#!/usr/bin/env python3 # -*- coding: utf-8 -*- r""" c_loop_NLO_match.py =================== THE NAMED PRODUCER. The FRG-2 NLO Litim shell-projection producer of the finite c_loop (the e^{-6 sigma} KK-Casimir wall coefficient) that the corpus REFERENCES (z_renormalized_c_loop.py's RETAINED_MODES comment: "matches c_loop_NLO_match.py spectrum") but does NOT contain. This file BUILDS that producer from first principles, target-blind, on canonical literature-justified grounds. It supersedes the partial reconstruction in gap04_litim_scheme_branch.py by constructing the actual NLO Litim THRESHOLD-FUNCTION shell projection over the five FRG-2 retained-mode shell masses (not the unbounded bare K6 Casimir tower), and it carries the relative-one-loop-sign analysis to its honest endpoint. ================================================================================ THE INTEGRITY CRUX (read twice) ================================================================================ There is a KNOWN TARGET the producer "should" reproduce: c_loop = +1.3637877e-5. CHOOSING O1/O2/O3 TO HIT THAT TARGET IS REVERSE-FITTING AND IS FORBIDDEN. Every O1/O2/O3 choice below is made on PRINCIPLED, canonical, literature-justified grounds (Litim 2001 PRD 64 105007 threshold functions l_n^d; the canonical FRG effective-potential / heat-kernel one-loop insertion; the corpus-pinned matching scale). All choices are COMMITTED in STEPS 1-3 BEFORE the target value is read. The target enters ONLY at STEP 4 as an INDEPENDENT validator. A sign is DERIVED-validated ONLY IF a non-reverse-fit principled construction BOTH gives the sign AND reproduces the magnitude (within the factor-2 fold band). NEVER force the favorable BRANCH-STANDS. -------------------------------------------------------------------------------- PROVENANCE / WHY THIS REPLACES AN EARLIER DRAFT -------------------------------------------------------------------------------- An earlier c_loop_NLO_match.py asserted that the relative-sign / BRANCH-STANDS reading is "DERIVED and does NOT require Chris," claiming to BREAK the owner-lock the corpus explicitly recorded. A held countersign and an independent rebuild found that claim does NOT hold, on four grounds documented below (integration direction skipped; finite part scheme-dependent; prior producer read the SAME scheme as RUNAWAY; the no-flip convention is selected by reference to the stored target's known + sign, which is circular). This version keeps the EXEMPLARY magnitude handling (honest, non-reverse-fit) and CORRECTS the sign claim to the corpus's standing owner-must-rule position. See STEP 3. ================================================================================ THE THREE OWNER-NAMED PIECES -- PRINCIPLED CHOICES (committed target-blind) ================================================================================ O1 SPECTRAL CUTOFF / SHELL-PROJECTION SCALE k. CHOICE: k = mu_match = M_13 (the corpus matching scale). In dimensionless M_13^2 units k^2 = 1, so the Litim threshold variable per retained mode is w_i = m_i^2 / k^2 = m_i^2 (m_i^2 the FRG-2 LO shell masses). JUSTIFICATION (corpus-pinned, NOT target-tuned): z_renormalized_c_loop.py evaluates every Z_i(mu_match) at EXACTLY mu_match = M_13, and the veff coefficient c_loop_Z is reported "at mu_match". The shell projection that feeds that re-weighting is therefore the projection AT k = mu_match. This is the ONLY scale the corpus actually fixes. The shell projection acts on the FIVE RETAINED-MODE shell masses (sigma_K6, rho_S2, chi_Y, theta_W, b_orb_sc) -- a FINITE set fixed by FRG-2 LO geometry/group theory -- NOT on the bare K6 Casimir tower (whose Litim sum is UNBOUNDED in the cutoff, the gap04_litim_scheme_branch demonstration; and whose smallest eigenvalue C2 >= 2 is never retained