#!/usr/bin/env python3 # -*- coding: utf-8 -*- r""" gap04_cloop_casimir_firstprinciples.py ====================================== Gap-04 FRG-2 lane, ELEGANT PATH: try to settle the c_loop owner-lock by reading the SIGN of c_loop's UNDERLYING heat-kernel density off a genuine first-principles SCALAR KK-tower Casimir / spectral-zeta computation -- the regularized zero-point sum on K6 x S^2 x S^1_Y whose K6-breathing-mode sigma-dependence yields the e^{-6 sigma} wall coefficient c_loop_Z = +1.3637e-5. A prior countersign (gap04_cloop_density_sign.py) left Gap-04 owner-locked because NO first-principles producer of c_loop was found in gap_04. A later lead flagged two scripts MISSED by that survey: severe_tests_paper/hc1_closure/symbolic/pub1_casimir.py severe_tests_paper/hc1_closure/symbolic/closure4_spectral_zeta.py as candidate scalar-Casimir machinery. This script CHECKS THE OBJECT before trusting it -- exactly the trap the task names: the prior survey's nearest hit (k6_zeta_casimir_insertion.py) was a DIFFERENT object (a Dirac/fermion-bundle LOCAL Seeley-DeWitt coefficient), and a matching "Casimir" name or a matching 1/R^6 (== e^{-6 sigma}) breathing exponent is NOT identity of object. CFCA discipline (METHOD_CFCA_June_14.md): - no-target-loading: NO observed value (A_s, Lambda_obs, r, eta_B, n_s, N_eff, PDG, Omega_DM, H_0, S_8) enters on any input side. The ONLY inputs are the frozen geometry (Einstein constants, dims) and group theory (SU(3) reps). Comparison to c_loop's frozen WALL VALUE happens only AFTER the object check. - NEVER force the favorable (well-stands) branch. If the flagged machinery does not actually execute the scalar KK-tower spectral-zeta sum, the honest outcome is MACHINERY-DIFFERENT-OBJECT-owner-locked -- the elimination is COMPLETE (the lead was checked and does not supply c_loop) but the debit STAYS BARE-COUNTED. We do not manufacture a sign we cannot compute. - elegance is the diagnostic: if the elegant route (read the sign off a real spectral-zeta sum) is blocked because no such sum is executed anywhere, that is the signal the constant is owner-gated, not a wall to brute-force through. ================================================================================ THE OBJECT, STATED PRECISELY (what a REAL c_loop producer must compute) ================================================================================ c_loop is the coefficient of the e^{-6 sigma} term in the Gap-04 FRG-4 V_eff (veff_coefficients_frg4.yaml, operator_exponents.c_loop_Z = "exp(-6 sigma)"). Here sigma is the K6 breathing mode: Vol(K6) = V_K6_0 * exp(+2 sigma), so the K6 curvature / first-Laplace eigenvalue scale as exp(-2 sigma), and a 4D one-loop Casimir energy density built from the K6 KK tower (mass^2 ~ lambda/R_K6^2 ~ exp(-2 sigma)) summed and zeta-regularized gives a leading piece ~ (1/R_K6)^6 = exp(-6 sigma). A genuine producer is therefore: c_loop = -(1/2) * (4 pi)^{-D/2} * zeta_{Delta_K6}(-1/2) [scalar KK tower] where zeta_{Delta_K6}(s) = sum over SU(3) reps (p,q) of deg(p,q) * [ C_2(p,q) / R_K6^2 ]^{-s} is the SCALAR spectral zeta on K6 = SU(3)/T^2 (eigenvalues = quadratic Casimir of the SU(3) reps appearing in L^2(SU(3)/T^2), degeneracy = dim of the rep), and the SIGN OF ITS UNDERLYING DENSITY is sign[ zeta_{Delta_K6}(-1/2) ] BEFORE the universal -(1/2) bosonic one-loop prefactor. * If that underlying density (zeta(-1/2)) is NEGATIVE -> under the shared global -(1/2) the positive-density a_4 cross-term flips relative sign -> c_a4 > 0 -> BRANCH-STANDS (the -sigma FRG-4 well stands). * If POSITIVE -> a_4 stays negative -> BRANCH-RUNAWAY (a_4 growth-8 dominates the c_loop growth-6 wall at -sigma -> V -> -inf). To READ OFF that sign honestly we need the zeta(-1/2) of the ACTUAL K6 scalar tower ACTUALLY SUMMED and ACTUALLY analytically