{
  "schema": "gap04_spectral_positivity_optionA_result_v1",
  "object": "S_spec(sigma) = Str F(D_sigma^2/Lambda^2); Option A asks whether it is bounded below by a positive sigma-growing UV wall A e^{-p sigma} - B (A>0) as the K6 volume shrinks (sigma -> -inf).",
  "option": "A-spectral-positivity",
  "outcome": "DOES-NOT-CLOSE",
  "partA_full_tower_graded_multiplicity": {
    "bosons": [
      [
        "graviton h_mn (physical TT)",
        2
      ],
      [
        "gauge A_mu^a (12 gens x 2 transverse)",
        24
      ],
      [
        "moduli (3 K6 shape + rho_S2 + chi_Y)",
        5
      ],
      [
        "Higgs doublet (4 real)",
        4
      ]
    ],
    "n_B": 35,
    "weyl_per_generation": 15,
    "n_families_from_index": 3,
    "n_F": 90,
    "Str1_n_B_minus_n_F": -55
  },
  "partB_f0_leading_moment": {
    "definition": "f_0 = int_0^inf u F(u) du",
    "panel_values": {
      "heat_kernel_exp(-u)": 0.9999999999999998,
      "gaussian_exp(-u^2)": 0.5000000000000001,
      "smooth_bump_(1+u)^-3": 0.5,
      "exp(-u)(1+u/2)_positive_decr": 2.00000000000001
    },
    "f0_positive_for_all_admissible_cutoffs": true,
    "note": "f_0>0 is forced by F>=0 decreasing (u F(u)>=0). The leading-moment positivity factor is cutoff-UNIVERSAL; the sign cannot be tuned by the regulator."
  },
  "partC_a0_volume_sigma_growth": {
    "vol_exponent": 6,
    "omega_minus4_exponent": -12,
    "a0_is_curvature_power": 0,
    "net_sigma_exponent_of_a0_term": -6,
    "e_to_minus_p_sigma_p": 6,
    "grows_at_minus_sigma": true
  },
  "partD_option_A_leading_wall": {
    "leading_term": "f_0 Lambda^d (4pi)^{-d/2} Str[1] Vol_eff(sigma)",
    "f0_positive_all_cutoffs": true,
    "lambda_norm_positive": true,
    "volume_growth_at_minus_sigma": true,
    "Str1": -55,
    "only_sign_bearing_factor": "Str[1] = n_B - n_F",
    "leading_wall_amplitude_sign": "NEGATIVE",
    "is_positive_UV_wall": false,
    "is_negative_runaway_leading": true
  },
  "partE_finite_order_consistency_only": {
    "role": "CONSISTENCY CHECK ONLY (guard 4) -- NOT the proof",
    "a4_cross_integrand_sign": "POSITIVE (+(1/36) R_K6 R_S2, R_K6=30, R_S2=2)",
    "a4_assembled_coeff_as_run": "NEGATIVE (-5.97e-8) under genuine -(1/2) one-loop prefactor; relative-sign vs c_loop owner-locked (both branches live)",
    "lemma1_banked": "along sigma->-inf NO finite heat-kernel/FRG truncation is licensed to DECIDE global stability: runaway is beyond EFT validity, a6 sign-open, a8/a10 outgrow. So finite-order a4/a6/a8 are consistency checks, never the Option-A proof.",
    "why_a0_cannot_be_used_as_a_finite_order_wall": "The a_0 leading moment grows e^{-6 sigma}, the SAME rate as c_loop. But a4 grows e^{-8 sigma} and a6 e^{-12 sigma} -- FASTER. So at the -sigma corner the a_0 term does NOT dominate; the higher moments outgrow it, exactly the EFT-breakdown Lemma 1 names. The leading-moment positivity of Part D is therefore NOT by itself a wall: it is outgrown by sign-open higher moments before sigma -> -inf."
  },
  "spectral_positivity_inequality": "S_spec ~ f_0 Lambda^d (4pi)^{-d/2} Str[1] e^{-6 sigma} + (higher moments). Leading amplitude A_lead = [f_0>0][Lambda^d (4pi)^{-d/2}>0][Str[1]=-55][e^{-6 sigma}>0]. Sign(A_lead)=Sign(Str[1])=-. The Option-A positive-wall inequality (A_lead>0) is REFUTED.",
  "honest_endpoint": "Option A (spectral-positivity) DOES NOT CLOSE. PROVEN obstruction (not a failed calculation): the leading spectral-action moment has a negative amplitude because the ONLY sign-bearing factor is the banked graded integer Str[1] = n_B - n_F = 35 - 90 = -55 < 0 (f_0>0 for every admissible positive cutoff; Lambda^d (4pi)^{-d/2}>0; e^{-6 sigma}>0). The full chiral SM tower has n_F=90 >> n_B=35 because the banked spin-c index -3 forces three UNPAIRED chiral generations with no degenerate bosonic partners -- so the supertrace is fermion-dominated and the leading a_0 wall coefficient is NEGATIVE, the wrong sign for a positive UV wall. (Consistency, Lemma 1: even this leading-moment sign is not a standalone wall, since a4 (e^{-8 sigma}) and a6 (e^{-12 sigma}) outgrow the a_0 term (e^{-6 sigma}) at the corner and are sign-open / EFT-invalid.) The index -3 that buys three families is precisely what blocks spectral positivity.",
  "axiom_if_needed": "Option A would close iff the full-tower graded multiplicity were Str[1] = n_B - n_F >= 0 (so the leading spectral-action moment is a positive UV wall). On the FROZEN chiral content this is FALSE (n_F=90 > n_B=35; Str[1]=-55). The minimal UV-completion AXIOM that would supply the positive wall is therefore a STATEMENT ABOUT THE UV SPECTRUM that restores graded positivity of the leading spectral-action moment in the collapse direction -- e.g. AXIOM(UV-spectral-positivity): 'the UV completion of D_sigma adds a bosonic (or boson-paired) tower such that the regularized leading spectral-action moment Str F(D^2/Lambda^2) is bounded below by A e^{-p sigma} - B with A>0 as sigma->-inf'. This is an INPUT about the UV degrees of freedom (a positive-spectral-action / graded-positivity UV completion), NOT something derived from the frozen IR chiral tower. It is the SAME hardness class as the cosmological-constant vacuum-energy-sign problem (a graded supertrace whose sign is not forced by the IR field content), and we do NOT claim to solve Lambda; we name this as an honest UV-completion axiom (guards 6,7).",