at the frozen k_R_K6^2 = 0.01). Using the finite retained-mode shell -- not the divergent tower -- is what makes the NLO shell projection a single finite number at fixed k, and it is corpus-read (the same five modes z_renorm uses), not chosen to hit 1.36e-5. O2 NLO THRESHOLD FUNCTION / FINITE SUBTRACTION. CHOICE: the canonical Litim optimised BOSONIC threshold function for the effective-potential (vacuum-energy) flow, l_0^d(w; eta) = (2/d) * (1 - eta/(d+2)) * 1/(1 + w), assembled into the one-loop vacuum-energy coefficient via the canonical FRG effective-potential-flow prefactor c_d = (1/2) * (4 pi)^{-d/2} / Gamma(d/2 + 1) ( == 2 v_d / d ), so that c_loop = c_d * sum_i g_i * (1 - eta/(d+2)) / (1 + w_i) = (1/2)(4 pi)^{-d/2} * sum_i g_i * l_0^d(w_i; eta). JUSTIFICATION (Litim 2001; canonical FRG EP flow): l_0^d is THE optimised dimensionless scalar threshold function; v_d = [2^{d+1} pi^{d/2} Gamma(d/2)]^{-1} is the standard FRG loop-momentum measure; c_d = 2 v_d/d is the standard EP-flow prefactor. The identity 2 v_d/d == (1/2)(4pi)^{-d/2}/Gamma(d/2+1) is VERIFIED in code so the (2/d) of l_0 and the 1/Gamma of c_d are not double-counted. DIMENSION d: the shell projection runs over the COMPACT internal space, so d = total_compact_dimension = 9 (K6 6 + S^2 2 + S^1_Y 1; frozen_inputs.yaml geometry.total_compact_dimension = 9). HONESTY GUARD (per the held countersign): d = 9 is a SELECTION, not a derivation -- the corpus's OWN gilkey/zeta producers use the BULK (4 pi)^{-13/2} one-loop prefactor for the 13D log-det. And the threshold- function ASSEMBLY (where the (2/d) and the 1/Gamma sit) is itself ambiguous at fixed d. We therefore REPORT the d=13 alternative and the four internally- canonical assembly placements as an explicit AMBIGUITY BAND, and we do NOT claim a unique canonical magnitude. O3 NLO RUNNING. CHOICE: eta = 1/(24 pi^2) per real scalar (the corpus C_ETA_LITIM), entering the threshold via the canonical (1 - eta/(d+2)) Litim NLO factor. JUSTIFICATION: Litim 2001 PRD 64 105007 Eq. (38); hard-coded in z_renormalized_c_loop.py. Sub-percent correction; agreed. ================================================================================ THE SIGN (the load-bearing Gap-04 deliverable) -- honest endpoint ================================================================================ The branch (BRANCH-STANDS vs BRANCH-RUNAWAY) is decided by the RELATIVE one-loop sign of c_a4 vs the c_loop wall, which equals sign(int_a4)/sign(int_loop), where int_loop is the SIGN OF c_loop's UNDERLYING heat-kernel density. * int_a4 = +(1/36) R_K6 R_S2 with R_K6 = +30, R_S2 = +2 -> POSITIVE, convention-invariant, manifest (gilkey_a4_cross_terms.py). * int_loop (the sign of c_loop's underlying density) is NOT banked anywhere in the corpus -- gilkey_a4_cross_terms.py records this EXPLICITLY as the owner-must-rule item (the CONVENTION_FREEZE slot for the a_4 sector is unfilled). It is NOT pinned by the principled construction. FOUR independent reasons the construction does NOT fix int_loop's sign: (i) INTEGRATION DIRECTION. The Wetterich EP flow partial_t u_k > 0 is the INSTANTANEOUS flow (bosonic, manifestly positive). But the accumulated vacuum-energy contribution integrates UV -> IR, i.e. t = log k DECREASING (dt < 0), so the accumulated DELTA-u carries the OPPOSITE sign to partial_t u. "partial_t u > 0" therefore does NOT fix the accumulated- density sign. (This is the specific flaw in the earlier draft.) (ii) SCHEME DEPENDENCE OF THE FINITE PART. The bare spectral zeta has a POLE at s = -1/2 (gap04_zeta_continuation_frg2.py): the residue sign is scheme-invariant (NEGATIVE) but the FINITE PART that actually sets the wall flips sign with the renormalization scale mu (at mu* ~ 0.061). The wall-setting sign is genuinely scheme-dependent. (iii) PRIOR-RECONSTRUCTION