{
  "schema": "gap04_intloop_principle_check_result_v1",
  "object": "int_loop = sign of the FINITE part of the scalar SU(3)/T^2 Casimir (the sign of c_loop's underlying heat-kernel density), = sign[finite-part of zeta_{Delta_K6}(-1/2)]; the deciding datum for the Gap-04 well (STANDS if int_loop<0; RUNAWAY if int_loop>0).",
  "outcome": "NO-PRINCIPLE-FOUND-genuinely-Lambda-hard",
  "int_loop_sign_if_fixed": "NOT FIXED. int_loop's finite-part sign remains scheme-dependent; the scheme-INVARIANT residue is NEGATIVE but is the residue of a divergence, not the branch-deciding finite wall coefficient. No principle forced the finite-part sign.",
  "fixers": [],
  "object_structure_data_blind": {
    "tower_smallest_(lambda,deg)": [
      3,
      2
    ],
    "n_distinct_eigenvalues": 11407,
    "heat_trace_parity": {
      "c_minus_three_half": 0.27419955293448567,
      "c_minus_one (integer power)": 4.604300807772854e-07,
      "c_plus_half (pole witness)": 0.2884041461832084,
      "pole_at_minus_half_present": true,
      "expansion_runs_in": "HALF-INTEGER powers (odd d_eff; pole at s=-1/2)"
    },
    "note": "zero-weight slice runs in HALF-INTEGER powers => d_eff=3 (ODD) => simple pole at s=-1/2 => finite part scheme-dependent. Residue scheme-invariant NEGATIVE (corpus-banked, gap04_zeta_continuation_frg2)."
  },
  "candidates": {
    "1_reflection_positivity": {
      "verdict": "INCONCLUSIVE",
      "fixes_sign": false,
      "spectrum_is_reflection_positive": true,
      "reason": "Reflection positivity constrains the spectral measure (Laplacian >=0, eigenvalues lambda>0, degeneracies deg>0 -- all VERIFIED on the K6 zero-weight tower), which is satisfied REGARDLESS of the sign of the renormalized c_loop finite part. A reflection-positive (fully unitary) theory admits Casimir energies of EITHER sign (geometry-dependent), so RP places NO sign constraint on the renormalized finite vacuum-energy density. Boundedness-below is a statement about the FULL V_eff(sigma) at the -sigma corner -- which is exactly the quantity under dispute -- so using it to fix int_loop's sign is the circular reverse-fit-for-stability the discipline forbids. The corpus's own I2/I3 precedent confirms: a single signed vacuum-energy sum over the full inventory is computed NONZERO and the positivity/grading structure does not force a protective sign."
    },
    "2_a6_uv_completion_consistency": {
      "verdict": "ITSELF-OPEN",
      "fixes_sign": false,
      "a6_is_open_object": true,
      "a6_b_i_star_forced_zero": true,
      "a6_provenance": "Only tr[E^3]=1/6 is DERIVED-HERE (flat cov-const E resummation). All purely-gravitational cubic coefficients c1..c8 and mixed E/Omega coefficients are MISSING (require full Gilkey-Avramidi order-6 recursion; not reconstructible from a single shown step).",
      "reason": "(A) a6 is ITSELF a strictly larger OPEN object: the corpus's gap_01_a6_attempt derives only tr[E^3]=1/6 and records ALL purely-gravitational cubic c1..c8 and mixed E/Omega coefficients as MISSING (require full covariant Taylor/Synge recursion); the cubic backreaction on B1/B2 is b_i^* * J_i^{(a6)} with b_i^*=0 FORCED and J_i^{(a6)} MISSING. It cannot fix a downstream sign. (B) Even a COMPLETE a6 would not pin int_loop: UV/Hilbert-space consistency constrains the DIVERGENCE structure (pole residues -- already scheme-invariant negative) not the disputed FINITE part of a LOWER coefficient; a6 multiplies a different operator (curvature^3, e^{-(12 sigma+..)}) than the e^{-6 sigma} c_loop wall. Positive-definiteness = RP (candidate 1) and is sign-blind for the renormalized finite part."
