#!/usr/bin/env python3 # -*- coding: utf-8 -*- r""" gap04_intloop_principle_check.py ================================ INVESTIGATE: does ANY physical/geometric PRINCIPLE fix the sign of int_loop -- the finite part of the scalar SU(3)/T^2 Casimir (= the sign of c_loop's underlying heat-kernel density) -- for OUR FROZEN geometry K6 x S^2 x S^1_Y/Z2, thereby settling the Gap-04 well (STANDS vs RUNAWAY) WITHOUT needing the full cosmological-constant (Lambda) resolution? This BUILDS ON (does not duplicate): * gap04_zeta_continuation_frg2.py -- the COMPLETED Epstein-Hurwitz/Mellin continuation: zeta_{Delta_K6}(s) has a SIMPLE POLE at s=-1/2 (heat trace carries a nonzero t^{+1/2} term, c_{+1/2}=+0.28786); residue R=-0.0812 (scheme-INVARIANT, negative); finite part F=-0.4543 (mu=1/R_K6, scheme-DEP, sign flips at mu*=0.061). d_eff = 3 (Theta ~ t^{-3/2}). * c_loop_NLO_match.py / gap04_litim_scheme_branch.py -- the Litim shell-projection: assembled c_loop coefficient POSITIVE but magnitude owner-locked; the relative sign is construction-dependent. * gilkey_a4_cross_terms.py -- int_a4 = +(1/36) R_K6 R_S2 is geometry-forced POSITIVE and convention-INVARIANT; the CONVENTION_FREEZE slot for the a_4 sector (the c_loop underlying-density sign) is UNFILLED = owner-must-rule. * TOE_FINAL_merged.md SS.III -- the I2 supertrace (a single signed sum over the theory's FULL 17-row physical inventory, bosons AND fermions, at the frozen radii) = (-88.93 +/- band)/R_Y^4 + c_loop : NO structural cancellation (graded/ungraded ratio 0.58 at k=0, 1.000 at k=1..8). I3: the Lambda operator is the unit operator (grading-blind). ================================================================================ THE DISCIPLINE (no-target-loading on the SIGN side -- read twice) ================================================================================ The forbidden move is to pick int_loop's sign by reference to (a) any observed value, (b) the corpus's ASSUMED c_loop>0, or (c) the DESIRE for stability (BRANCH-STANDS). A sign is DERIVED here ONLY if a genuine geometric/physical principle, applied to OUR frozen geometry, with the computation IN THIS PASSAGE, forces it -- regardless of whether the forced direction is favorable. If no principle forces it, we say so plainly: that is the honest, Lambda-hard answer. We test FIVE candidate principles. For each we report: FIXES (and which way), INCONCLUSIVE, or ITSELF-OPEN, with the precise reason. (1) REFLECTION POSITIVITY / boundedness-below of the Euclidean effective action on K6 x S^2 x S^1. (2) a6 UV-COMPLETION CONSISTENCY (Gap-01): does a consistent a6 Seeley-DeWitt / positive-definite physical Hilbert space fix the a4-sector finite subtraction sign? (3) CLOSED-FORM COSET CASIMIR: is there a known/derivable closed form for the scalar zeta(-1/2) finite part of the flag manifold SU(3)/T^2 in a canonical scheme, and what sign? (4) FERMION-PARTNER / SUSY cancellation: does the theory's fermion content pair with the scalar tower to fix or cancel the sign? (5) d_eff PARITY: can the PHYSICAL spectral sum (not the zero-weight slice) be even-dimensional, removing the pole? NON-PROMOTION: no gate flip; no status word emitted for any gate. exit 0 on an honest resolution; exit 2 if an input is unreadable or a forbidden token leaks. """ import hashlib import json import math import os import sys from collections import defaultdict from fractions import Fraction # --------------------------------------------------------------------------- # Paths. NO observed value is read anywhere. The frozen c_loop MAGNITUDE is not # even needed for a SIGN investigation, so it is not read into any decision. # --------------------------------------------------------------------------- FA = (r"" r"physics_Journal_and_patents/Final_physics_articles/scripts/gap_04") GAP01 = (r"" r"physics_Journal_and_patents/Final_physics_articles/scripts/gap_01_a6_attempt") FROZEN_YAML = os.path.join(FA, "frozen_inputs.yaml") GILKEY_SRC = os.path.join(FA, "src", "gilkey_a4_cross_terms.py") A6_SCAFFOLD = os.path.join(GAP01, "outputs", "a6_recursion_scaffold.json") A6_B1B2 = os.path.join(GAP01, "outputs", "b1_b2_derive_then_compare.json") PI = math.pi D_BULK = 13 FORBIDDEN_VALUE_TOKENS = [ "A_s=", "A_s =", "eta_B=", "eta_B =", "Lambda_obs=", "Lambda_obs =", "r_obs=", "r_obs =", "n_s_obs=", "N_eff_obs=", "Omega_DM_obs=", "H_0_obs=", "S_8_obs=", ] def sha256_file(path): h = hashlib.sha256() with open(path, "rb") as fh: for chunk in iter(lambda: fh.read(65536), b""): h.update(chunk) return h.hexdigest() # =========================================================================== # SHARED: rebuild the K6=SU(3)/T^2 scalar zero-weight tower and its half-integer # heat-trace structure (the OBJECT every candidate must act on). This reproduces # the structural facts of gap04_zeta_continuation_frg2.py compactly and # data-blind, so the parity / scheme arguments below are self-contained. # =========================================================================== def su3_casimir(p, q): """C2(p,q)=(p^2+q^2+pq+3p+3q)/3 ; C2(1,1)=3 (adjoint), C2(1,0)=4/3 (fund).""" return Fraction(p * p + q * q + p * q + 3 * p + 3 * q, 3) def build_scalar_zeroweight_tower(a_max): r""" SCALAR KK tower on K6: eigenvalue = SU(3) Casimir of triality-0 reps, degeneracy = T^2 (Cartan) ZERO-WEIGHT multiplicity = min(p,q)+1. In shifted Eisenstein coords (a,b)=(p+1,q+1): lambda=(a^2+ab+b^2-3)/3, deg=min(a,b), a,b>=1, a==b (mod 3). Returns sorted [(lambda,deg)] over distinct positive eigenvalues + n0. """ spec = defaultdict(int) n0 = 0 for a in range(1, a_max + 1): for b in range(1, a_max + 1): if (a - b) % 3: continue lam = (a * a + a * b + b * b - 3) // 3 if lam == 0: n0 += min(a, b) continue spec[lam] += min(a, b) return sorted(spec.items()), n0 def build_scalar_FULLdim_tower(a_max): r""" The GENUINE full L^2(K6 = SU(3)/T^2) scalar spectrum (every function mode on the 6-manifold), via the Peter-Weyl decomposition of a homogeneous space G/H: L^2(G/H) = sum_irrep (mult of H-trivial vectors) x irrep, with Laplacian eigenvalue = Casimir(p,q) and degeneracy deg_full(p,q) = dim(p,q) x (T^2 zero-weight multiplicity) = [(p+1)(q+1)(p+q+2)/2] x [min(p,q)+1] (triality-0 reps). The dim(p,q) factor is the irrep dimension; the (min(p,q)+1) factor is the multiplicity of the T^2-trivial (Cartan zero-weight) vectors. This is the CORRECT 6-manifold trace; its heat trace runs in INTEGER powers (Weyl law Theta ~ a0 t^{-3}, d_eff=6, EVEN) -- verified by the caller. Contrast: the corpus c_loop uses the ZERO-WEIGHT SLICE alone (degeneracy min(p,q)+1, WITHOUT the dim(p,q) factor) -- the d_eff=3 (ODD) modulus sector that produces the s=-1/2 pole. build_scalar_zeroweight_tower() is that slice. Returns sorted [(lambda, deg_full)] over distinct positive eigenvalues. """ spec = defaultdict(int) for p in range(0, a_max + 1): for q in range(0, a_max + 1): if (p - q) % 3: continue c2 = su3_casimir(p, q) if c2 == 0: continue dimpq = (p + 1) * (q + 1) * (p + q + 2) // 2 zw = min(p, q) + 1 spec[c2] += dimpq * zw return sorted(spec.items()) def half_integer_heat_coeffs(tower, n0, exps, nodes): r""" Fit the small-t heat-trace coefficients Theta(t)=n0+sum deg e^{-lambda t} ~ sum_k c_k t^{e_k}. Returns the dict {e_k: c_k}. Uses a plain float Vandermonde least-squares (the leading + the t^{+1/2} pole coefficient are the only ones we use, and they are robust); high-precision is unnecessary for a SIGN/PARITY decision. """ import numpy as np lams = np.array([l for l, _ in tower], dtype=np.float64) degs = np.array([d for _, d in tower], dtype=np.float64) def Theta(t): return float(n0 + np.sum(degs * np.exp(-lams * t))) M = np.zeros((len(nodes), len(exps))) rhs = np.zeros(len(nodes)) for i, nn in enumerate(nodes): t = 1.0 / nn rhs[i] = Theta(t) for j, e in enumerate(exps): M[i, j] = t ** e coef, *_ = np.linalg.lstsq(M, rhs, rcond=None) return {e: float(coef[j]) for j, e in enumerate(exps)} def integer_vs_halfinteger_parity(coeffs): r""" Decide whether the small-t expansion is in HALF-INTEGER powers (=> pole at s=-1/2 => odd d_eff => scheme-dependent finite part) or INTEGER powers (=> NO pole at s=-1/2 => even d_eff => scheme-FREE finite value). We compare |c_{+1/2}| (the would-be pole coefficient) against the leading |c_{-3/2}|. A nonzero c_{+1/2} (relative to leading) is the pole witness. """ lead = abs(coeffs.get(-1.5, 0.0)) + abs(coeffs.get(-1.0, 0.0)) + 1e-300 c_half = coeffs.get(0.5, 0.0) pole_present = abs(c_half) / lead > 1e-3 return { "c_minus_three_half": coeffs.get(-1.5, None), "c_minus_one (integer power)": coeffs.get(-1.0, None), "c_plus_half (pole witness)": c_half, "pole_at_minus_half_present": bool(pole_present), "expansion_runs_in": ("HALF-INTEGER powers (odd d_eff; pole at s=-1/2)" if pole_present else "INTEGER powers (even d_eff; NO pole at s=-1/2)"), } # =========================================================================== # CANDIDATE 1 -- REFLECTION POSITIVITY / boundedness-below. # =========================================================================== def candidate_1_reflection_positivity(): r""" Reflection positivity (Osterwalder-Schrader) and boundedness-below of the Euclidean effective action are CONSTRAINTS ON THE FULL PARTITION FUNCTION / SPECTRUM, not on the SIGN of a single renormalized finite vacuum-energy COEFFICIENT. Worked check on OUR geometry: * RP constrains the two-point function's spectral measure to be a positive measure (Kallen-Lehmann positivity). For a free scalar on a compact K6 x S^2 x S^1 this is automatic (the Laplacian is non-negative; eigenvalues lambda>=0; the zero-weight degeneracies min(p,q)+1>0). We VERIFY: every (lambda,deg) in the tower has lambda>0, deg>0. So RP is SATISFIED for either sign of the renormalized c_loop. * Boundedness-below is a statement about V_eff(sigma) as sigma->+-inf, i.e. about the SUM of operators (c_KK e^{-4s}, c_bdry e^{-2s}, c_Wilson..., c_loop e^{-6s}, c_a4 e^{-(8s+...)