continued. This script builds the tower data-blind to (i) define the object concretely and (ii) test whether the flagged candidates execute this very sum. ================================================================================ OBJECT CHECK of the flagged candidates (the decisive step) ================================================================================ We parse pub1_casimir.py and closure4_spectral_zeta.py and test, mechanically, whether either one EXECUTES the scalar KK-tower spectral-zeta sum defined above: (C1) loops over the SU(3) rep tower (p,q) building C_2(p,q) and degeneracies; (C2) forms zeta_{Delta_K6}(s) and analytically continues to s=-1/2; (C3) derives the K6 coefficient FROM that continuation (not a typed-in rational); (C4) isolates the SCALAR breathing-mode e^{-6 sigma} piece as a single term (not a boson+fermion species-counted flux-balance stabilization radius). If any of C1-C4 fails, the candidate does NOT produce the scalar c_loop wall and its K6 number cannot supply the underlying-density sign. 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 re import sys from fractions import Fraction # --------------------------------------------------------------------------- # Paths to the flagged candidate machinery (read-only; object check). # --------------------------------------------------------------------------- SYM = (r"" r"physics_Journal_and_patents/severe_tests_paper/hc1_closure/symbolic") PUB1 = os.path.join(SYM, "pub1_casimir.py") CLOSURE4 = os.path.join(SYM, "closure4_spectral_zeta.py") K6ZETA = os.path.join(SYM, "k6_zeta_casimir_insertion.py") # prior-survey near-hit # Gap-04 frozen artifacts (for the c_loop WALL value -- read ONLY after the # object check, and only the geometry/coefficient, never an observed anchor). FA = (r"" r"physics_Journal_and_patents/Final_physics_articles/scripts/gap_04") VEFF_YAML = os.path.join(FA, "outputs", "veff_coefficients_frg4.yaml") GILKEY_JSON = os.path.join(FA, "outputs", "gilkey_a4_cross_terms.json") # --------------------------------------------------------------------------- # FROZEN inputs (geometry + group theory ONLY; no observed value). # --------------------------------------------------------------------------- R_K6_0 = 30.0 # scalar curvature * unit vol, K6 = SU(3)/T^2 (6 * Einstein +5) R_S2_0 = 2.0 # round S^2 (2 * Einstein +1) A4_R2_MASTER = Fraction(5, 360) # convention-invariant Gilkey a_4 R^2 coeff A4_CROSS_PREFACTOR = 2 * A4_R2_MASTER # = 1/36 D_BULK = 13 HK_NORM = 1.0 / (math.pow(4.0 * math.pi, D_BULK / 2.0)) # (4 pi)^{-D/2} > 0 GLOBAL_ONE_LOOP = -0.5 # universal bosonic one-loop magnitude 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() # =========================================================================== # PART A. Build the SCALAR K6 = SU(3)/T^2 KK tower data-blind (the OBJECT). # =========================================================================== # L^2(SU(3)/T^2) decomposes into SU(3) irreps; for a scalar the Laplace # eigenvalue on the homogeneous space is the quadratic Casimir C_2(p,q) of the # irrep (in the normalization where the adjoint (1,1) has C_2 = ... ), with # multiplicity = dim(p,q) times the count of T^2-invariant (zero-weight) vectors. # We use the standard SU(3) Casimir and Weyl dimension formulas. This block # DEFINES the object; we do NOT need the full continuation to make the object # DECISION (the candidates do not even reach this loop), but we execute enough # of it to show what a real producer's inner loop looks like. def su3_dim(p, q): """Weyl dimension of the SU(3) irrep with Dynkin labels (p, q).""" return (p + 1) * (q + 1) * (p + q + 2) // 2 def su3_casimir(p, q): """Quadratic Casimir C_2(p,q) of SU(3) (standard normalization, adjoint=3).""" # C_2 = (1/3)(p^2 + q^2 + p q) + (p + q) [normalization with C_2(1,0)=4/3, # C_2(1,1)=3 for the adjoint] return Fraction(p * p + q * q + p * q, 3) + (p + q) def t2_zero_weight_multiplicity(p, q): """ Number of zero-weight (T^2-invariant) states in the SU(3) irrep (p,q): the multiplicity of the scalar KK mode of that rep on SU(3)/T^2. For SU(3) the zero-weight multiplicity of (p,q) equals min(p,q)+1 when (p-q) % 3 == 0, else 0 (only reps with triality 0 contain a zero weight). """ if (p - q) % 3 != 0: return 0 return min(p, q) + 1 def build_scalar_k6_tower(n_max): """ Build the first chunk of the SCALAR K6 spectral data: list of (lambda_unit = C_2(p,q), degeneracy = mult * dim-on-coset). Returns the partial spectral data; this is the inner object a genuine zeta_{Delta_K6}(s) sum would continue. We expose it so the object is concrete, not rhetorical. """ tower = [] for p in range(n_max + 1): for q in range(n_max + 1): mult = t2_zero_weight_multiplicity(p, q) if mult == 0: continue lam = su3_casimir(p, q) # eigenvalue (in 1/R_K6^2 units) if lam == 0: continue # zero mode (the (0,0) trivial rep) deg = mult # scalar coset KK degeneracy tower.append((p, q, float(lam), deg)) return tower def partial_zeta_K6(tower, s): """ Partial (non-continued) spectral zeta sum sum_n deg_n * lambda_n^{-s}. For Re(s) large this converges; at s = -1/2 it DIVERGES and REQUIRES analytic continuation (Mellin / Epstein-Hurwitz on the SU(3) Casimir lattice) -- which is exactly the multi-week object NOT executed by the candidates. We compute it at a convergent s only to witness the tower is real and the continuation is nontrivial. """ total = 0.0 for (_p, _q, lam, deg) in tower: total += deg * (lam ** (-s)) return total # =========================================================================== # PART B. OBJECT CHECK: do the flagged candidates EXECUTE this sum? # =========================================================================== def candidate_object_check(src_text): """ Mechanically test C1-C4 against a candidate's source. Returns dict of booleans + evidence. """ # C1: loops over SU(3) rep tower building C_2(p,q) and degeneracies. # Require an actual for-loop over rep labels AND a Casimir computation # in code (not merely a comment). Comments containing "C_2(p,q)" do NOT # count. code_lines = [] for ln in src_text.splitlines(): stripped = ln.split("#", 1)[0] # drop trailing comments code_lines.append(stripped) code = "\n".join(code_lines) has_rep_loop = bool( re.search(r"for\s+\(?\s*p\s*,\s*q", code) # for p, q in ... or re.search(r"for\s+p\s+in\s+range.*\n.*for\s+q\s+in\s+range", code) ) computes_casimir_in_code = bool( re.search(r"(C_?2|casimir).*=.*(p|q)", code, re.IGNORECASE) and ("p*p" in code or "p**2" in code or "p * p" in code or "p*q" in code or "p * q" in code) ) C1 = has_rep_loop and computes_casimir_in_code # C2: forms a spectral zeta zeta_K6(s) and continues to s = -1/2. # Require either an explicit zeta(-0.5)/zeta(-1/2) continuation on the K6 # tower, or use of a Riemann/Hurwitz/Epstein zeta continuation on K6. forms_zeta_K6 = bool(re.search(r"zeta_K6\s*\(", code)) continues_minus_half = ( "-1/2" in code or "-0.5" in code or "(-0.5)" in code ) and forms_zeta_K6 uses_zeta_continuation = bool( re.search(r"(special\.)?zeta\s*\(", code) or "mpmath" in code or "zetac" in code or "hurwitz" in code.lower() ) C2 = (forms_zeta_K6 and continues_minus_half) or ( forms_zeta_K6 and uses_zeta_continuation) # C3: derives the K6 coefficient FROM a continuation (not a typed-in rational). # Detect the tell-tale hardcoded rationals 1/945, 31/15120 used as the K6 # coefficient; if present AND no continuation, C3 fails. hardcoded_k6 = ("945" in code) or ("15120" in code) C3 = (not hardcoded_k6) and (C1 or uses_zeta_continuation) # C4: isolates the SCALAR breathing-mode e^{-6 sigma} piece as a SINGLE # scalar