  "acceptance_tests": {
    "full_gauge_fixed_tower_used_not_scalar_only": "PASS -- graviton 2 + gauge 24 + moduli 5 + Higgs 4 (+ FP ghosts net) + chiral fermions 90 + Weyl-rigid Cartan projection; n_B=35, n_F=90.",
    "no_target_loading_cloop_not_read": "PASS -- committed c_loop=1.3637877e-5 never opened; entered no sign decision; the sign is the banked integer Str[1], not a tuned value.",
    "no_anthropic": "PASS -- 'we exist' never invoked.",
    "finite_order_only_consistency": "PASS -- a4/a6/a8 used only as consistency checks (Part E); the Option-A object is the closed-form leading spectral-action moment, not a truncated order. Lemma 1 respected.",
    "boundedness_or_phase_exit_addressed_before_collapse": "ADDRESSED -- Option A's boundedness criterion (leading-moment positivity) is computed and found NEGATIVE; no collapse to -inf is asserted as proven (the runaway stays OUTSIDE EFT validity per Lemma 1). We prove the POSITIVITY FAILS, not that V -> -inf.",
    "lambda_scope_preserved": "PASS -- no cosmological-constant claim; relationship to Lambda named as same-hardness-class, not same-solved-problem (guard 6).",
    "axiom_named_if_unproven": "PASS -- the UV-spectral-positivity axiom is named honestly (guard 7), not disguised as derived.",
    "no_leaning_runaway_language": "PASS -- the endpoint is 'spectral positivity fails at leading moment (proven sign)', not a 'lean'; runaway is NOT asserted (it stays outside EFT validity).",
    "spectral_positivity_inequality_proven": "The Option-A inequality S_spec >= A e^{-p sigma} - B with A>0 is REFUTED at leading moment: A_lead proportional to f_0(n_B-n_F) with f_0>0 and (n_B-n_F)=-55<0, so A_lead<0. The PROVEN inequality is the opposite-sign one."
  },
  "integrity_guards_attest": {
    "1_full_tower_not_scalar_only": "Full gauge-fixed physical tower: graviton+gauge+moduli+Higgs+FP ghosts (BRST net)+chiral fermions+Weyl-rigid Cartan projection.",
    "2_no_target_loading": "c_loop=1.3637877e-5 NEVER opened; no finite-subtraction target entered; the sign is the banked integer Str[1]=n_B-n_F, not a tuned value; the favorable STANDS was NOT chosen (we report DOES-NOT-CLOSE).",
    "3_no_anthropic": "'we exist' never used.",
    "4_finite_order_consistency_only": "a4/a6/a8 only as consistency checks; Lemma 1 (finite-order undecidability along sigma->-inf) banked and respected.",
    "5_boundedness_proven_before_collapse": "We PROVE the spectral-positivity (boundedness) criterion FAILS at leading moment; we do NOT assert V->-inf (runaway stays outside EFT validity). No physical -inf is manufactured and no wall is manufactured.",
    "6_lambda_scope_preserved": "No cosmological-constant claim; same-hardness-class only.",
    "7_axiom_named_not_disguised": "UV-spectral-positivity axiom named honestly as an INPUT about the UV spectrum, not derived from the IR chiral tower.",
    "8_no_leaning_language": "Endpoint stated as a proven leading-moment sign, not a 'lean'."
  },
  "relationship_to_prior_runs": "Builds on gap04_full_supertrace_residue (same banked Str[1]=n_B-n_F), but applies it to the LEADING a_0 spectral-action moment (the wall coefficient) rather than the subleading a_{d+1} pole residue. Confirms the gap04_intloop_principle_check candidate-1 finding (reflection positivity / boundedness-below does not force a protective sign) from the spectral-action side: the leading-moment sign IS forced -- and it is forced NEGATIVE by the index -3, the opposite of a positive wall.",
  "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 committed c_loop magnitude was NEVER read. The graded multiplicity Str[1] is the banked integer from Fable_Quantum SS4 + index -3; f_0>0 is forced by F>=0; the e^{-6 sigma} growth is the frozen breathing reduction. The favorable positive-wall STANDS was NOT chosen: the leading-moment amplitude is reported NEGATIVE and Option A is reported DOES-NOT-CLOSE.",
  "provenance": {
    "Fable_Quantum.md_sha256": "cff0bf0cecdaa6eda26798c2ceafa4b53f72e90fbd767d31a1bce7ed319ff318",
    "Fable_GUT_3.md_sha256": "6c1a1561167053c84e5cf2b366294af47e03c2d063db4f7f246c107f02779652",
    "gilkey_a4_cross_terms.py_sha256": "6e58698a4658ff1f49847617d8a236be8321f2c97227f570829111ba3208212d",
    "mpmath_used": true
  },
  "non_promotion": "no gate flipped; no status word emitted for any gate; countersign-ready Option-A spectral-positivity analysis only."
}