{
  "schema": "gap04_oneloop_consistency_result_v2",
  "purpose": "Apply the corpus's GENUINE single one-loop log-det convention to the a_4 cross-term and report the sign c_a4 takes -- without forcing the favorable well-stands outcome.",
  "resolution": "INDETERMINATE-owner-must-rule",
  "resolution_reason": "The GENUINE single one-loop convention is the global bosonic -(1/2)(4pi)^{-D/2}, but a global prefactor does NOT fix the SIGN of c_a4 relative to c_loop -- each term's sign = (global -1/2) x (its OWN integrand). Acting on the a_4 cross integrand (MANIFESTLY POSITIVE: (1/36)*R_K6*R_S2 with R_K6,R_S2 > 0) the genuine -(1/2) gives c_a4 NEGATIVE (-5.97e-08), the as-run value. To make c_a4 POSITIVE one must ASSERT c_loop's stored +value already absorbed a -(1/2) acting on a NEGATIVE underlying density and match a_4 to it; but c_loop's underlying-density sign is NEVER established in the corpus (z_renormalized_c_loop.py reads c_loop as a frozen inherited number and applies no -(1/2); TOE_FINAL records c_loop as OPEN / not computed from first principles, positivity merely assumed, guarded by F6). So the relative one-loop sign is NOT resolvable data-blind: BOTH branches are live (BRANCH-RUNAWAY c_a4<0 -> real -sigma runaway since a_4 growth 8 dominates c_loop growth 6, threatening F1; BRANCH-STANDS c_a4>0 only if the owner rules c_loop's underlying density negative). This is an OWNER-MUST-RULE, exactly as gap04_convention_audit.py ORIGINALLY recorded and gap04_disjointness.py flagged (convention_dependent = owner-locked). We do NOT force 'stands'.",
  "genuine_single_convention": "V^(1-loop) = -(1/2)(4 pi)^{-D/2} sum_n a_n. The -(1/2)(4pi)^{-D/2} prefactor is GLOBAL to every a_n, but does NOT force same-sign: sign(c_n) = sign(-1/2) x sign(own integrand). c_a4's integrand (1/36)*R_K6*R_S2 > 0 => c_a4 NEGATIVE under the genuine convention.",
  "provenance_hashes": {
    "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86",
    "gilkey_a4_cross_terms.json": "86ea94dff5efdb295432dfcd494bae18aebc0b0bf44db181d090ddb29fe10845",
    "veff_coefficients_frg4.yaml": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac",
    "gilkey_a4_cross_terms.py_FIXED": "6e58698a4658ff1f49847617d8a236be8321f2c97227f570829111ba3208212d",
    "gilkey_a4_cross_terms.py.pre_oneloop_fix_2026-06-14.bak": "8975d36c3e0cd5331aca1def94d423c4031ddaf2f4c7e9064f079b9f203f289c"
  },
  "frozen_inputs_used": {
    "R_K6_0": 30.0,
    "R_S2_0": 2.0,
    "a4_R2_master_coeff_5_over_360": 0.013888888888888888,
    "a4_cross_prefactor_1_over_36": 0.027777777777777776,
    "D_bulk": 13,
    "factor_4pi_to_minus_D_over_2": 7.163675151870211e-08,
    "genuine_global_one_loop_prefactor": -0.5
  },
  "computation": {
    "a4_cross_density_pre_oneloop": 1.193945858645035e-07,
    "a4_cross_density_sign": "+",
    "c_a4_under_genuine_global_minus_half": -5.969729293225175e-08,
    "c_a4_BRANCH_RUNAWAY_value": -5.969729293225175e-08,
    "c_a4_BRANCH_STANDS_value": 5.969729293225175e-08,
    "c_loop_wall_value_frozen": 1.3637166327064904e-05,
    "c_loop_wall_positive_loadbearing_F6": true,
    "c_loop_underlying_density_sign_banked": false,
    "arithmetic_check": "c_a4 = (+/-)0.5*(4pi)^-6.5*(1/36)*30*2 = (+/-)5.97e-08"
  },
  "disjointness_dominance_at_neg_sigma": {
    "growth_a4_cross": 8.0,
    "growth_c_loop_wall": 6.0,
    "a4_dominates_neg_sigma_corner": true,
    "consequence_if_c_a4_negative": "a_4 (growth 8) dominates c_loop (growth 6) at the -sigma corner; with c_a4 < 0 the dominant term drives V -> -inf, a REAL separate 3-modulus -sigma runaway threatening F1."
