{
  "schema": "gap04_reason_hunt_result_v1",
  "purpose": "CFCA reason-hunt over Gap-04's residual numbers: find a reason for each ('just is' is the floor, not the first answer) and classify DERIVED / EARNED-IRREDUCIBLE / BARE-COUNTED with the economy ledger.",
  "provenance_hashes": {
    "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86",
    "veff_coefficients_frg4.yaml": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac",
    "gilkey_a4_cross_terms.json": "86ea94dff5efdb295432dfcd494bae18aebc0b0bf44db181d090ddb29fe10845"
  },
  "entries": [
    {
      "number": "a_4 master R^2 coefficient = +5/360 (cross prefactor 2*5/360 = 1/36)",
      "value": "1/72 (=1/72); cross 1/36",
      "classification": "DERIVED",
      "reason_or_why_not": "Number/group-theoretic + convention-INVARIANT. The +5/360 is the fixed coefficient of R^2 in the Gilkey/Seeley-DeWitt a_4 heat-kernel master formula (Gilkey 1995 Eq. 4.1.7); it is a universal property of the Laplace-type operator, the same rational in BOTH the D=-(nabla^2+E) and D=-(nabla^2)-E conventions (R^2 is even in curvature). The cross prefactor 1/36 = 2*(5/360) where the 2 is the binomial cross-term coefficient of (R_1+R_2+R_3)^2. No dial; a theorem-fixed rational. NOT 'just is'.",
      "outputs_explained": "Sets the magnitude and (with the registry) the sign of c_a4_K6_S2; the same master formula governs every a_4 insertion in the FRG-4 expansion."
    },
    {
      "number": "curvature registry R_K6_0 = +30, R_S2_0 = +2 (single-signed)",
      "value": "R_K6_0 = 6*5 = 30 ; R_S2_0 = 2*1 = 2",
      "classification": "DERIVED",
      "reason_or_why_not": "Geometry-FORCED. R_i = n_i * lambda_i with the FROZEN Einstein constants of the frozen geometry: K6 = SU(3)/T^2 (F_3 flag) has the normal Einstein metric Ric = +5g (Besse, Einstein Manifolds, Tab. 7.107; Castellani-D'Auria-Fre), n=6 -> R=30; round S^2 has Ric = +g, n=2 -> R=2. Both POSITIVE because both are positively-curved Einstein factors -- no dial once the geometry K6 x S^2 x S^1_Y is frozen. The registry is convention-consistent (single curvature-sign convention across the corpus). NOT 'just is': forced by the chosen compact geometry, which is itself fixed upstream by the 3-family index.",
      "outputs_explained": "Pins the a_4 cross-density POSITIVE; sets R_K6 (Casimir/KK scale) feeding c_KK and the e^{-4 sigma} wall; same registry feeds the first-Laplace eigenvalues used in the shell projection."
    },
    {
      "number": "one-loop normalization (4 pi)^{-D/2} with D = 13",
      "value": "7.163675e-08 (= (4 pi)^{-6.5})",
      "classification": "DERIVED",
      "reason_or_why_not": "Number-theoretic normalization (a recognizable (4 pi)^{-D/2}) x a geometry-FORCED exponent. The (4 pi)^{-D/2} is the universal Gaussian momentum-integration measure of a D-dimensional one-loop determinant (proper-time / heat-kernel normalization); D = 13 is FORCED by the frozen total signature (12,1). No dial. NOT 'just is'.",
      "outputs_explained": "Common one-loop measure for c_a4 AND every other heat-kernel coefficient in the FRG-4 expansion; not specific to one term."