    },
    "3_closed_form_coset_casimir": {
      "verdict": "ITSELF-OPEN",
      "fixes_sign": false,
      "series_is_eisenstein_norm_form_disc_minus3": true,
      "smallest_(lambda,deg)": [
        3,
        2
      ],
      "heat_trace_parity": {
        "c_minus_three_half": 0.27419955293448567,
        "c_minus_one (integer power)": 4.604300807772854e-07,
        "c_plus_half (pole witness)": 0.2884041461832084,
        "pole_at_minus_half_present": true,
        "expansion_runs_in": "HALF-INTEGER powers (odd d_eff; pole at s=-1/2)"
      },
      "reason": "The zero-weight series is sum min(a,b) (a^2+ab+b^2)^{-s} -- the MIN-WEIGHTED Eisenstein/Loeschian form (norm form of the Eisenstein integers Z[omega], discriminant -3; smallest level (1,1) adjoint C2=3, deg 2 -- VERIFIED). Closed forms in the literature (Epstein 1903; Chowla-Selberg) give (i) the UNWEIGHTED Epstein zeta and (ii) special s-values via Dedekind-eta / Lerch transcendents -- NEITHER delivers a scheme-FREE finite value at s=-1/2 for the min-WEIGHTED sum. The min(a,b) weight breaks the modular closed form, and the continuation hits a POLE at s=-1/2 (nonzero t^{+1/2} heat coefficient -- VERIFIED here), so only the residue (scheme-invariant negative) is closed-form; the branch-deciding FINITE part remains scheme-dependent. No closed form fixes the sign."
    },
    "4_fermion_susy_cancellation": {
      "verdict": "INCONCLUSIVE",
      "fixes_sign": false,
      "theory_is_supersymmetric": false,
      "I2_supertrace_cancels": false,
      "reason": "The frozen content is the chiral Standard Model (3 families from APS index +3 on K6=SU(3)/T^2; non-SUSY), so there is NO boson-fermion degeneracy to enforce Str 1 = Str m^2 = Str m^4 = 0. The corpus's OWN I2 supertrace (TOE_FINAL SS.III) already computed the signed bose-minus-fermi vacuum-energy sum over the full 17-row inventory at the frozen radii: Str rho = (-88.93 +/- band)/R_Y^4 + c_loop, graded/ungraded ratio 0.58 at k=0 (NOT zero) -- NO structural cancellation at any order. The chiral fermion tower (spin-c on non-spin K6, a DIFFERENT bundle than the scalar Casimir tower) does not pair off or fix the scalar finite-part sign, and without exact degeneracy it does not even cancel the s=-1/2 pole -- it only renormalizes the (already scheme-invariant) residue. Fermions/SUSY do not fix int_loop."
    },
    "5_d_eff_parity": {
      "verdict": "INCONCLUSIVE",
      "fixes_sign": false,
      "zero_weight_slice_d_eff": 3,
      "zero_weight_slice_parity": "HALF-INTEGER powers (odd d_eff; pole at s=-1/2)",
      "full_L2_tower_leading_power_p": 3.0,
      "full_L2_tower_parity_witness": {
        "a0_coeff_of_t_minus_3_integer_volume_term": 0.4999951738554541,
        "coeff_of_t_minus_2p5_half_integer_odd_witness": 0.00023133666748974963,
        "half_integer_term_negligible_vs_integer_leading": true,
        "note": "genuine L^2(K6) trace (deg = dim(p,q) x zero-weight-mult, Peter-Weyl) runs in INTEGER powers: t^-3 coeff > 0, t^-2.5 coeff ~ 0 => d_eff=6 EVEN, pole-free at s=-1/2. The corpus c_loop uses the zero-weight SLICE (deg = zero-weight-mult alone) which is d_eff=3 ODD and pole-ful."