DISAGREEMENT. gap04_litim_scheme_branch.py read the SAME Litim scheme family's leading order as BRANCH-RUNAWAY (the opposite, UNFAVORABLE branch). Two corpus reconstructions already disagree with a STANDS reading. (iv) CIRCULARITY. Selecting "the self-consistent convention is the one in which stored c_loop = +1.36e-5 needs no -(1/2) flip" uses the target's KNOWN + sign to fix the construction -- a no-target-loading violation ON THE SIGN side. Forbidden. ENDPOINT: the relative sign is CONSTRUCTION/CONVENTION-DEPENDENT. Equally principled readings give opposite branches (int_loop > 0 -> RUNAWAY; int_loop < 0 -> STANDS). This is exactly the owner-must-rule situation the corpus recorded. We default HOLD; we do NOT force BRANCH-STANDS and we do NOT claim the sign is derived. 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. This file is STAGED here; promotable to the corpus gap_04/src ONLY if validated + owner-signed. Validation FAILS (magnitude mismatch + sign owner-locked), so it is NOT promotable as a sign-derivation. """ import hashlib import json import math import os import sys # --------------------------------------------------------------------------- # Paths. The frozen magnitude is read ONLY at STEP 4 (post-hoc validator). # --------------------------------------------------------------------------- FA = (r"" r"physics_Journal_and_patents/Final_physics_articles/scripts/gap_04") FROZEN_YAML = os.path.join(FA, "frozen_inputs.yaml") ZRENORM_SRC = os.path.join(FA, "src", "z_renormalized_c_loop.py") GILKEY_SRC = os.path.join(FA, "src", "gilkey_a4_cross_terms.py") VEFF_YAML = os.path.join(FA, "outputs", "veff_coefficients_frg4.yaml") PI = math.pi FOUR_PI_SQ = (4.0 * PI) ** 2 D_BULK = 13 # full bulk dimension (gilkey/zeta one-loop prefactor) D_COMPACT = 9 # K6(6) + S^2(2) + S^1_Y(1); the shell-projection dim GLOBAL_ONE_LOOP = -0.5 # Frozen FRG-2 c_loop magnitude -- VALIDATOR target ONLY (read at STEP 4). C_LOOP_FRG2_TARGET = 1.3637877214788921e-05 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() # =========================================================================== # Retained shell masses -- the FIVE FRG-2 LO retained modes, read off # z_renormalized_c_loop.py's RETAINED_MODES seeds and recovered as EXACT # rationals over (4 pi)^2. These VALUES are the FRG-2 LO coefficients # (geometry/group-forced), fixed BEFORE any c_loop value is seen. # =========================================================================== SEED_SIGMA = 0.009498860966469166 # -> 3/2 / (4 pi)^2 (c_KK universality) SEED_THETA = 4.875756906074309e-05 # -> Wilson harmonic / (4 pi)^2 SEED_BORB = 0.012665147955292222 # -> 2 / (4 pi)^2 (|c_bdry|) RETAINED_MODES = [ {"name": "sigma_K6", "g_dof": 1, "m2": SEED_SIGMA}, {"name": "rho_S2", "g_dof": 1, "m2": SEED_SIGMA}, {"name": "chi_Y", "g_dof": 1, "m2": SEED_SIGMA}, {"name": "theta_W", "g_dof": 1, "m2": SEED_THETA}, {"name": "b_orb_sc", "g_dof": 1, "m2": SEED_BORB}, ] ETA_LITIM = 1.0 / (24.0 * PI * PI) # O3 (Litim 2001 Eq. 38) # =========================================================================== # O2 building blocks -- Litim threshold + canonical EP-flow prefactor. # =========================================================================== def v_d(d): """Standard FRG loop-momentum measure v_d = [2^{d+1} pi^{d/2} Gamma(d/2)]^{-1}.""" return 1.0 / (2.0 ** (d + 1) * PI ** (d / 2.0) * math.gamma(d / 2.0)) def c_d_EPflow(d): """Canonical FRG effective-potential-flow prefactor c_d = 2 v_d / d == (1/2)(4 pi)^{-d/2} / Gamma(d/2 + 1).""" return 2.0 * v_d(d) / d def l0_litim(w, d, eta): """Litim optimised bosonic threshold function l_0^d(w; eta) = (2/d)(1 - eta/(d+2)) / (1 + w) (Litim 2001).""" return (2.0 / d) * (1.0 - eta / (d + 2.0)) / (1.0 + w) def c_loop_principled(d, eta): """The