    },
    {
      "number": "one-loop log-det prefactor: global MAGNITUDE -(1/2) [DERIVED] vs RELATIVE sign of c_a4 w.r.t. c_loop [OWNER-LOCKED]",
      "value": "-1/2 global magnitude is universal; the RELATIVE one-loop sign of c_a4 vs the c_loop wall is NOT fixed data-blind",
      "classification": "BARE-COUNTED",
      "reason_or_why_not": "PARTIALLY DERIVED, but the LOAD-BEARING part is OWNER-LOCKED, so this entry is honestly counted as a bare (undecided) debit until the owner rules. The MAGNITUDE -(1/2) IS derived: it is the universal bosonic one-loop log-det prefactor (V^(1-loop) = -(1/2)(4pi)^{-D/2} sum_n a_n), GLOBAL to every a_n. BUT 'global prefactor' does NOT force two coefficients to the same SIGN: sign(c_n) = sign(-1/2) x sign(that term's OWN integrand). c_a4's integrand (1/36)*R_K6*R_S2 is POSITIVE, so the genuine global -(1/2) yields c_a4 NEGATIVE. Making c_a4 POSITIVE (to match the c_loop wall) REQUIRES asserting c_loop's stored +value already absorbed a -(1/2) acting on a NEGATIVE underlying density -- a sign NEVER established in the corpus (z_renormalized_c_loop.py reads c_loop as a frozen inherited number; TOE_FINAL records c_loop OPEN, positivity merely assumed, F6-guarded). So the RELATIVE one-loop sign is NOT Noether/geometry-FORCED -- it is the genuine OWNER-MUST-RULE item (gap04_convention_audit original verdict; gap04_disjointness convention_dependent=True). Classifying it 'DERIVED' would be the manufactured step the countersign caught. Honest floor: 'just is' / owner-locked until the c_loop underlying-density sign is banked.",
      "outputs_explained": "Once RULED, it sets the relative sign of the a_4 cross-term in V_eff and decides whether the FRG-4 -sigma well stands (BRANCH-STANDS) or fails as a separate runaway (BRANCH-RUNAWAY). Until ruled it explains nothing else on its own -> a pure debit, counted against the ~5-6 budget."
    },
    {
      "number": "c_loop_Z = +1.3637e-5 (the e^{-6 sigma} KK-Casimir wall)",
      "value": "1.3637166327064904e-05 (M_13^4 units), POSITIVE",
      "classification": "EARNED-IRREDUCIBLE",
      "reason_or_why_not": "Its SIGN is DERIVED (geometry-forced positive vacuum energy of the KK shell + the global one-loop convention). Its MAGNITUDE is NOT reducible to a closed form data-blind -- it is the output of an FRG-2 NLO shell-projection / Z-renormalization integral over the retained mode spectrum (Wetterich flow, Litim regulator), a genuine computation, not a closed rational. So it remains an irreducible computed input -- BUT it EARNS its keep (>= 2 independent outputs).",
      "outputs_explained": "ECONOMY-EARNING (>=2): (a) it is the small-sigma wall that closes F6 / blocks the sigma -> -inf runaway in Gap-04; AND (b) it gates the A_s-absolute normalization (the As-absolute row is BLOCKED on K13-2 + c_loop) and the Gap-09 scale in the cascade. Two+ independent downstream outputs -> nets POSITIVE on the ledger."
    },
    {
      "number": "c_bdry = -0.012665 (Dai-Freed 4D + orbifold-bordism half)",
      "value": "-0.012665147955292222 (M_13^4 units), NEGATIVE",
      "classification": "EARNED-IRREDUCIBLE",
      "reason_or_why_not": "Its SIGN is DERIVED/geometry-forced: the boundary term is fixed by the Dai-Freed / Freed-Hopkins orbifold-bordism anomaly class of the S^1_Y/Z_2 fixed points (2 fixed points, boundary_anomaly_class in the frozen geometry) -- negative, and CLOSED in Fable-Latest. Its MAGNITUDE is a computed bordism/eta-invariant output, not a closed rational data-blind, so it stays an irreducible computed input -- but it EARNS its keep.",
      "outputs_explained": "ECONOMY-EARNING (>=2): (a) c_bdry<0 is what makes the -sigma well robust ('well unconditional in c_loop' rides on c_bdry<0, v10); AND (b) it sets the e^{-2 sigma - 2 chi} term coupling sigma and chi in V_eff, entering the joint Hessian / F8 Higgs-Wilson vacuum check. Two+ independent roles -> nets POSITIVE."
