{
  "schema": "gap04_higher_operator_wall_result_v1",
  "object": "The -sigma (shrinking-internal) corner of V(sigma,rho,chi): is the asymptotic fate set by a FORCED higher-operator (a6, e^{-12 sigma}) wall, a forced non-perturbative wall, or is the corner outside EFT validity (perturbatively undecidable)?",
  "outcome": "EFT-BREAKDOWN-at-corner-perturbatively-undecidable",
  "operator_tower": [
    {
      "name": "c_KK (K6 zero-mode)",
      "exp_sigma": 4,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 4,
      "order": "a0/KK",
      "coeff_sign": "POSITIVE (FORCED)",
      "sign_status": "geometry-forced (zero-mode universality, +)"
    },
    {
      "name": "c_Wilson (Hosotani)",
      "exp_sigma": 4,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 4,
      "order": "Wilson",
      "coeff_sign": "NEGATIVE (FORCED, but cos(theta_W) bounded)",
      "sign_status": "geometry-forced; |c_Wilson| ~ 5e-5 << c_KK"
    },
    {
      "name": "c_bdry (Dai-Freed)",
      "exp_sigma": 2,
      "exp_rho": 0,
      "exp_chi": 2,
      "growth_minus_sigma": 2,
      "order": "boundary",
      "coeff_sign": "NEGATIVE (FORCED)",
      "sign_status": "geometry-forced (orbifold bordism, -)"
    },
    {
      "name": "c_loop (KK-Casimir wall)",
      "exp_sigma": 6,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 6,
      "order": "a_loop (zeta(-1/2))",
      "coeff_sign": "POSITIVE assembled; UNDERLYING DENSITY SIGN DISPUTED",
      "sign_status": "OWNER-LOCKED: assembled coeff +, but the branch-deciding underlying-density sign (int_loop) is scheme-dependent (zeta has a POLE at s=-1/2; residue - but finite part flips with mu). NOT geometry-forced."
    },
    {
      "name": "c_a4_K6_S2 (cross-term)",
      "exp_sigma": 8,
      "exp_rho": 4,
      "exp_chi": 1,
      "growth_minus_sigma": 8,
      "order": "a4 cross",
      "coeff_sign": "int_a4 = +(1/36)R_K6 R_S2 FORCED +; RELATIVE sign vs c_loop DISPUTED",
      "sign_status": "int_a4 magnitude POSITIVE & convention-invariant (+(1/36)*30*2). But the OVERALL operator sign depends on the SAME unfilled CONVENTION_FREEZE / int_loop sign that c_loop rides on. As-run code: c_a4 = -5.97e-8 (NEGATIVE) -> dominates c_loop -> RUNAWAY. Owner-locked, both live."
    },
    {
      "name": "a6 (cubic-curvature, pure K6)",
      "exp_sigma": 12,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 12,
      "order": "a6 (Seeley-DeWitt)",
      "coeff_sign": "OPEN (see a6 focus): leading tr[E^3]=+1/6 FORCED +, but 8 purely-gravitational c1..c8 + mixed coeffs MISSING",
      "sign_status": "a6 cubic coefficient sign is NOT geometry-forced as a whole: only tr[E^3]=+1/6 is derived; the c1..c8 purely-gravitational cubic coefficients (R^3, R|Ric|^2, R|Riem|^2, tr Ric^3, ... , Riem^3) and the K6-block cubic Riemann contractions (K5..K8) are MISSING. Sign of the DOMINANT contraction undetermined."
    },
    {
      "name": "a8 (curvature^4, pure K6)",
      "exp_sigma": 14,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 14,
      "order": "a8 (Seeley-DeWitt)",
      "coeff_sign": "OPEN (higher Seeley-DeWitt; even a6 unsupplied)",
      "sign_status": "a8 coefficients are a strictly larger open object than a6 (which is itself open). Sign undetermined."
    },
    {
      "name": "a10 (curvature^5, pure K6)",
      "exp_sigma": 16,
      "exp_rho": 0,
      "exp_chi": 0,
      "growth_minus_sigma": 16,
      "order": "a10 (Seeley-DeWitt)",
      "coeff_sign": "OPEN (higher Seeley-DeWitt; even a6 unsupplied)",
      "sign_status": "a10 coefficients are a strictly larger open object than a6 (which is itself open). Sign undetermined."