PRINCIPLED FRG-2 NLO Litim shell-projection c_loop (committed O1/O2/O3): c_loop = (1/2)(4 pi)^{-d/2} * sum_i g_i * l_0^d(w_i; eta) = c_d * sum_i g_i * (1 - eta/(d+2)) / (1 + w_i), w_i = m_i^2. Returns (c_loop, threshold_core_sum).""" half_pref = 0.5 * (4.0 * PI) ** (-d / 2.0) s = sum(m["g_dof"] * l0_litim(m["m2"], d, eta) for m in RETAINED_MODES) return half_pref * s, s def assembly_band(d, eta): """Four internally-canonical threshold-assembly placements at FIXED d, used to expose the assembly ambiguity HONESTLY (per the held countersign). Each is a defensible reading of "where the (2/d) and 1/Gamma live"; they are NOT all correct simultaneously, and reporting the spread is the honest CFCA move.""" core_full = sum(m["g_dof"] * (2.0 / d) * (1.0 - eta / (d + 2.0)) / (1.0 + m["m2"]) for m in RETAINED_MODES) # core carries (2/d) core_bare = sum(m["g_dof"] * (1.0 - eta / (d + 2.0)) / (1.0 + m["m2"]) for m in RETAINED_MODES) # core without (2/d) half = 0.5 * (4.0 * PI) ** (-d / 2.0) plain = (4.0 * PI) ** (-d / 2.0) return { # A: EP-flow prefactor c_d on the (2/d)-free core (folds (2/d) into c_d). "A_EPflow_c_d_times_core_bare": c_d_EPflow(d) * core_bare, # B: COMMITTED O2 -- (1/2)(4pi)^{-d/2} * sum l_0^d (l_0 carries its own (2/d)). "B_half_times_core_full": half * core_full, # C: (1/2)(4pi)^{-d/2} on the (2/d)-free core. "C_half_times_core_bare": half * core_bare, # D: plain heat-kernel (4pi)^{-d/2} on the (2/d)-free core. "D_plain_times_core_bare": plain * core_bare, } # =========================================================================== # The a_4 relative-sign / branch logic (geometry-forced; target-blind). # =========================================================================== def a4_and_relative_sign(): R_K6_0, R_S2_0 = 30.0, 2.0 a4_prefactor = 1.0 / 36.0 hk_norm_13 = (4.0 * PI) ** (-D_BULK / 2.0) int_a4 = a4_prefactor * R_K6_0 * R_S2_0 # POSITIVE, convention-invariant c_a4_genuine_minus_half = GLOBAL_ONE_LOOP * hk_norm_13 * int_a4 # NEGATIVE (= -5.97e-8) return { "int_a4_sign": "POSITIVE", "int_a4_value": int_a4, "c_a4_under_genuine_minus_half": c_a4_genuine_minus_half, # -5.97e-8 "abs_c_a4": abs(c_a4_genuine_minus_half), "relative_sign_rule": ( "relative sign(c_a4)/sign(c_loop) = sign(int_a4)/sign(int_loop). " "int_a4 = +(1/36)R_K6 R_S2 is POSITIVE & convention-invariant. " "int_loop (sign of c_loop's UNDERLYING density) is NOT banked; the " "branch rides entirely on it."), "int_loop_status_NOT_pinned": { "i_integration_direction": ( "Wetterich partial_t u > 0 is INSTANTANEOUS; accumulated UV->IR " "(dt<0) carries OPPOSITE sign -> does NOT fix accumulated-density " "sign."), "ii_scheme_dependence": ( "bare zeta has POLE at s=-1/2; residue NEGATIVE (invariant) but " "finite part FLIPS with mu (mu*~0.061) -> wall sign scheme-dep."), "iii_prior_disagreement": ( "gap04_litim_scheme_branch read SAME scheme family leading order " "as BRANCH-RUNAWAY (opposite branch)."), "iv_circularity": ( "choosing the no-flip convention by reference to stored c_loop=+ " "uses the target's sign -> circular, forbidden."), }, "branch_if_int_loop_POSITIVE": ( "c_a4 SAME relative sign as c_loop. With genuine -(1/2) on positive " "int_a4, c_a4 = -5.97e-8 < 0; growth-8 operator dominates growth-6 " "c_loop wall at the -sigma corner -> V -> -inf -> BRANCH-RUNAWAY."), "branch_if_int_loop_NEGATIVE": ( "c_a4 OPPOSITE relative sign to c_loop -> c_a4 = +5.97e-8 > 0, a " "positive wall -> the -sigma well stands -> BRANCH-STANDS."), } def main(): for p in (FROZEN_YAML, ZRENORM_SRC, GILKEY_SRC, VEFF_YAML): if not os.path.exists(p): sys.stderr.write("REFUSE(exit2): missing input %s\n" % p) return 2 frozen_text = open(FROZEN_YAML, "r", encoding="utf-8").read() zrenorm_src = open(ZRENORM_SRC, "r", encoding="utf-8").read() gilkey_src = open(GILKEY_SRC, "r", encoding="utf-8").read() veff_text = open(VEFF_YAML, "r", encoding="utf-8").read() for name, txt in (("frozen_inputs.yaml", frozen_text), ("z_renormalized_c_loop.py", zrenorm_src), ("gilkey_a4_cross_terms.py", gilkey_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 # ===================================================================== # STEP 1 -- commit O1/O2/O3 + verify the corpus pins (target-blind). # ===================================================================== seeds_in_src = (str(SEED_SIGMA) in zrenorm_src and str(SEED_THETA) in zrenorm_src and str(SEED_BORB) in zrenorm_src) mu_match_pinned = ("MU_MATCH_GEV" in zrenorm_src and "Z_at_mu_match" in zrenorm_src) num_sigma = SEED_SIGMA * FOUR_PI_SQ num_borb = SEED_BORB * FOUR_PI_SQ sigma_is_three_halves = abs(num_sigma - 1.5) < 1e-12 borb_is_two = abs(num_borb - 2.0) < 1e-12 eta_in_src = ("24.0 * math.pi * math.pi" in zrenorm_src) identity_ok = all( abs(c_d_EPflow(d) - 0.5 * (4.0 * PI) ** (-d / 2.0) / math.gamma(d / 2.0 + 1.0)) < 1e-15 * abs(c_d_EPflow(d)) for d in (9, 13)) O1 = ("k = mu_match = M_13 (corpus-pinned: z_renorm evaluates Z at mu_match); " "w_i = m_i^2 in M_13^2 units; projection over the FIVE FRG-2 retained " "shell masses (finite), NOT the unbounded bare K6 Casimir tower.") O2 = ("Litim optimised bosonic threshold l_0^d(w;eta)=(2/d)(1-eta/(d+2))/(1+w) " "assembled with canonical EP-flow prefactor c_d=2 v_d/d " "=(1/2)(4pi)^{-d/2}/Gamma(d/2+1); d = total_compact_dimension = 9. " "d=9 is a SELECTION (corpus gilkey/zeta use bulk 13); band reported.") O3 = ("eta = 1/(24 pi^2) per real scalar (Litim 2001 Eq. 38), via the NLO " "factor (1 - eta/(d+2)). Sub-percent.") # ===================================================================== # STEP 2 -- compute the principled c_loop (magnitude + sign), target-blind. # ===================================================================== c_loop_d9, core_d9 = c_loop_principled(D_COMPACT, ETA_LITIM) c_loop_d13, core_d13 = c_loop_principled(D_BULK, ETA_LITIM) band_d9 = assembly_band(D_COMPACT, ETA_LITIM) band_d13 = assembly_band(D_BULK, ETA_LITIM) c_loop_committed = c_loop_d9 c_loop_committed_sign = "POSITIVE" if c_loop_committed > 0 else "NEGATIVE" sign_robust_positive = (c_loop_d9 > 0 and c_loop_d13 > 0 and all(v > 0 for v in band_d9.values()) and all(v > 0 for v in band_d13.values())) # ===================================================================== # STEP 3 -- the relative one-loop sign + branch (do NOT force STANDS). # The relative sign rides on int_loop (sign of c_loop's underlying density), # which is NOT banked and is scheme-dependent (four reasons above). Equally # principled readings give opposite branches. We report the leading-order # reading (the UNFAVORABLE one) honestly and decline to break the owner-lock. # ===================================================================== a4 = a4_and_relative_sign() leading_order_branch = "BRANCH-RUNAWAY" # assembled coeff positive -> int_loop+ reading branch_selected = "still-owner-locked" relative_sign_derived = False # construction/convention-dependent # ===================================================================== # STEP 4 -- VALIDATE against the frozen magnitude (read NOW, post-hoc). # ===================================================================== target = C_LOOP_FRG2_TARGET ratio_committed = c_loop_committed / target reproduces_committed = (0.5 < abs(ratio_committed) < 2.0) ratio_band = {} for tag, val in list(band_d9.items()): ratio_band["d9_" + tag] = val / target for tag, val in list(band_d13.items()): ratio_band["d13_" + tag] = val / target any_in_fold = any(0.5 < abs(r) < 2.0 for r in ratio_band.values()) nearest = min(ratio_band.values(), key=lambda r: abs(math.log(abs(r)))) magnitude_reproduces_without_reverse_fit = bool(reproduces_committed) # ===================================================================== # OUTCOME (honest; do NOT force the favorable branch). # ===================================================================== outcome = "CONSTRUCTION-DEPENDENT-sign-owner-must-rule" magnitude_validation = ( "Committed principled c_loop (O1+O2+O3, EP-flow d=9) = %.6e, ratio to " "target = %.4f -> OUTSIDE the factor-2 fold (does NOT reproduce). The + " "SIGN of the assembled coefficient is reproduced (robustly: sum of " "positive terms x positive prefactor), but the magnitude is NOT " "reproduced under the committed canonical normalization. Across the four " "internally-canonical assemblies x {d=9,d=13} the ratio spans %.2e..%.2f " "(>~10^4); the target sits inside this wide band but NO single principled " "prescription lands cleanly in [0.5,2.0] (nearest = %.3f, just outside). " "Hitting 1.36e-5 exactly would require hand-picking the assembly = " "reverse-fit, which is FORBIDDEN and DECLINED." % (c_loop_committed, ratio_committed, min(abs(r) for r in ratio_band.values()), max(abs(r) for r in ratio_band.values()), abs(nearest))) well_verdict = ( "OWNER-LOCKED / CONDITIONAL. The FRG-2 NLO Litim shell-projection is now " "constructed from first principles target-blind: O1 (k=mu_match, w_i=m_i^2 " "over the five FRG-2 retained shell masses), O2 (Litim l_0^d threshold + " "canonical EP-flow prefactor c_d=2 v_d/d, d=9), O3 (eta=1/(24 pi^2)). The " "assembled c_loop is POSITIVE (robustly) but its MAGNITUDE does NOT " "reproduce 1.3637877e-5 under the committed canonical normalization " "(ratio 0.039); across equally-principled assemblies the magnitude is " "unpinned (>~10^4 span), so the magnitude is genuinely owner-locked and " "we DECLINE to reverse-fit it. The load-bearing DELIVERABLE -- the " "relative one-loop SIGN of c_a4 vs c_loop that decides STANDS vs RUNAWAY " "-- is NOT derived: it rides on the sign of c_loop's UNDERLYING density, " "which the corpus records as NOT banked (gilkey_a4_cross_terms.py: " "owner-must-rule) and which is NOT pinned by the construction for four " "independent reasons -- (i) the Wetterich partial_t u > 0 is " "instantaneous, the accumulated UV->IR contribution carries the opposite " "sign; (ii) the bare zeta has a POLE at s=-1/2 and the finite part flips " "with mu; (iii) the prior gap04_litim_scheme_branch read the SAME scheme " "family as BRANCH-RUNAWAY; (iv) selecting the no-flip convention by " "reference to the stored + target is circular. Equally principled " "readings give opposite branches (int_loop>0 -> RUNAWAY; int_loop<0 -> " "STANDS). The leading-order reading is the UNFAVORABLE RUNAWAY. We " "DEFAULT HOLD and do NOT force BRANCH-STANDS. The relative sign is " "CONSTRUCTION-DEPENDENT = owner-must-rule (Chris).") result = { "schema": "c_loop_NLO_match_result_v1", "object": ( "c_loop = (1/2)(4 pi)^{-d/2} sum_i g_i l_0^d(w_i; eta) -- the FRG-2 NLO " "Litim shell-projection over the five retained shell masses; the " "e^{-6 sigma} KK-Casimir wall coefficient."), "outcome": outcome, "step1_principled_choices_committed_target_blind": { "O1_spectral_cutoff_scale": O1, "O2_NLO_threshold_prescription": O2, "O3_NLO_running": O3, "corpus_pins": { "O1_mu_match_pinned_in_zrenorm": bool(mu_match_pinned), "O1_retained_seeds_match_zrenorm": bool(seeds_in_src), "O2_EPflow_identity_2vd_over_d_eq_half_4pi_minus_d2_over_Gamma": bool(identity_ok), "O3_eta_1_over_24pi2_in_zrenorm": bool(eta_in_src), "shell_mass_sigma_numerator_is_3_over_2": bool(sigma_is_three_halves), "shell_mass_borb_numerator_is_2": bool(borb_is_two), }, "shell_masses_M13_2_units": {m["name"]: m["m2"] for m in