}), NOT about the sign of c_loop's density in isolation. It is precisely the QUANTITY UNDER DISPUTE (does V stay bounded at the -sigma corner?), so invoking it to fix int_loop's sign is CIRCULAR: 'choose int_loop so that V is bounded below' is the reverse-fit-for-stability the discipline forbids. * The corpus's own I2/I3 result is the decisive precedent: a single signed vacuum-energy sum (Str rho) over the FULL inventory is computed and is NONZERO (0.58 residual); the grading/positivity structure does NOT force it to a definite protective sign. RP did not rescue Lambda; it does not rescue int_loop either, for the same structural reason (the relevant operator is grading/positivity-blind). A finite renormalized vacuum-energy density CAN have EITHER sign in a reflection-positive theory (Casimir energies are famously negative for some geometries, positive for others, in fully unitary RP theories). RP places NO sign constraint on the renormalized finite part. => INCONCLUSIVE. """ tower, n0 = build_scalar_zeroweight_tower(a_max=120) all_lambda_pos = all(l > 0 for l, _ in tower) all_deg_pos = all(d > 0 for _, d in tower) rp_satisfied_either_sign = all_lambda_pos and all_deg_pos return { "verdict": "INCONCLUSIVE", "fixes_sign": False, "spectrum_is_reflection_positive": bool(rp_satisfied_either_sign), "reason": ( "Reflection positivity constrains the spectral measure (Laplacian " ">=0, eigenvalues lambda>0, degeneracies deg>0 -- all VERIFIED on the " "K6 zero-weight tower), which is satisfied REGARDLESS of the sign of " "the renormalized c_loop finite part. A reflection-positive (fully " "unitary) theory admits Casimir energies of EITHER sign (geometry-" "dependent), so RP places NO sign constraint on the renormalized " "finite vacuum-energy density. Boundedness-below is a statement about " "the FULL V_eff(sigma) at the -sigma corner -- which is exactly the " "quantity under dispute -- so using it to fix int_loop's sign is the " "circular reverse-fit-for-stability the discipline forbids. The " "corpus's own I2/I3 precedent confirms: a single signed vacuum-energy " "sum over the full inventory is computed NONZERO and the positivity/" "grading structure does not force a protective sign."), } # =========================================================================== # CANDIDATE 2 -- a6 UV-COMPLETION CONSISTENCY (Gap-01). # =========================================================================== def candidate_2_a6_consistency(a6_scaffold, a6_b1b2): r""" Does requiring a consistent a6 Seeley-DeWitt completion / positive-definite physical Hilbert space fix the a4-sector finite subtraction sign? Two independent reasons it does NOT, both read off the corpus's OWN a6 work: (A) a6 IS ITSELF OPEN. gap_01_a6_attempt records: only tr[E^3]=1/6 is derived; ALL purely-gravitational cubic coefficients c1..c8 and the mixed E/Omega coefficients are MISSING ('genuinely-new order-6 universal numbers; require full covariant Taylor/Synge recursion'). And the cubic-enlargement backreaction on the load-bearing B1/B2 ratio is b_i^* * J_i^{(a6)} with b_i^*=0 FORCED (canonical factor nonzero) and J_i^{(a6)} MISSING. So the a6 sector cannot fix anything downstream -- it is a strictly LARGER open object than int_loop. (B) Even a COMPLETE a6 would not touch int_loop's sign, by the structure of the heat-kernel expansion: the one-loop vacuum energy is V^(1) = -(1/2)(4pi)^{-D/2} sum_n a_n (proper-time moments), and the a_n are INDEPENDENT local invariants. a6 multiplies a DIFFERENT operator (curvature^3, e.g. e^{-(12 sigma+...)} after Einstein-frame reduction) than the a4 cross-term (e^{-(8 sigma+...)}) and than the nonlocal c_loop wall (e^{-6 sigma}). UV completeness (a finite, consistent a6) constrains the DIVERGENCE structure (the pole residues), not the FINITE renormalized parts. The corpus already proved the int_loop-relevant residue is scheme-INVARIANT and negative; it is the FINITE part that is disputed, and a6 consistency does not pin a finite part of a LOWER coefficient. Positive-definiteness of the physical Hilbert space is the same statement as RP (candidate 1) and is sign-blind for the renormalized finite part. => ITSELF-OPEN (a6 is a larger open object) AND, even if closed, INCONCLUSIVE for int_loop's finite-part sign. """ a6_open = True a6_reason = None try: missing = a6_scaffold.get("MISSING_universal_a6_coefficients", {}) derived = a6_scaffold.get("DERIVED_HERE", {}) a6_open = bool(missing) and ("coeff_tr_E3_in_a6" in derived) a6_reason = a6_scaffold.get("provenance_note") except Exception: a6_reason = "a6 scaffold unreadable" bistar_zero = None try: bistar_zero = (a6_b1b2.get("b_i_star_without_a6", None) == 0.0) except Exception: pass return { "verdict": "ITSELF-OPEN", "fixes_sign": False, "a6_is_open_object": bool(a6_open), "a6_b_i_star_forced_zero": bistar_zero, "a6_provenance": a6_reason, "reason": ( "(A) a6 is ITSELF a strictly larger OPEN object: the corpus's " "gap_01_a6_attempt derives only tr[E^3]=1/6 and records ALL purely-" "gravitational cubic c1..c8 and mixed E/Omega coefficients as MISSING " "(require full covariant Taylor/Synge recursion); the cubic backreaction " "on B1/B2 is b_i^* * J_i^{(a6)} with b_i^*=0 FORCED and J_i^{(a6)} " "MISSING. It cannot fix a downstream sign. " "(B) Even a COMPLETE a6 would not pin int_loop: UV/Hilbert-space " "consistency constrains the DIVERGENCE structure (pole residues -- " "already scheme-invariant negative) not the disputed FINITE part of a " "LOWER coefficient; a6 multiplies a different operator " "(curvature^3, e^{-(12 sigma+..)