term -- NOT a boson+fermion species-counted multi-factor # flux-balance radius minimization. Detect the flux-balance signature. flux_balance = bool( re.search(r"minimize\s*\(", code) or "V_flux" in code or "V_full" in code or "R_K6_stab" in code ) species_counted = ("N_f" in code) or ("N_b" in code) or ("c_K6_spinor" in code) isolates_scalar_wall = (not flux_balance) and (not species_counted) C4 = isolates_scalar_wall return { "C1_loops_su3_rep_tower_and_computes_C2_in_code": C1, "C1_has_rep_loop": has_rep_loop, "C1_computes_casimir_in_code": computes_casimir_in_code, "C2_forms_and_continues_zeta_K6_to_minus_half": C2, "C2_forms_zeta_K6": forms_zeta_K6, "C2_uses_zeta_continuation": uses_zeta_continuation, "C3_derives_K6_coeff_from_continuation_not_typed_rational": C3, "C3_hardcoded_1over945_or_31over15120_present": hardcoded_k6, "C4_isolates_scalar_e_minus_6sigma_wall_single_term": C4, "C4_is_flux_balance_radius_minimization": flux_balance, "C4_is_boson_fermion_species_counted": species_counted, "produces_scalar_c_loop_wall": bool(C1 and C2 and C3 and C4), } def read_c_loop_wall(veff_text): for ln in veff_text.splitlines(): s = ln.strip() if s.startswith("c_loop_Z:"): try: return float(s.split(":", 1)[1].split("#")[0].strip()) except ValueError: return None return None def main(): # ---- locate inputs (refuse if missing) -------------------------------- for p in (PUB1, CLOSURE4, VEFF_YAML, GILKEY_JSON): if not os.path.exists(p): sys.stderr.write("REFUSE(exit2): missing input %s\n" % p) return 2 pub1_src = open(PUB1, "r", encoding="utf-8").read() closure4_src = open(CLOSURE4, "r", encoding="utf-8").read() veff_text = open(VEFF_YAML, "r", encoding="utf-8").read() gilkey = json.load(open(GILKEY_JSON, "r", encoding="utf-8")) # forbidden-value firewall on everything we read. for name, txt in (("pub1_casimir.py", pub1_src), ("closure4_spectral_zeta.py", closure4_src), ("veff_coefficients_frg4.yaml", veff_text)): 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" % (name, leaked)) return 2 # ===================================================================== # PART A. Build the scalar K6 tower (the OBJECT, data-blind). # ===================================================================== tower = build_scalar_k6_tower(n_max=12) # Witness the tower is real and the continuation is nontrivial: at a # convergent s the partial sum is finite & positive; at s=-1/2 it diverges # (so a genuine producer MUST analytically continue -- the deferred step). z_at_2 = partial_zeta_K6(tower, s=2.0) # convergent witness n_modes_listed = len(tower) smallest_modes = sorted(tower, key=lambda r: r[2])[:6] # ===================================================================== # PART B. OBJECT CHECK on the flagged candidates. # ===================================================================== pub1_check = candidate_object_check(pub1_src) closure4_check = candidate_object_check(closure4_src) pub1_is_object = pub1_check["produces_scalar_c_loop_wall"] closure4_is_object = closure4_check["produces_scalar_c_loop_wall"] any_candidate_is_object = pub1_is_object or closure4_is_object # ===================================================================== # AFTER the object check: read the c_loop WALL value (geometry side only). # ===================================================================== c_loop_wall = read_c_loop_wall(veff_text) c_a4_on_disk = float( gilkey["a4_cross_terms"]["K6_x_S2"]["coefficient_value_M13_4_units"]) # ===================================================================== # THE HONEST RESOLUTION (do NOT force a sign / a branch). # ===================================================================== if any_candidate_is_object: # Only reachable if a candidate ACTUALLY executes the scalar KK-tower # spectral-zeta sum and continues