RETAINED_MODES}, "eta_litim": ETA_LITIM, }, "step2_principled_c_loop_target_blind": { "c_loop_committed_EPflow_d9": c_loop_committed, "c_loop_committed_sign": c_loop_committed_sign, "threshold_core_sum_d9": core_d9, "c_loop_EPflow_d13": c_loop_d13, "threshold_core_sum_d13": core_d13, "assembly_band_d9": band_d9, "assembly_band_d13": band_d13, "sign_robustly_positive_across_band": bool(sign_robust_positive), }, "step3_relative_sign_and_branch": { "a4_and_relative_sign": a4, "leading_order_reading": leading_order_branch, "leading_order_note": ( "assembled c_loop coefficient POSITIVE -> if int_loop inherited " "positive, c_a4 keeps sign -> c_a4 < 0 -> RUNAWAY. Reported " "honestly; NOT forced to STANDS."), "relative_sign_derived": relative_sign_derived, "branch_selected": branch_selected, "why_not_derived": ( "the relative sign rides on the sign of c_loop's UNDERLYING " "density (int_loop), which the corpus records as NOT banked " "(owner-must-rule) and which is scheme-dependent (zeta pole at " "s=-1/2; finite part flips with mu; instantaneous Wetterich flow " "sign != accumulated-density sign; prior reconstruction read " "RUNAWAY). Choosing the no-flip convention by reference to the " "stored + target is circular/forbidden."), }, "step4_magnitude_validation_post_hoc": { "performed_after_sign_and_structure": True, "frozen_c_loop_FRG2_target": target, "committed_ratio_to_target": ratio_committed, "committed_reproduces_within_factor_2": bool(reproduces_committed), "ratio_band_all_principled_assemblies": ratio_band, "any_assembly_in_fold": bool(any_in_fold), "nearest_ratio": nearest, "magnitude_reproduces_without_reverse_fit": magnitude_reproduces_without_reverse_fit, "verdict": magnitude_validation, }, "cloop_magnitude": ( "committed principled (EP-flow d=9) = %.6e (POSITIVE); magnitude " "owner-locked / unpinned across equally-principled assemblies " "(~10^4 span); does NOT reproduce 1.36e-5 without reverse-fit." % c_loop_committed), "cloop_sign": ("POSITIVE for the assembled coefficient (robust). The " "underlying-density sign that the BRANCH needs is NOT " "pinned (scheme-dependent / not banked)."), "relative_sign": ( "CONSTRUCTION-DEPENDENT. sign(c_a4)/sign(c_loop)=sign(int_a4)/sign(int_loop); " "int_a4=+ (manifest) but int_loop (c_loop underlying density sign) is " "NOT banked and scheme-dependent. int_loop>0 -> RUNAWAY; int_loop<0 -> " "STANDS. NOT derived."), "branch_selected": branch_selected, "well_verdict": well_verdict, "magnitude_validation": magnitude_validation, "reverse_fit_attestation": ( "O1/O2/O3 were committed in STEPS 1-3 (this docstring + code) BEFORE " "the target 1.3637877e-5 was read at STEP 4. The shell masses are the " "corpus FRG-2 LO rationals (3/2/(4pi)^2, 2/(4pi)^2, Wilson/(4pi)^2), " "geometry/group-forced; none equals the target or a simple multiple. " "The committed canonical normalization (EP-flow d=9) MISMATCHES the " "target (ratio 0.039); the closer assembly (d9 C ~ 2.06) is DECLINED " "(it is not THE canonical placement and choosing it to land nearer the " "target would be reverse-fit). The hard-coded target does NOT enter " "STEPS 1-3. MAGNITUDE side is genuinely target-blind. The SIGN side is " "explicitly NOT claimed derived precisely BECAUSE deriving it would " "require selecting the no-flip convention by reference to the target's " "known + sign -- the circular reverse-fit we refuse. (An earlier draft " "made exactly that circular selection and claimed STANDS as derived; " "this version corrects it to owner-must-rule.)"), "what_requires_chris": ( "(SIGN -- the load-bearing deliverable) The sign of c_loop's " "UNDERLYING heat-kernel density (int_loop) -- the unfilled " "CONVENTION_FREEZE slot for the a_4 sector -- which fixes the relative " "one-loop sign of c_a4 vs c_loop and hence the branch (STANDS vs " "RUNAWAY). This is owner-must-rule per gilkey_a4_cross_terms.py and is " "NOT broken here. " "(MAGNITUDE / O1 / O2) The exact NLO threshold-assembly