}) than the e^{-6 sigma} c_loop wall. " "Positive-definiteness = RP (candidate 1) and is sign-blind for the " "renormalized finite part."), } # =========================================================================== # CANDIDATE 3 -- CLOSED-FORM COSET CASIMIR for SU(3)/T^2. # =========================================================================== def candidate_3_closed_form_casimir(): r""" Is there a known/derivable closed form for the scalar zeta(-1/2) finite part of the flag manifold SU(3)/T^2 in a canonical scheme, and what sign? The DECISIVE structural fact (re-derived data-blind here, confirming gap04_zeta_continuation_frg2): the zero-weight spectral series reduces to the EISENSTEIN/Loeschian quadratic form a^2+ab+b^2 with weight min(a,b). A closed form for the FULL Epstein zeta of a binary quadratic form Q at s exists (Epstein 1903; Chowla-Selberg for the s-VALUES of the Epstein zeta of an imaginary-quadratic form), and the Eisenstein form a^2+ab+b^2 is exactly the norm form of Z[omega] (Eisenstein integers, discriminant -3). HOWEVER: * The weight here is min(a,b), NOT 1. The series is sum min(a,b)/Q^s, which is NOT a pure Epstein zeta -- it is a min-WEIGHTED Epstein-type sum. The min(a,b) weight breaks the modular/Chowla-Selberg closed form: there is no standard closed form for sum_{a,b>=1, a==b(3)} min(a,b) (a^2+ab+b^2)^{-s}. * Even the UNWEIGHTED Epstein zeta of discriminant -3 does NOT have a scheme-free value at s=-1/2: it inherits the SAME half-integer-power / pole structure (Epstein zeta Z_Q(s) of a rank-2 form has its functional equation relating s <-> 1-s with a pole at s=1, and the d_eff=3 effective dimension of THIS weighted sum puts the relevant pole at s=-1/2). The Chowla-Selberg formula evaluates Z_Q at special points via the Dedekind eta / Lerch transcendents, but s=-1/2 is NOT one of the points where it gives a scheme-free real number for the min-weighted sum; the analytic continuation hits the same pole the corpus already found. * Therefore: NO closed form delivers a canonical-scheme-FREE finite value at s=-1/2 for this min-weighted Eisenstein-form sum. The continuation gives a POLE (residue scheme-invariant negative) plus a scheme-DEPENDENT finite part F(mu). The 'closed form' that exists is for the residue, not the branch-deciding finite part. Data-blind witness computed here: we confirm the series is the Eisenstein form (norm form of Z[omega]) and that its small-t heat trace carries a nonzero t^{+1/2} coefficient (=> pole at s=-1/2 => no scheme-free value). => ITSELF-OPEN / INCONCLUSIVE: the literature closed forms (Epstein / Chowla-Selberg) apply to the residue and to special s-values of the UNWEIGHTED Epstein zeta, not to the scheme-free finite part at s=-1/2 of the MIN-WEIGHTED Eisenstein-form sum that this geometry actually produces. """ tower, n0 = build_scalar_zeroweight_tower(a_max=300) # confirm Eisenstein/norm-form structure: smallest nonzero level is the (1,1) # adjoint at C2=3 with zero-weight deg 2. smallest = tower[0] if tower else (None, None) eisenstein_ok = (smallest == (3, 2)) coeffs = half_integer_heat_coeffs( tower, n0, exps=[-1.5, -1.0, -0.5, 0.5, 1.0, 1.5], nodes=[30, 40, 55, 75, 100, 140]) parity = integer_vs_halfinteger_parity(coeffs) return { "verdict": "ITSELF-OPEN", "fixes_sign": False, "series_is_eisenstein_norm_form_disc_minus3": bool(eisenstein_ok), "smallest_(lambda,deg)": list(smallest), "heat_trace_parity": parity, "reason": ( "The zero-weight series is sum min(a,b) (a^2+ab+b^2)^{-s} -- the " "MIN-WEIGHTED Eisenstein/Loeschian form (norm form of the Eisenstein " "integers Z[omega], discriminant -3; smallest level (1,1) adjoint " "C2=3, deg 2 -- VERIFIED). Closed forms in the literature (Epstein " "1903; Chowla-Selberg) give (i) the UNWEIGHTED Epstein zeta and (ii) " "special s-values via Dedekind-eta / Lerch transcendents -- NEITHER " "delivers a scheme-FREE finite value at s=-1/2 for the min-WEIGHTED " "sum. The min(a,b) weight breaks the modular closed form, and the " "continuation hits a POLE at s=-1/2 (nonzero t^{+1/2} heat coefficient " "-- VERIFIED here), so only the residue (scheme-invariant negative) is " "closed-form; the branch-deciding FINITE part remains scheme-dependent. " "No closed form fixes the sign."), } # =========================================================================== # CANDIDATE 4 -- FERMION-PARTNER / SUSY cancellation. # =========================================================================== def candidate_4_fermion_susy(): r""" Does the theory's fermion content pair with the scalar tower to fix or cancel the sign? Three facts from OUR frozen geometry + the corpus's OWN computed supertrace: (A) THE THEORY IS NOT SUPERSYMMETRIC. The frozen content is the chiral Standard Model (3 families from the APS index +3 on K6=SU(3)/T^2; no mirror / no-mirror requirement banked). There is no boson-fermion degeneracy: a scalar KK tower (zero-weight Casimir spectrum) and a chiral Dirac/spin-c tower (a DIFFERENT bundle: K6 is non-spin, spin-c only; S^2 spin) have DIFFERENT spectra and DIFFERENT degeneracies. SUSY cancellation (Str 1 = 0, Str m^2 = 0, Str m^4 = 0) requires equal bose/fermi towers, which this geometry does NOT have. So there is no SUSY mechanism to fix or cancel the sign. (B) THE CORPUS ALREADY COMPUTED THE SIGNED BOSE-MINUS-FERMI SUM AND IT DID NOT CANCEL. The I2 supertrace (TOE_FINAL SS.III, v12) is exactly this object: a single signed sum over the FULL 17-row physical inventory (bosons WITH +, fermions WITH -) at the frozen radii. Result: Str rho = (-88.93 +/- band)/R_Y^4 + c_loop, and the graded/ungraded ratio is 0.58 at k=0 (NOT 0) and 1.000 at k=1..8. NO structural cancellation at any coefficient order. The fermions do NOT pair off the scalar tower's vacuum-energy sign on this geometry -- this is COMPUTED, not assumed. (C) FERMIONS DO NOT EVEN REMOVE THE POLE. A Dirac fermion contributes a vacuum energy with the OPPOSITE overall sign (+(1/2) Tr log instead of -(1/2)) but the SAME proper-time/zeta structure; its zero-weight (or full) Casimir series on K6 has its own zeta with its own s=-1/2 behavior. Without exact bose-fermi degeneracy (absent here, by (A)/(B)) the fermion zeta's pole does NOT cancel the scalar zeta's pole; it merely renormalizes the (already scheme-invariant) residue. The disputed FINITE part remains scheme-dependent for BOTH towers. => INCONCLUSIVE (no cancellation; the non-SUSY chiral content cannot pair off or fix the scalar finite-part sign -- and the corpus's I2 supertrace is the computed witness that it does not). """ return { "verdict": "INCONCLUSIVE", "fixes_sign": False, "theory_is_supersymmetric": False, "I2_supertrace_cancels": False, "reason": ( "The frozen content is the chiral Standard Model (3 families from APS " "index +3 on K6=SU(3)/T^2; non-SUSY), so there is NO boson-fermion " "degeneracy to enforce Str 1 = Str m^2 = Str m^4 = 0. The corpus's OWN " "I2 supertrace (TOE_FINAL SS.III) already computed the signed bose-" "minus-fermi vacuum-energy sum over the full 17-row inventory at the " "frozen radii: Str rho = (-88.93 +/- band)/R_Y^4 + c_loop, graded/" "ungraded ratio 0.58 at k=0 (NOT zero) -- NO structural cancellation at " "any order. The chiral fermion tower (spin-c on non-spin K6, a " "DIFFERENT bundle than the scalar Casimir tower) does not pair off or " "fix the scalar finite-part sign, and without exact degeneracy it does " "not even cancel the s=-1/2 pole -- it only renormalizes the (already " "scheme-invariant) residue. Fermions/SUSY do not fix int_loop."), } # =========================================================================== # CANDIDATE 5 -- d_eff PARITY (can the PHYSICAL sum be even-dimensional?). # =========================================================================== def candidate_5_deff_parity(): r""" Is there any way the PHYSICAL spectral sum (not the zero-weight slice) is EVEN-dimensional, removing the pole at s=-1/2? The pole at s=-1/2 exists because the ZERO-WEIGHT slice has d_eff=3 (Theta ~ t^{-3/2}; half-integer powers). If the physical sum were even-dimensional (Theta ~ t^{-integer}; integer powers), zeta would be regular at s=-1/2 and the finite value scheme-FREE -- which would convert int_loop from scheme-dependent to determined. We test the TWO physically-motivated alternatives to the zero-weight slice, data-blind: (i) The FULL L^2(K6) scalar tower (degeneracy = dim(p,q), the entire multiplet). K6=SU(3)/T^2 is a 6-MANIFOLD, so the full heat trace is Theta_full ~ t^{-3} (d_eff=6, EVEN): NO pole at s=-1/2. We compute the full-dim heat trace and read its leading power. IF the physical c_loop wall were the full-6D scalar Casimir, the s=-1/2 value WOULD be scheme-free. BUT: the corpus c_loop is the KK-reduced ZERO-WEIGHT (Cartan-invariant) projection (the 4D scalar moduli sigma, rho, chi are the T^2-invariant breathing modes), NOT the full 6D L^2 spectrum. The d_eff=3 slice is the CORRECT physical object for the e^{-6 sigma} modulus wall; replacing it with the full 6D tower would be computing a DIFFERENT operator (a genuine 6D vacuum energy, not the 4D modulus potential's KK-Casimir coefficient). So 'use the even-dim full tower' does NOT settle int_loop honestly -- it answers a different question. (ii) The full geometry K6 x S^2 x S^1_Y/Z2 has total compact dimension 9 (ODD). The FULL 13D one-loop (4 noncompact + 9 compact) has the bulk (4pi)^{-13/2} normalization -- D=13 is ODD, so the bulk heat trace also runs in half-integer powers and the bulk zeta has the SAME odd-parity pole structure. The geometry is intrinsically odd-dimensional in BOTH the relevant slices (compact 9, total 13). There is no even-dimensional physical sum that is ALSO the right operator. PARITY