    },
    {
      "name": "V_NP (D-brane/NS5 instanton)",
      "exp_sigma": null,
      "exp_rho": null,
      "exp_chi": null,
      "growth_minus_sigma": "S_inst -> 0 at -sigma (no wall)",
      "order": "non-perturbative",
      "coeff_sign": "OPEN/OUTSIDE-CORPUS (requires brane stack / flux data)",
      "sign_status": "S_inst ~ Vol(cycle) GROWS at +sigma, SHRINKS at -sigma. e^{-S_inst} suppresses the +sigma corner (the named B-UQFC-14-NP-1 +modulus wall candidate) but at -sigma the action -> 0 and the dilute-instanton expansion BREAKS. No controlled forced-positive p>8 -sigma wall. Sign and magnitude OUTSIDE the frozen corpus."
    },
    {
      "name": "V_flux (flux quantization)",
      "exp_sigma": null,
      "exp_rho": null,
      "exp_chi": null,
      "growth_minus_sigma": "~ +Vol^{-2} (grows at -sigma)",
      "order": "non-perturbative (flux)",
      "coeff_sign": "POSITIVE by quantization, but power-law not p>8 exponential",
      "sign_status": "A quantized flux energy ~ n^2 / Vol(cycle) ~ e^{-c sigma} with POSITIVE coefficient (n^2>0) and GROWS at -sigma -- but it is a POWER-LAW (curvature-like, low growth rate, c ~ a few), NOT a p>8 exponential, so it does NOT outrun a6's e^{-12 sigma}. And the flux SECTOR (which cycles, which quanta) is OUTSIDE the frozen corpus."
    }
  ],
  "operator_tower_summary": "-sigma corner growth ladder (operator : sigma-growth at sigma->-inf): c_bdry e^{-2s} (+2) ; c_KK/c_Wilson e^{-4s} (+4) ; c_loop e^{-6s} (+6, underlying-density sign DISPUTED) ; a4 K6xS2 e^{-(8s+4r+c)} (+8, relative sign DISPUTED, as-run NEGATIVE) ; a6 cubic e^{-12s} (+12, sign OPEN) ; a8 e^{-14s}, a10 e^{-16s} (+14,+16, sign OPEN, even-larger open objects). Non-perturbative: instanton/wrapped-brane actions GROW at +sigma (no -sigma wall); flux ~ n^2/Vol is positive but low-growth power-law.",
  "fastest_operator_within_EFT": "a6 (cubic-curvature Seeley-DeWitt, pure-K6, e^{-12 sigma}) is the FASTEST-growing operator at the -sigma corner among the perturbative tower (growth +12 > a4 +8 > c_loop +6). Its coefficient sign is NOT geometry-forced (only tr[E^3]=+1/6 derived; c1..c8 + K6-block K5..K8 MISSING). a8/a10 grow even faster (+14/+16) and are larger open objects -- which is ITSELF the signal that the curvature expansion is not convergent here.",
  "fastest_perturbative_growth": {
    "name": "a6 (cubic-curvature, pure K6)",
    "growth": 12
  },
  "tower_has_unbounded_growth_ladder": true,
  "a6_focus": {
    "derived_positive_piece": {
      "tr_E3_coeff": "1/6",
      "is_positive": true,
      "note": "tr[E^3]=+1/6 is the ONLY a6 coefficient derived in-passage (flat cov-const E resummation). It is POSITIVE but it is the potential-sector piece; on the graviton bundle E->Riem on Sym^2 requires the MISSING endomorphism matrix, so even tr[E^3]'s CONTRIBUTION sign/magnitude to C_a6 is not pinned."