    },
    {
      "number": "c_KK = c_S2 = c_KK_S1Y = +0.0094989 (zero-mode universality)",
      "value": "0.009498860966469166 (all three equal), POSITIVE",
      "classification": "EARNED-IRREDUCIBLE",
      "reason_or_why_not": "PARTIALLY DERIVED: the EQUALITY of the three is geometry/group-FORCED -- it is the zero-mode universality of the FRG-2 LO coefficient (one universal constant, not three independent dials; that they coincide is a Noether-style universality, not a coincidence). The shared MAGNITUDE is an FRG-2 LO computed output, not a closed rational data-blind. So it is one irreducible computed constant (not three) that EARNS its keep.",
      "outputs_explained": "ECONOMY-EARNING: ONE constant explains THREE wall coefficients (c_KK e^{-4 sigma}, c_S2 e^{-4 rho}, c_KK_S1Y e^{-4 chi}) by universality -> credit 3 >= 1 debit. Strongly net-positive."
    },
    {
      "number": "c_Wilson = -4.876e-5 (Hosotani n=1 harmonic)",
      "value": "-4.875756906074309e-05 (M_13^4 units), NEGATIVE",
      "classification": "EARNED-IRREDUCIBLE",
      "reason_or_why_not": "Its STRUCTURE is DERIVED: it is the n=1 harmonic of the Hosotani Wilson-line (gauge-holonomy) potential cos(theta_W) e^{-4 sigma} -- a group-theoretic object (first Fourier harmonic on the Wilson-line circle), Noether-meaningful (gauge holonomy). Its magnitude is an FRG-2 computed output. Irreducible computed input that EARNS keep.",
      "outputs_explained": "ECONOMY-EARNING (>=2): (a) sets the theta_W extremum (theta_W = 0) used in the V_eff and the Hessian; AND (b) feeds the joint Higgs-Wilson vacuum check (F8). Net-positive."
    },
    {
      "number": "c_cross_K6_S2_FRG2 = 0 (cross-term first arises at FRG-4)",
      "value": "0 at FRG-2 truncation",
      "classification": "DERIVED",
      "reason_or_why_not": "Geometry/truncation-FORCED zero. At FRG-2 the heat-kernel expansion stops below the a_4 curvature^2 order, so the K6 x S2 cross term (which lives in a_4) is identically absent. NOT a tuned zero -- a forced consequence of the truncation order. NOT 'just is'.",
      "outputs_explained": "Explains why the cross-term is a NEW FRG-4 object and was absent in the \u00a7II single-sigma potential (the v10 reconciliation)."
    }
  ],
  "economy_ledger": {
    "pure_debit_budget_approx": "5-6",
    "bare_counted_count": 1,
    "bare_counted_numbers": [
      "one-loop log-det prefactor: global MAGNITUDE -(1/2) [DERIVED] vs RELATIVE sign of c_a4 w.r.t. c_loop [OWNER-LOCKED]"
    ],
    "derived_count": 4,
    "earned_irreducible_count": 4,
    "irreducibles_total": 5,
    "ledger_verdict": "Most residual Gap-04 numbers are DERIVED (reason found, no longer 'just is') or EARNED-IRREDUCIBLE (each explains >= 2 independent outputs, netting positive): the +5/360 master coeff, the curvature registry, the (4pi)^{-D/2} normalization, and the FRG-2 zero (DERIVED); c_loop, c_bdry, the zero-mode universal constant, and c_Wilson (EARNED-IRREDUCIBLE). The SINGLE exception, surfaced by the countersign, is the RELATIVE one-loop sign of c_a4 vs the c_loop wall (entry 4): it is NOT geometry/Noether-FORCED and is NOT yet earned -- it is the genuine OWNER-MUST-RULE item, honestly BARE-COUNTED (1 pure debit) until the owner banks the sign of c_loop's underlying heat-kernel density. So the pure-debit budget (~5-6) carries exactly ONE Gap-04 debit, the owner-locked relative one-loop sign; everything else pays its keep. This is the honest floor: classifying that relative sign 'DERIVED' was the manufactured step the countersign rejected."
  },
  "no_target_loading": true,
  "non_promotion": "no gate flipped; countersign-input only"
}