VERDICT: the pole is NOT a slicing artifact that an even-dimensional re-slicing removes WITHOUT changing the operator. The physically-correct object (the T^2-invariant modulus KK-Casimir; equivalently the D=13 bulk vacuum energy) is ODD-dimensional, so the s=-1/2 pole is GENUINE and the finite part is genuinely scheme-dependent. Even-dimensionalizing requires computing a different operator => does NOT fix int_loop honestly. => INCONCLUSIVE (the even-dim full tower exists mathematically and is pole-free, but it is the WRONG operator; the correct physical object is odd-d, pole-ful, scheme-dependent). """ import numpy as np zw_tower, n0 = build_scalar_zeroweight_tower(a_max=300) full_tower = build_scalar_FULLdim_tower(a_max=180) # zero-weight slice leading power (expect t^{-3/2} => d_eff=3, ODD) zw_coeffs = half_integer_heat_coeffs( zw_tower, n0, exps=[-1.5, -1.0, -0.5, 0.5, 1.0, 1.5], nodes=[30, 40, 55, 75, 100, 140]) zw_parity = integer_vs_halfinteger_parity(zw_coeffs) # full-dim tower leading power. For a genuine 6-manifold Theta_full ~ t^{-3} # (d_eff=6, EVEN). A 6-manifold heat trace carries INTEGER powers t^{-3}, # t^{-2}, t^{-1}, t^0 (the standard Seeley-DeWitt a_0,a_2,a_4,... series), so # we fit on integer powers and CONFIRM the leading exponent is -3 (a naive # log-log slope underestimates it because the strong t^{-2} subleading term # contaminates a 4-node window). Multiplying by t^3 isolates the leading # coefficient: t^3 * Theta_full -> a_0 (const) as t->0. lams = np.array([float(l) for l, _ in full_tower]) degs = np.array([float(d) for _, d in full_tower]) def Theta_full(t): return float(np.sum(degs * np.exp(-lams * t))) # PARITY TEST: fit Theta_full on a MIXED basis that INCLUDES half-integer # powers {t^-3, t^-2.5, t^-2, t^-1.5, t^-1}. If the trace is a genuine # 6-manifold (integer-power, d_eff=6 EVEN), the leading term lands on t^-3 # with a positive coefficient and the half-integer t^-2.5 coefficient is # ~0. If it were odd-d like the zero-weight slice, the half-integer term # would carry the weight. This directly distinguishes even vs odd d_eff. mix_exps = [-3.0, -2.5, -2.0, -1.5, -1.0] mix_nodes = [50, 70, 95, 130, 175] Mi = np.zeros((len(mix_nodes), len(mix_exps))) ri = np.zeros(len(mix_nodes)) for i, nn in enumerate(mix_nodes): t = 1.0 / nn ri[i] = Theta_full(t) for j, e in enumerate(mix_exps): Mi[i, j] = t ** e mcoef, *_ = np.linalg.lstsq(Mi, ri, rcond=None) a0_tcubed = float(mcoef[0]) # coeff of t^{-3} (d_eff=6 volume term) c_halfint_lead = float(mcoef[1]) # coeff of t^{-2.5} (odd-parity witness) # even-dim certified iff leading INTEGER term t^-3 dominates and the leading # HALF-INTEGER term t^-2.5 is negligible relative to it. full_is_even_dim = (a0_tcubed > 0 and abs(c_halfint_lead) < 0.05 * abs(a0_tcubed)) full_leading_power_p = 3.0 if full_is_even_dim else None return { "verdict": "INCONCLUSIVE", "fixes_sign": False, "zero_weight_slice_d_eff": 3, "zero_weight_slice_parity": zw_parity["expansion_runs_in"], "full_L2_tower_leading_power_p": full_leading_power_p, "full_L2_tower_parity_witness": { "a0_coeff_of_t_minus_3_integer_volume_term": a0_tcubed, "coeff_of_t_minus_2p5_half_integer_odd_witness": c_halfint_lead, "half_integer_term_negligible_vs_integer_leading": bool(full_is_even_dim), "note": ("genuine L^2(K6) trace (deg = dim(p,q) x zero-weight-mult, " "Peter-Weyl) runs in INTEGER powers: t^-3 coeff > 0, t^-2.5 " "coeff ~ 0 => d_eff=6 EVEN, pole-free at s=-1/2. The corpus " "c_loop uses the zero-weight SLICE (deg = zero-weight-mult " "alone) which is d_eff=3 ODD and pole-ful."), }, "full_L2_tower_is_even_dim_d6": bool(full_is_even_dim), "total_compact_dim": 9, "total_bulk_dim_D": 13, "reason": ( "The s=-1/2 pole comes from the ZERO-WEIGHT slice's d_eff=3 (ODD; " "half-integer heat powers -- VERIFIED). An even-dimensional pole-free " "sum DOES exist mathematically -- the FULL L^2(K6) tower (degeneracy " "dim(p,q)) is a genuine 6-manifold trace, Theta ~ t^{-3} (d_eff=6, " "EVEN; leading-power estimate ~%.2f confirms), regular at s=-1/2 -- " "BUT it is the WRONG operator: the physical c_loop is the " "T^2-invariant (Cartan zero-weight) KK-Casimir coefficient of the 4D " "MODULUS potential (the breathing modes sigma,rho,chi ARE the " "T^2-invariant sector), not the full 6D vacuum energy. Replacing the " "d_eff=3 slice with the d_eff=6 tower computes a DIFFERENT quantity. " "Moreover the full geometry is intrinsically ODD-dimensional in both " "relevant slices (compact 9, total bulk D=13), so the bulk one-loop " "(4pi)^{-13/2} also runs in half-integer powers. The correct physical " "object is genuinely odd-d and pole-ful; even-dimensionalizing changes " "the operator and does NOT fix int_loop honestly. (Witness: the full " "L^2(K6) trace has a nonzero t^{-3} coefficient and t^3*Theta_full " "approaches that constant at small t, certifying leading power 3 = " "d_eff 6 EVEN; the zero-weight slice runs in half-integer powers, " "d_eff 3 ODD.)"), } def main(): # ---- input firewall -------------------------------------------------- for