placement and " "the shell-projection dimension (d=9 compact vs d=13 bulk) that pin " "the magnitude to 1.3637877e-5 -- unpinned across equally-principled " "choices. (O3 is sub-percent and agreed.)"), "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 construction is built " "from frozen geometry + SU(3)/T^2 group theory + the corpus FRG-2 LO " "coefficients (exact rationals over (4 pi)^2) + Litim 2001 threshold " "functions ALONE. The frozen magnitude 1.3637877e-5 was read ONLY at " "STEP 4 (post-hoc validator), strictly AFTER O1/O2/O3, the c_loop " "magnitude+sign, the relative-sign analysis, and the branch logic were " "fixed -- and ONLY as a comparison, never as an input. The favorable " "BRANCH-STANDS was NOT forced: the leading-order reading is the " "UNFAVORABLE RUNAWAY, the magnitude is reported as an honest mismatch, " "and the relative sign is reported CONSTRUCTION-DEPENDENT = " "owner-must-rule. The SIGN was NOT derived via the target's known + " "sign (which would be no-target-loading on the sign side)."), "provenance": { "frozen_inputs.yaml": sha256_file(FROZEN_YAML), "z_renormalized_c_loop.py": sha256_file(ZRENORM_SRC), "gilkey_a4_cross_terms.py": sha256_file(GILKEY_SRC), "veff_coefficients_frg4.yaml": sha256_file(VEFF_YAML), }, "non_promotion": ( "no gate flipped; no status word emitted for any gate. STAGED " "producer; promotable to gap_04/src ONLY if validated + owner-signed. " "Validation FAILS (magnitude mismatch + sign owner-locked), so NOT " "promotable as a sign-derivation."), } 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, "c_loop_NLO_match_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("c_loop_NLO_match.py -- FRG-2 NLO Litim shell-projection (the named producer)") print("=" * 78) print("OBJECT: c_loop = (1/2)(4pi)^{-d/2} sum_i g_i l_0^d(w_i;eta) [e^{-6 sigma} wall]") print("-" * 78) print("STEP 1 principled O1/O2/O3 (committed BEFORE target):") print(" O1 k=mu_match, w_i=m_i^2, FIVE retained shell masses (corpus-pinned)") print(" mu_match pinned in z_renorm : %s ; seeds match : %s" % (mu_match_pinned, seeds_in_src)) print(" shell num 3/2: %s ; num 2: %s" % (sigma_is_three_halves, borb_is_two)) print(" O2 Litim l_0^d + EP-flow c_d=2v_d/d, d=9 (SELECTION; band reported)") print(" EP-flow identity 2v_d/d==(1/2)(4pi)^-d/2/Gamma : %s" % identity_ok) print(" O3 eta=1/(24pi^2) in z_renorm : %s" % eta_in_src) print("-" * 78) print("STEP 2 principled c_loop (target-blind):") print(" committed (EP-flow d=9) : %+.6e [%s]" % (c_loop_committed, c_loop_committed_sign)) print(" threshold core sum d=9 : %.6f" % core_d9) print(" assembly band d=9 : %s" % {k: "%.3e" % v for k, v in band_d9.items()}) print(" assembly band d=13 : %s" % {k: "%.3e" % v for k, v in band_d13.items()}) print(" sign robustly positive : %s" % sign_robust_positive) print("-" * 78) print("STEP 3 relative sign + branch (NOT forced):") print(" int_a4 POSITIVE (manifest) ; int_loop (c_loop underlying density) NOT banked") print(" int_loop>0 -> RUNAWAY ; int_loop<0 -> STANDS (scheme-dependent, 4 reasons)") print(" leading-order reading : %s (UNFAVORABLE)" % leading_order_branch) print(" relative_sign_derived : %s" % relative_sign_derived) print(" branch_selected : %s" % branch_selected) print("-" * 78) print("STEP 4 magnitude validation (read AFTER sign/structure):") print(" frozen target : %.6e" % target) print(" committed ratio : %.4f (in fold: %s)" % (ratio_committed, reproduces_committed)) print(" nearest principled ratio: %.3f (just outside [0.5,2.0])" % abs(nearest)) print(" reproduces w/o reverse-fit : %s" % magnitude_reproduces_without_reverse_fit) print("-" * 78) print("OUTCOME : %s" % outcome) print("BRANCH : %s" % branch_selected) print("artifact:", out_path) print("=" * 78) return 0 if __name__ == "__main__": sys.exit(main())