zeta_{Delta_K6}(-1/2). It does not # (proven below by C1-C4). Kept honest, never forced. outcome = "DERIVED-via-flagged-machinery (unreachable: candidates do " outcome += "not execute the sum)" machinery_applies = ("YES -- a flagged candidate executed the scalar " "KK-tower spectral-zeta sum") density_sign = "(would be read from zeta_{Delta_K6}(-1/2))" branch = "(decided by that sign)" debit = "DERIVED" else: outcome = "MACHINERY-DIFFERENT-OBJECT-owner-locked" machinery_applies = ( "NO. pub1_casimir.py and closure4_spectral_zeta.py do NOT compute " "the scalar KK-Casimir wall that produces the e^{-6 sigma} c_loop " "coefficient. They are a DIFFERENT object: a multi-factor " "boson+fermion species-counted flux-balance RADIUS-STABILIZATION " "(minimize c_K6/R^6 + c_S2/R^2 + c_S1/R + flux), with the K6 " "coefficient TYPED IN as a self-disclaimed rational (c_K6_scalar = " "1/945 'estimated'; c_K6_spinor = -31/15120 'rough estimate'). " "Neither loops over the SU(3) rep tower (no C_2(p,q) computed in " "code -- it appears only in a comment), neither forms zeta_{Delta_K6}" "(s) nor continues it to s=-1/2, and neither isolates the single " "scalar e^{-6 sigma} wall term. The 1/R_K6^6 == e^{-6 sigma} " "breathing exponent coincides, but exponent-match is NOT " "object-identity (c_a4 e^{-(8s+4r+x)} and c_KK e^{-4s} also scale " "as breathing powers and are different objects).") density_sign = "GATED-ON-OPEN (UNBANKED -- not supplied by the flagged machinery)" branch = "still-owner-locked" debit = ("STAYS BARE-COUNTED (the relative-one-loop-sign debit in " "gap04_reason_hunt.py is NOT promoted to DERIVED: the flagged " "machinery was checked and does NOT supply the scalar c_loop " "underlying-density sign)") result = { "schema": "gap04_cloop_casimir_firstprinciples_result_v1", "purpose": ( "Try the elegant path: read the sign of c_loop's underlying " "heat-kernel density off a genuine first-principles scalar KK-tower " "spectral-zeta Casimir computation; FIRST verify the flagged " "machinery (pub1_casimir.py / closure4_spectral_zeta.py) is that " "object."), "outcome": outcome, "machinery_applies": machinery_applies, "c_loop_underlying_density_sign": density_sign, "branch_selected": branch, "debit_reclassification": debit, "object_definition_the_real_producer_must_compute": { "c_loop_operator": "c_loop_Z * exp(-6 sigma) in V_eff (veff_coefficients_frg4.yaml)", "sigma_is": "K6 breathing mode: Vol(K6)=V_K6_0*exp(+2 sigma) -> R_K6 ~ exp(-2 sigma) -> Casimir ~ (1/R_K6)^6 = exp(-6 sigma)", "producer": "c_loop = -(1/2)*(4 pi)^{-D/2}*zeta_{Delta_K6}(-1/2) [scalar KK tower]", "zeta_K6": "sum_(p,q) deg(p,q) * [C_2(p,q)/R_K6^2]^{-s}, deg from T^2-zero-weight mult, eigenvalue = SU(3) Casimir", "underlying_density_sign_is": "sign[ zeta_{Delta_K6}(-1/2) ] BEFORE the -(1/2) prefactor", }, "part_A_scalar_K6_tower_built_data_blind": { "n_max": 12, "n_triality0_modes_listed": n_modes_listed, "smallest_modes_(p,q,C2,deg)": smallest_modes, "partial_zeta_K6_at_s=2_convergent_witness": z_at_2, "note": ( "The tower is real and the eigenvalues are the SU(3) Casimirs of " "triality-0 reps (the (1,1) adjoint at C_2=3 is the first nonzero " "scalar KK level). At s=-1/2 this sum DIVERGES and REQUIRES " "Epstein-Hurwitz / Mellin analytic continuation on the SU(3) " "Casimir lattice -- precisely the multi-week object the candidates " "explicitly DEFER ('Full spectral zeta function ... 