      },
      "full_L2_tower_is_even_dim_d6": true,
      "total_compact_dim": 9,
      "total_bulk_dim_D": 13,
      "reason": "The s=-1/2 pole comes from the ZERO-WEIGHT slice's d_eff=3 (ODD; half-integer heat powers -- VERIFIED). An even-dimensional pole-free sum DOES exist mathematically -- the FULL L^2(K6) tower (degeneracy dim(p,q)) is a genuine 6-manifold trace, Theta ~ t^{-3} (d_eff=6, EVEN; leading-power estimate ~%.2f confirms), regular at s=-1/2 -- BUT it is the WRONG operator: the physical c_loop is the T^2-invariant (Cartan zero-weight) KK-Casimir coefficient of the 4D MODULUS potential (the breathing modes sigma,rho,chi ARE the T^2-invariant sector), not the full 6D vacuum energy. Replacing the d_eff=3 slice with the d_eff=6 tower computes a DIFFERENT quantity. Moreover the full geometry is intrinsically ODD-dimensional in both relevant slices (compact 9, total bulk D=13), so the bulk one-loop (4pi)^{-13/2} also runs in half-integer powers. The correct physical object is genuinely odd-d and pole-ful; even-dimensionalizing changes the operator and does NOT fix int_loop honestly. (Witness: the full L^2(K6) trace has a nonzero t^{-3} coefficient and t^3*Theta_full approaches that constant at small t, certifying leading power 3 = d_eff 6 EVEN; the zero-weight slice runs in half-integer powers, d_eff 3 ODD.)"
    }
  },
  "any_principle_fixes_sign": false,
  "reflection_positivity": "Reflection positivity constrains the spectral measure (Laplacian >=0, eigenvalues lambda>0, degeneracies deg>0 -- all VERIFIED on the K6 zero-weight tower), which is satisfied REGARDLESS of the sign of the renormalized c_loop finite part. A reflection-positive (fully unitary) theory admits Casimir energies of EITHER sign (geometry-dependent), so RP places NO sign constraint on the renormalized finite vacuum-energy density. Boundedness-below is a statement about the FULL V_eff(sigma) at the -sigma corner -- which is exactly the quantity under dispute -- so using it to fix int_loop's sign is the circular reverse-fit-for-stability the discipline forbids. The corpus's own I2/I3 precedent confirms: a single signed vacuum-energy sum over the full inventory is computed NONZERO and the positivity/grading structure does not force a protective sign.",
  "a6_consistency": "(A) a6 is ITSELF a strictly larger OPEN object: the corpus's gap_01_a6_attempt derives only tr[E^3]=1/6 and records ALL purely-gravitational cubic c1..c8 and mixed E/Omega coefficients as MISSING (require full covariant Taylor/Synge recursion); the cubic backreaction on B1/B2 is b_i^* * J_i^{(a6)} with b_i^*=0 FORCED and J_i^{(a6)} MISSING. It cannot fix a downstream sign. (B) Even a COMPLETE a6 would not pin int_loop: UV/Hilbert-space consistency constrains the DIVERGENCE structure (pole residues -- already scheme-invariant negative) not the disputed FINITE part of a LOWER coefficient; a6 multiplies a different operator (curvature^3, e^{-(12 sigma+..)}) than the e^{-6 sigma} c_loop wall. Positive-definiteness = RP (candidate 1) and is sign-blind for the renormalized finite part.",
  "closed_form_casimir": "The zero-weight series is sum min(a,b) (a^2+ab+b^2)^{-s} -- the MIN-WEIGHTED Eisenstein/Loeschian form (norm form of the Eisenstein integers Z[omega], discriminant -3; smallest level (1,1) adjoint C2=3, deg 2 -- VERIFIED). Closed forms in the literature (Epstein 1903; Chowla-Selberg) give (i) the UNWEIGHTED Epstein zeta and (ii) special s-values via Dedekind-eta / Lerch transcendents -- NEITHER delivers a scheme-FREE finite value at s=-1/2 for the min-WEIGHTED sum. The min(a,b) weight breaks the modular closed form, and the continuation hits a POLE at s=-1/2 (nonzero t^{+1/2} heat coefficient -- VERIFIED here), so only the residue (scheme-invariant negative) is closed-form; the branch-deciding FINITE part remains scheme-dependent. No closed form fixes the sign.",
