, time origin, polarization basis, or asymptotic frame. Each contribution must be transported:

[ \mathsf A^{(i)}_{\star}

U_i,\mathsf A^{(i)}_{\rm local},U_i^{-1}. ]

Only then is

meaningful.

The controlled two-tensor certificate found:

[ \epsilon_{\rm transported}

1.050e-16, ]

while direct addition without transport produced

[ \epsilon_{\rm untransported}

0.462. ]

This is not a cosmetic coordinate issue. Wrong-frame addition can manufacture apparent correction rank, apparent curvature, and apparent failure of superposition.

Preferred IMRI frames

A practical implementation uses several nested frames:

• a local regular/self-force frame around the secondary;
• an osculating-geodesic or co-moving orbital frame;
• a co-rotating modal frame that removes the carrier ;
• a frozen asymptotic Bondi/BMS frame;
• a detector or observer frame applied only after the physical waveform is assembled.

The transport maps are part of the coupling algebra. Their derivatives contribute to and to the connection curvature. They cannot be applied as an undocumented plotting convention after compression.

Cost model

Let:

• be the number of independently solved waveform or response channels;
• be samples per independent function;
• be the expensive PDE, self-force, Teukolsky, or NR-derived label cost;
• (r) be the reduced correction rank;
• (d) be the surviving invariant master dimension;
• be the inexpensive output reconstruction cost.