  },
  "branches": {
    "BRANCH_STANDS": {
      "condition": "owner rules c_loop's UNDERLYING heat-kernel density NEGATIVE, so the shared single one-loop treatment flips a_4's relative sign to +",
      "c_a4": 5.969729293225175e-08,
      "effect": "c_a4 a POSITIVE wall; -sigma well STANDS at FRG-4.",
      "named_falsifier": "F1_ext: if the owner banks c_loop underlying density negative AND a true interior critical point is found, the well stands; refuted if no such point exists."
    },
    "BRANCH_RUNAWAY": {
      "condition": "the genuine global -(1/2) acts on both terms' own positive densities (c_a4 keeps its as-run NEGATIVE sign)",
      "c_a4": -5.969729293225175e-08,
      "effect": "c_a4 < 0 and DOMINATES the -sigma corner (growth 8 > 6); V -> -inf: a REAL separate 3-modulus -sigma runaway row.",
      "named_falsifier": "F1: any unbounded direction with V_eff -> -inf. Under this branch the -sigma FRG-4 corner FAILS F1 at decision grade unless a non-perturbative wall (B-UQFC-14-NP-1) supplies a barrier."
    }
  },
  "deciding_fact_for_owner": "The SIGN of c_loop's underlying heat-kernel density (equivalently: is c_loop a -(1/2) log-det of a NEGATIVE-density object, or a positive Casimir wall under a DIFFERENT bookkeeping?). It is NOT banked anywhere (the CONVENTION_FREEZE a_4-sector slot is unfilled). Banking it resolves the relative one-loop sign and selects one branch.",
  "prereg_is_self_contradictory_not_a_tiebreaker": "frozen_inputs.yaml gilkey_a4_protocol carries the explicit -(1/2) in einstein_frame_reduction (=> c_a4 NEGATIVE) AND 'POSITIVE sign' in honest_sign_assessment (dropping the -(1/2); referring to the DENSITY sign, attached to exp(-(14 sigma+6 rho+2 chi)) which does NOT match the computed operator exp(-(8 sigma+4 rho+chi))). The prereg's 'POSITIVE' is the density sign, not a one-loop-consistent c_a4 prediction; citing it as corroboration of a positive c_a4 selects the favorable half of a self-contradictory spec. NOT used as a tie-breaker here.",
  "implication_for_well": "The -sigma FRG-4 well's survival is CONDITIONAL on the owner-locked relative one-loop sign. It is NOT forced to stand (the genuine convention gives c_a4 NEGATIVE on its own positive integrand), and it is NOT forced to fail (the relative sign vs c_loop is unbanked). Owner must rule; both branches carry named falsifiers.",
  "implication_for_v10": "INDEPENDENT of this ruling: v10's Section-II 'well UNCONDITIONAL in c_loop' is TRUE AS STATED and needs NO erratum. It quantifies over the SINGLE-sigma potential V(sigma)=c_KK e^-4s + c_bdry e^-2s + c_Wilson cos(theta_W) e^-4s + c_loop e^-6s, whose well-robustness rides on c_bdry < 0 (no value of c_loop in the declared lane threatens the well). The a_4 cross-term exp(-(8s+4r+chi)) is a SEPARATE 3-modulus FRG-4 object NOT present in the \u00a7II potential, so v10's claim does not quantify over it. The v10 'no erratum' reconciliation is correct; it does NOT, however, imply the FRG-4 3-modulus well stands -- that is the owner-locked item above.",
  "no_target_loading": true,
  "non_promotion": "no gate flipped; source edit is backed up and reversible (the as-run -1/2 is restored, module is self-consistent); this artifact is a countersign-READY both-branches packet the owner signs."
}