  "fermion_cancellation": "The frozen content is the chiral Standard Model (3 families from APS index +3 on K6=SU(3)/T^2; non-SUSY), so there is NO boson-fermion degeneracy to enforce Str 1 = Str m^2 = Str m^4 = 0. The corpus's OWN I2 supertrace (TOE_FINAL SS.III) already computed the signed bose-minus-fermi vacuum-energy sum over the full 17-row inventory at the frozen radii: Str rho = (-88.93 +/- band)/R_Y^4 + c_loop, graded/ungraded ratio 0.58 at k=0 (NOT zero) -- NO structural cancellation at any order. The chiral fermion tower (spin-c on non-spin K6, a DIFFERENT bundle than the scalar Casimir tower) does not pair off or fix the scalar finite-part sign, and without exact degeneracy it does not even cancel the s=-1/2 pole -- it only renormalizes the (already scheme-invariant) residue. Fermions/SUSY do not fix int_loop.",
  "d_eff_parity": "The s=-1/2 pole comes from the ZERO-WEIGHT slice's d_eff=3 (ODD; half-integer heat powers -- VERIFIED). An even-dimensional pole-free sum DOES exist mathematically -- the FULL L^2(K6) tower (degeneracy dim(p,q)) is a genuine 6-manifold trace, Theta ~ t^{-3} (d_eff=6, EVEN; leading-power estimate ~%.2f confirms), regular at s=-1/2 -- BUT it is the WRONG operator: the physical c_loop is the T^2-invariant (Cartan zero-weight) KK-Casimir coefficient of the 4D MODULUS potential (the breathing modes sigma,rho,chi ARE the T^2-invariant sector), not the full 6D vacuum energy. Replacing the d_eff=3 slice with the d_eff=6 tower computes a DIFFERENT quantity. Moreover the full geometry is intrinsically ODD-dimensional in both relevant slices (compact 9, total bulk D=13), so the bulk one-loop (4pi)^{-13/2} also runs in half-integer powers. The correct physical object is genuinely odd-d and pole-ful; even-dimensionalizing changes the operator and does NOT fix int_loop honestly. (Witness: the full L^2(K6) trace has a nonzero t^{-3} coefficient and t^3*Theta_full approaches that constant at small t, certifying leading power 3 = d_eff 6 EVEN; the zero-weight slice runs in half-integer powers, d_eff 3 ODD.)",
  "well_verdict": "OWNER-LOCKED / Lambda-hard -- NOT closed by any principle checked. The five candidate principles were tested against OUR frozen geometry K6 x S^2 x S^1_Y, target-blind, and NONE fixes the sign of int_loop (the finite part of the scalar SU(3)/T^2 Casimir = the sign of c_loop's underlying density) WITHOUT the full Lambda resolution: (1) reflection positivity is satisfied for EITHER sign of the renormalized finite part (RP constrains the spectral measure, not the renormalized vacuum-energy sign; Casimir energies of either sign are RP-consistent) and boundedness-below is the disputed quantity itself (circular); (2) a6 UV-completion is ITSELF a strictly larger open object (only tr[E^3]=1/6 derived; cubic c1..c8 + mixed coeffs MISSING; b_i^*=0 forced) and even if complete would constrain the divergence structure (already-scheme-invariant residue), not the disputed finite part of a lower coefficient; (3) NO closed form delivers a scheme-free finite value at s=-1/2 for the MIN-WEIGHTED Eisenstein-form sum the geometry produces (Epstein / Chowla-Selberg cover the residue and the unweighted zeta's special s-values, not the min-weighted finite part); the continuation hits the same s=-1/2 pole; (4) the theory is non-SUSY chiral SM (no bose-fermi degeneracy) and the corpus's OWN I2 supertrace already computed the signed bose-minus-fermi vacuum-energy sum and found NO cancellation (ratio 0.58 at k=0); (5) an even-dimensional pole-free sum exists (the full L^2(K6) 6D tower, Theta ~ t^{-3}) but it is the WRONG operator -- the physical c_loop is the T^2-invariant d_eff=3 modulus KK-Casimir, and both relevant slices (compact 9, bulk D=13) are intrinsically ODD-dimensional, so the pole is GENUINE. ENDPOINT: the int_loop sign is genuinely scheme-/owner-locked. Settling the Gap-04 well STANDS-vs-RUNAWAY without the full Lambda resolution is NOT achieved by any principle checked. This is the honest Lambda-hard answer; the favorable STANDS was NOT forced.",
  "geometry_specific_finding": "On OUR frozen geometry, the scalar Casimir tower is the MIN-WEIGHTED Eisenstein/Loeschian form (norm form of the Eisenstein integers Z[omega], disc -3), restricted to the T^2 zero-weight (Cartan-invariant) sublattice. This sublattice has d_eff=3 (ODD), forcing a simple pole at s=-1/2 -- the geometric ROOT CAUSE of int_loop's scheme dependence. The min(a,b) zero-weight degeneracy is what breaks the Chowla-Selberg closed form, and the odd parity is shared by the 9D compact and 13D bulk slices, so it is not removable by re-slicing to the right operator. Two corpus-specific facts seal the result: (a) int_a4 = +(1/36) R_K6 R_S2 with R_K6=+30, R_S2=+2 is geometry-forced POSITIVE and convention-invariant, so the WHOLE branch rides on int_loop's sign and nothing else; (b) the I2 supertrace over this geometry's full 17-row inventory is computed NONZERO (0.58), so the fermions provably do not cancel it. The pole's odd parity is intrinsic to SU(3)/T^2's zero-weight Casimir spectrum -- a geometry-specific, not generic, obstruction.",
  "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). No reference was made to the corpus's ASSUMED c_loop>0 sign, nor to the DESIRE for stability, to pick int_loop's sign -- that circular reverse-fit is exactly what this check refuses. Every candidate was evaluated by a genuine geometric/physical principle applied to the frozen geometry, with the spectral computation (Eisenstein tower, heat-trace parity, full-dim tower power) performed IN THIS PASSAGE. The frozen c_loop MAGNITUDE was not read into any decision (a SIGN investigation does not need it). The favorable BRANCH-STANDS was NOT forced: the honest endpoint is that NO principle fixes the sign, which leaves the UNFAVORABLE possibility (RUNAWAY) equally live and the well owner-locked.",
  "what_requires_chris": "The named FRG-2 NLO Litim shell-projection SCHEME (renormalization scale mu + shell-by-shell finite subtraction) that fixes the FINITE part of zeta_{Delta_K6}(-1/2) at s=-1/2 -- the branch-deciding int_loop sign and the e^{-6 sigma} c_loop wall coefficient. This is the open object B-UQFC-14-FRG-2 and the unfilled CONVENTION_FREEZE slot in gilkey_a4_cross_terms.py. NONE of the five principles checked (reflection positivity, a6 consistency, closed-form coset Casimir, fermion/SUSY cancellation, d_eff parity) substitutes for that scheme choice; each fails for a precise, geometry-specific reason recorded above. The sign remains owner-must-rule.",
  "provenance": {
    "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86",
    "gilkey_a4_cross_terms.py": "6e58698a4658ff1f49847617d8a236be8321f2c97227f570829111ba3208212d",
    "a6_recursion_scaffold.json": "e79df30272495d23d819b6faca8e4165c5fda2c5ee7246b0bb1eee30dbf2de3d",
    "b1_b2_derive_then_compare.json": "3bba676ff0af2fff8c99fc3a9857e589b62e2ee37638bcdf640168e3a0bfa219"
  },
  "non_promotion": "no gate flipped; no status word emitted for any gate. STAGED principle-check; the honest endpoint is NO-PRINCIPLE-FOUND = genuinely Lambda-hard. Not promotable as a sign-derivation."
}