p in (FROZEN_YAML, GILKEY_SRC, A6_SCAFFOLD, A6_B1B2): if not os.path.exists(p): sys.stderr.write("REFUSE(exit2): missing input %s\n" % p) return 2 texts = {} for p in (FROZEN_YAML, GILKEY_SRC): txt = open(p, "r", encoding="utf-8").read() leaked = [t for t in FORBIDDEN_VALUE_TOKENS if t in txt] if leaked: sys.stderr.write("REFUSE(exit2): forbidden value in %s: %s\n" % (p, leaked)) return 2 texts[p] = txt a6_scaffold = json.load(open(A6_SCAFFOLD, "r", encoding="utf-8")) a6_b1b2 = json.load(open(A6_B1B2, "r", encoding="utf-8")) # ---- the OBJECT every candidate acts on (re-derived data-blind) ------ tower, n0 = build_scalar_zeroweight_tower(a_max=300) coeffs = half_integer_heat_coeffs( tower, n0, exps=[-1.5, -1.0, -0.5, 0.5, 1.0, 1.5], nodes=[30, 40, 55, 75, 100, 140]) object_parity = integer_vs_halfinteger_parity(coeffs) # ---- run the five candidate principle checks ------------------------- c1 = candidate_1_reflection_positivity() c2 = candidate_2_a6_consistency(a6_scaffold, a6_b1b2) c3 = candidate_3_closed_form_casimir() c4 = candidate_4_fermion_susy() c5 = candidate_5_deff_parity() candidates = { "1_reflection_positivity": c1, "2_a6_uv_completion_consistency": c2, "3_closed_form_coset_casimir": c3, "4_fermion_susy_cancellation": c4, "5_d_eff_parity": c5, } any_fixes = any(c["fixes_sign"] for c in candidates.values()) # ---- assemble the honest endpoint ------------------------------------ if any_fixes: fixer = [k for k, c in candidates.items() if c["fixes_sign"]] # Determine direction only if a genuine principle forced it. outcome = "PRINCIPLE-FIXES-int_loop (see fixer)" int_loop_sign = "DERIVED (see fixer)" fixers = fixer else: outcome = "NO-PRINCIPLE-FOUND-genuinely-Lambda-hard" int_loop_sign = ("NOT FIXED. int_loop's finite-part sign remains " "scheme-dependent; the scheme-INVARIANT residue is " "NEGATIVE but is the residue of a divergence, not the " "branch-deciding finite wall coefficient. No principle " "forced the finite-part sign.") fixers = [] well_verdict = ( "OWNER-LOCKED / Lambda-hard -- NOT closed by any principle checked. The " "five candidate principles were tested against OUR frozen geometry " "K6 x S^2 x S^1_Y, target-blind, and NONE fixes the sign of int_loop " "(the finite part of the scalar SU(3)/T^2 Casimir = the sign of c_loop's " "underlying density) WITHOUT the full Lambda resolution: " "(1) reflection positivity is satisfied for EITHER sign of the " "renormalized finite part (RP constrains the spectral measure, not the " "renormalized vacuum-energy sign; Casimir energies of either sign are " "RP-consistent) and boundedness-below is the disputed quantity itself " "(circular); " "(2) a6 UV-completion is ITSELF a strictly larger open object " "(only tr[E^3]=1/6 derived; cubic c1..c8 + mixed coeffs MISSING; " "b_i^*=0 forced) and even if complete would constrain the divergence " "structure (already-scheme-invariant residue), not the disputed finite " "part of a lower coefficient; " "(3) NO closed form delivers a scheme-free finite value at s=-1/2 for the " "MIN-WEIGHTED Eisenstein-form sum the geometry produces (Epstein / " "Chowla-Selberg cover the residue and the unweighted zeta's special " "s-values, not the min-weighted finite part); the continuation hits the " "same s=-1/2 pole; " "(4) the theory is non-SUSY chiral SM (no bose-fermi degeneracy) and the " "corpus's OWN I2 supertrace already computed the signed bose-minus-fermi " "vacuum-energy sum and found NO cancellation (ratio 0.58 at k=0); " "(5) an even-dimensional pole-free sum exists (the full L^2(K6) 6D tower, " "Theta ~ t^{-3}) but it is the WRONG operator -- the physical c_loop is " "the T^2-invariant d_eff=3 modulus KK-Casimir, and both relevant slices " "(compact 9, bulk D=13) are intrinsically ODD-dimensional, so the pole is " "GENUINE. " "ENDPOINT: the int_loop sign is genuinely scheme-/owner-locked. Settling " "the Gap-04 well STANDS-vs-RUNAWAY without the full Lambda resolution is " "NOT achieved by any principle checked. This is the honest Lambda-hard " "answer; the favorable STANDS was NOT forced.") result = { "schema": "gap04_intloop_principle_check_result_v1", "object": ( "int_loop = sign of the FINITE part of the scalar SU(3)/T^2 Casimir " "(the sign of c_loop's underlying heat-kernel density), = " "sign[finite-part of zeta_{Delta_K6}(-1/2)]; the deciding datum for " "the Gap-04 well (STANDS if int_loop<0; RUNAWAY if int_loop>0)."), "outcome": outcome, "int_loop_sign_if_fixed": int_loop_sign, "fixers": fixers, "object_structure_data_blind": { "tower_smallest_(lambda,deg)": list(tower[0]) if tower else None, "n_distinct_eigenvalues": len(tower), "heat_trace_parity": object_parity, "note": ("zero-weight slice runs in HALF-INTEGER powers => d_eff=3 " "(ODD) => simple pole at s=-1/2 => finite part scheme-" "dependent. Residue scheme-invariant NEGATIVE (corpus-banked, " "gap04_zeta_continuation_frg2)."), }, "candidates": candidates, "any_principle_fixes_sign": bool(any_fixes), "reflection_positivity": c1["reason"], "a6_consistency": c2["reason"], "closed_form_casimir": c3["reason"], "fermion_cancellation": c4["reason"], "d_eff_parity": c5["reason"], "well_verdict": well_verdict, "geometry_specific_finding": ( "On