2-4 week " "dedicated calculation'). We did NOT continue it here because no " "data-blind continuation produces the FROZEN inherited c_loop " "magnitude without the owner's FRG-2 NLO shell-projection scheme; " "asserting a sign from a half-built continuation would be the " "manufactured favorable-branch step the countersign exists to catch."), }, "part_B_object_check_of_flagged_candidates": { "pub1_casimir.py": pub1_check, "closure4_spectral_zeta.py": closure4_check, "decisive_findings": [ "C1 FAILS both: no for-loop over the SU(3) (p,q) rep tower with a " "C_2(p,q) computed in code; 'C_2(p,q)' appears only in a docstring " "comment.", "C2 FAILS both: neither forms zeta_{Delta_K6}(s) nor continues to " "s=-1/2; no Riemann/Hurwitz/Epstein zeta continuation is invoked.", "C3 FAILS both: the K6 coefficient is the typed-in self-disclaimed " "rational (1/945 'estimated', 31/15120 'rough estimate'), NOT a " "continuation output.", "C4 FAILS both: each is a boson+fermion species-counted " "(N_b, N_f=45) multi-factor flux-balance RADIUS minimization " "(minimize c_K6/R^6 + c_S2/R^2 + c_S1/R + V_flux), not a single " "isolated scalar e^{-6 sigma} wall term.", ], }, "numerical_disjointness_after_object_check": { "candidate_c_K6_total_sign_required_for_their_flux_balance": "NEGATIVE (they require c_K6<0 to stabilize)", "candidate_c_K6_total_approx": "~ -0.08 (12*(1/945) - 45*(31/15120))", "c_loop_wall_value": c_loop_wall, "c_loop_wall_sign": "POSITIVE (+1.3637e-5)", "disjoint": ( "Different SIGN, different MAGNITUDE (candidate ~ -8e-2 vs c_loop " "+1.4e-5, 3+ orders apart), different DEFINITION (combined " "boson+fermion species count vs FRG-2 NLO Litim-regulator scalar " "shell projection). The candidate K6 number is NOT c_loop."), }, "prior_survey_nearest_hit_reconfirmed_different": ( "k6_zeta_casimir_insertion.py remains a DIFFERENT object: it closes " "only the LOCAL Dirac/fermion-bundle Seeley-DeWitt a_4 coefficient at " "the Einstein point and explicitly 'does not claim to compute the full " "finite nonlocal determinant'. It is neither the scalar KK-Casimir " "wall nor the flux-balance stabilizer."), "object_check": ( "The flagged machinery computes a fermion-and-boson species-counted " "multi-factor flux-balance radius stabilization with a TYPED-IN, " "self-disclaimed K6 rational (1/945, 31/15120) -- NOT the scalar " "KK-Casimir wall (e^{-6 sigma}, the breathing-mode zero-point sum). " "k6_zeta_casimir_insertion.py is a third, also-different object (local " "Dirac-bundle Seeley-DeWitt coefficient). None executes the scalar " "zeta_{Delta_K6}(-1/2) sum whose sign decides the branch."), "well_verdict": ( "OWNER-LOCKED / CONDITIONAL, UNCHANGED -- but the elimination is now " "COMPLETE: the two flagged scalar-Casimir candidates (and the prior " "near-hit) have been checked and do NOT supply the scalar c_loop " "underlying-density sign. The -sigma FRG-4 3-modulus well's survival " "stays gated on the named OPEN first-principles object " "B-UQFC-14-FRG-2. The favorable branch was NOT forced (the genuine " "global -(1/2) on a_4's own positive density gives c_a4 NEGATIVE " "= %.3e -> runaway); the well is NOT forced to fail (the relative " "sign vs c_loop is unbanked). Both branches stay live with named " "falsifiers." ) % c_a4_on_disk, "what_requires_chris": ( "A REAL first-principles c_loop producer must: (1) enumerate the " "SCALAR K6=SU(3)/T^2 KK tower in CODE -- loop over SU(3) reps (p,q) of " "triality 0, eigenvalue = C_2(p,q)/R_K6^2, degeneracy = T^2-zero-weight " "multiplicity (built data-blind here in Part A); (2) form the spectral " "zeta zeta_{Delta_K6}(s) = sum deg*lambda^{-s} and ANALYTICALLY CONTINUE " "it (Epstein-Hurwitz / Mellin on the SU(3) Casimir lattice, MS-bar " "scheme matching the FRG-2 Litim regulator) to s = -1/2; (3) apply the " "universal -(1/2)(4 pi)^{-D/2} one-loop prefactor and READ OFF " "sign[zeta_{Delta_K6}(-1/2)] = the underlying-density sign; (4) confirm " "it reproduces the frozen c_loop magnitude (1.3637877e-5) under the " "FRG-2 NLO shell-projection scheme, closing B-UQFC-14-FRG-2. None of " "the flagged machinery does steps 1-3; the K6 zeta(-1/2) continuation " "is the deferred multi-week object."), "provenance_hashes": { "pub1_casimir.py": sha256_file(PUB1), "closure4_spectral_zeta.py": sha256_file(CLOSURE4), "k6_zeta_casimir_insertion.py": sha256_file(K6ZETA), "veff_coefficients_frg4.yaml": sha256_file(VEFF_YAML), "gilkey_a4_cross_terms.json": sha256_file(GILKEY_JSON), }, "frozen_inputs_used": { "R_K6_0": R_K6_0, "R_S2_0": R_S2_0, "a4_cross_prefactor_1_over_36": float(A4_CROSS_PREFACTOR), "D_bulk": D_BULK, "factor_4pi_to_minus_D_over_2": HK_NORM, "global_one_loop_prefactor": GLOBAL_ONE_LOOP, "c_loop_wall_value_read_after_object_check": c_loop_wall, }, "no_target_loading": True, "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). The scalar K6 tower in " "Part A is built from the frozen geometry (Einstein constant +5, dim 6) " "and SU(3) group theory ALONE. The c_loop WALL value was read ONLY " "after the object check, and only as a geometry-side coefficient for " "the disjointness witness -- never as an input to a derivation. The " "favorable (well-stands) branch was NOT forced: no sign was asserted " "for zeta_{Delta_K6}(-1/2); the well is left owner-locked because the " "deciding object is not supplied by the flagged machinery."), "non_promotion": ( "no gate flipped; no status word emitted for any gate; " "countersign-ready object-mismatch analysis only."), } 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_cloop_casimir_firstprinciples_result.json") with open(out_path, "w", encoding="utf-8") as fh: json.dump(result, fh, indent=2) # ---- decision-grade packet to stdout ---------------------------------- print("=" * 74) print("Gap-04 c_loop FIRST-PRINCIPLES scalar KK-Casimir attempt (FRG-2 lane)") print("=" * 74) print("OBJECT (real producer): c_loop = -(1/2)(4pi)^{-D/2} zeta_{Delta_K6}(-1/2)") print(" sigma = K6 breathing mode -> Casimir ~ (1/R_K6)^6 = exp(-6 sigma)") print("-" * 74) print("PART A scalar K6=SU(3)/T^2 tower built data-blind:") print(" triality-0 KK modes listed (n_max=12): %d" % n_modes_listed) print(" first nonzero levels (p,q,C2,deg): %s" % ", ".join("(%d,%d;%.3g;x%d)" % (p, q, c, d) for (p, q, c, d) in smallest_modes[:4])) print(" partial zeta_K6(s=2) convergent witness: %.6g" % z_at_2) print(" zeta_K6(-1/2): DIVERGES -> requires Epstein-Hurwitz continuation") print(" (the deferred multi-week object; NOT executed here") print(" and NOT in the candidates)") print("-" * 74) print("PART B object check of flagged candidates:") for nm, chk in (("pub1_casimir.py", pub1_check), ("closure4_spectral_zeta.py", closure4_check)): print(" %s produces scalar c_loop wall: %s" % (nm, chk["produces_scalar_c_loop_wall"])) print(" C1 rep-tower+C2 in code: %s | C2 zeta_K6 continued: %s | " "C3 not-typed-rational: %s | C4 isolated scalar wall: %s" % (chk["C1_loops_su3_rep_tower_and_computes_C2_in_code"], chk["C2_forms_and_continues_zeta_K6_to_minus_half"], chk["C3_derives_K6_coeff_from_continuation_not_typed_rational"], chk["C4_isolates_scalar_e_minus_6sigma_wall_single_term"])) print("-" * 74) print("c_loop wall value (read AFTER object check): %s [POSITIVE]" % c_loop_wall) print("candidate c_K6_total ~ -0.08 [NEGATIVE, 3+ orders larger] -> DISJOINT") print("-" * 74) print("OUTCOME : %s" % outcome) print("MACHINERY APPLIES: %s" % ("NO -- different object" if not any_candidate_is_object else "YES")) print("DENSITY SIGN: %s" % density_sign) print("BRANCH : %s" % branch) print("DEBIT : %s" % ("STAYS BARE-COUNTED" if not any_candidate_is_object else "DERIVED")) print("-" * 74) print("ELIMINATION COMPLETE: flagged scalar-Casimir machinery checked and does") print(" NOT supply c_loop. Real producer requires the K6 zeta(-1/2)") print(" continuation (steps 1-4 in result.what_requires_chris) =") print(" named open object B-UQFC-14-FRG-2 (owner = Chris).") print("artifact:", out_path) print("=" * 74) return 0 if __name__ == "__main__": sys.exit(main())