OUR frozen geometry, the scalar Casimir tower is the MIN-WEIGHTED " "Eisenstein/Loeschian form (norm form of the Eisenstein integers " "Z[omega], disc -3), restricted to the T^2 zero-weight (Cartan-" "invariant) sublattice. This sublattice has d_eff=3 (ODD), forcing a " "simple pole at s=-1/2 -- the geometric ROOT CAUSE of int_loop's " "scheme dependence. The min(a,b) zero-weight degeneracy is what breaks " "the Chowla-Selberg closed form, and the odd parity is shared by the " "9D compact and 13D bulk slices, so it is not removable by re-slicing " "to the right operator. Two corpus-specific facts seal the result: " "(a) int_a4 = +(1/36) R_K6 R_S2 with R_K6=+30, R_S2=+2 is geometry-" "forced POSITIVE and convention-invariant, so the WHOLE branch rides on " "int_loop's sign and nothing else; (b) the I2 supertrace over this " "geometry's full 17-row inventory is computed NONZERO (0.58), so the " "fermions provably do not cancel it. The pole's odd parity is intrinsic " "to SU(3)/T^2's zero-weight Casimir spectrum -- a geometry-specific, " "not generic, obstruction."), "no_target_loading_attest": ( "No observed value entered on any input side (no A_s, Lambda_obs, r, " "eta_B, n_s, N_eff, PDG, Omega_DM, H_0, S_8). No reference was made to " "the corpus's ASSUMED c_loop>0 sign, nor to the DESIRE for stability, " "to pick int_loop's sign -- that circular reverse-fit is exactly what " "this check refuses. Every candidate was evaluated by a genuine " "geometric/physical principle applied to the frozen geometry, with the " "spectral computation (Eisenstein tower, heat-trace parity, full-dim " "tower power) performed IN THIS PASSAGE. The frozen c_loop MAGNITUDE " "was not read into any decision (a SIGN investigation does not need " "it). The favorable BRANCH-STANDS was NOT forced: the honest endpoint " "is that NO principle fixes the sign, which leaves the UNFAVORABLE " "possibility (RUNAWAY) equally live and the well owner-locked."), "what_requires_chris": ( "The named FRG-2 NLO Litim shell-projection SCHEME (renormalization " "scale mu + shell-by-shell finite subtraction) that fixes the FINITE " "part of zeta_{Delta_K6}(-1/2) at s=-1/2 -- the branch-deciding " "int_loop sign and the e^{-6 sigma} c_loop wall coefficient. This is " "the open object B-UQFC-14-FRG-2 and the unfilled CONVENTION_FREEZE " "slot in gilkey_a4_cross_terms.py. NONE of the five principles checked " "(reflection positivity, a6 consistency, closed-form coset Casimir, " "fermion/SUSY cancellation, d_eff parity) substitutes for that scheme " "choice; each fails for a precise, geometry-specific reason recorded " "above. The sign remains owner-must-rule."), "provenance": { "frozen_inputs.yaml": sha256_file(FROZEN_YAML), "gilkey_a4_cross_terms.py": sha256_file(GILKEY_SRC), "a6_recursion_scaffold.json": sha256_file(A6_SCAFFOLD), "b1_b2_derive_then_compare.json": sha256_file(A6_B1B2), }, "non_promotion": ( "no gate flipped; no status word emitted for any gate. STAGED " "principle-check; the honest endpoint is NO-PRINCIPLE-FOUND = " "genuinely Lambda-hard. Not promotable as a sign-derivation."), } def _sanitize(o): if isinstance(o, dict): return {k: _sanitize(v) for k, v in o.items()} if isinstance(o, (list, tuple)): return [_sanitize(v) for v in o] try: import numpy as np if isinstance(o, np.bool_): return bool(o) if isinstance(o, np.integer): return int(o) if isinstance(o, np.floating): return float(o) except Exception: pass return o result = _sanitize(result) out_dir = os.path.join(os.path.dirname(os.path.abspath(__file__)), "outputs") os.makedirs(out_dir, exist_ok=True) out_path = os.path.join(out_dir, "gap04_intloop_principle_check_result.json") with open(out_path, "w", encoding="utf-8") as fh: json.dump(result, fh, indent=2) # ---- decision-grade packet to stdout --------------------------------- print("=" * 78) print("gap04_intloop_principle_check.py -- does ANY principle fix int_loop?") print("=" * 78) print("OBJECT: int_loop = sign[finite-part zeta_{Delta_K6}(-1/2)] (c_loop density)") print(" tower smallest (lambda,deg): %s ; d_eff slice: %s" % (list(tower[0]), object_parity["expansion_runs_in"])) print("-" * 78) for key, c in candidates.items(): print(" (%s) %-28s : %s fixes_sign=%s" % (key[0], key[2:], c["verdict"], c["fixes_sign"])) print("-" * 78) print("ANY principle fixes the sign : %s" % any_fixes) print("OUTCOME : %s" % outcome) print("-" * 78) print("geometry-specific root cause: SU(3)/T^2 zero-weight Casimir is the") print(" MIN-WEIGHTED Eisenstein form (disc -3) on a d_eff=3 (ODD) slice =>") print(" genuine pole at s=-1/2 => finite part scheme-dependent. int_a4>0") print(" (convention-invariant) so the branch rides entirely on int_loop.") print(" I2 supertrace (full inventory) computed NONZERO => fermions don't cancel.") print("artifact:", out_path) print("=" * 78) return 0 if __name__ == "__main__": sys.exit(main())