    },
    "cubic_invariants_computed": {
      "K1_R3": 32768,
      "K2_R_Ric2": 4864,
      "K3_R_Riem2": 2048,
      "K4_trRic3": 752
    },
    "K5_K6_block_missing_geometry": true,
    "K7_K6_block_missing_geometry": true,
    "weyl2_K6_in_corpus_scalar_data": null,
    "weyl_flag": "corpus scalar |Riem|^2(K6)=60 implies |Weyl|^2=0 (max-sym value), but true SU(3)/T^2 has |Weyl|^2 != 0 -> K5..K8 NOT determined by scalar data -> MISSING_GEOMETRY",
    "missing_purely_gravitational_c1_c8": [
      "c1_R3",
      "c2_R_Ric2",
      "c3_R_Riem2",
      "c4_trRic3",
      "c5_Ric.Ric.Riem",
      "c6_Ric.Riem.Riem",
      "c7_Riem3_I1",
      "c8_Riem3_I2"
    ],
    "missing_mixed_E_Omega": [
      "tr[E^2 R]",
      "tr[E Omega^2]",
      "tr[Omega_{MN}Omega^{NR}Omega_R^M]",
      "tr[E R_{..}-contraction]",
      "..."
    ],
    "cubic_coeffs_are_sign_mixed": true,
    "a6_sign_geometry_forced": false,
    "verdict": "a6's e^{-12 sigma} coefficient sign is NOT geometry-forced. Only the potential-sector tr[E^3]=+1/6 is derived (and even its contribution to C_a6 needs the missing graviton endomorphism E-matrix). The DOMINANT pieces are the purely-gravitational cubic-curvature contractions c1..c8 of {R^3, R|Ric|^2, R|Riem|^2, tr Ric^3, Ric.Ric.Riem, Ric.Riem.Riem, Riem^3(I1), Riem^3(I2)} -- whose UNIVERSAL coefficients are a SIGN-MIXED rational set (like a4's +5/-2/+2) and are MISSING, and whose K6-block VALUES (K5..K8) are MISSING_GEOMETRY (true flag manifold has Weyl != 0). The leading tr[E^3]=+1/6 being positive does NOT settle the sign of the full a6 coefficient, because the cubic gravitational sector can carry either sign. a6 sign is OPEN, not convenient-positive."
  },
  "a6_sign_status": "a6's e^{-12 sigma} coefficient sign is NOT geometry-forced. Only the potential-sector tr[E^3]=+1/6 is derived (and even its contribution to C_a6 needs the missing graviton endomorphism E-matrix). The DOMINANT pieces are the purely-gravitational cubic-curvature contractions c1..c8 of {R^3, R|Ric|^2, R|Riem|^2, tr Ric^3, Ric.Ric.Riem, Ric.Riem.Riem, Riem^3(I1), Riem^3(I2)} -- whose UNIVERSAL coefficients are a SIGN-MIXED rational set (like a4's +5/-2/+2) and are MISSING, and whose K6-block VALUES (K5..K8) are MISSING_GEOMETRY (true flag manifold has Weyl != 0). The leading tr[E^3]=+1/6 being positive does NOT settle the sign of the full a6 coefficient, because the cubic gravitational sector can carry either sign. a6 sign is OPEN, not convenient-positive.",
  "eft_validity_analysis": {
    "curvature_reaches_cutoff_sigma_unitvol": 1.7005986908310777,
    "curvature_note": "Unit-volume curvature R_K6_0=30 already exceeds M_13^2 for sigma below ~+1.70 in unit-vol normalization; the meaningful invariant breakdown is the operator-ratio crossing (test 2), which is normalization-light.",
    "c_a6_bracket": {
      "floor": 5.969729293225176e-09,
      "nominal": 1.6301340790033548e-05,
      "note": "magnitude bracket only; SIGN unknown. Uses shared one-loop prefactor x a4-like rational x computed cubic invariant scale (K1=R^3)."
    },
    "sigma_a4_equals_cloop": -2.7156337438401494,
    "sigma_a6_equals_a4_nominal": 1.4024289914042833,
    "sigma_a6_equals_a4_floor": -0.5756462732485115,
    "sigma_a6_equals_cloop_nominal": 0.02974141298947247,
    "sigma_EFT_breakdown_onset": 1.4024289914042833,
    "EFT_breakdown_note": "EFT (curvature/derivative expansion) stops converging for sigma below ~1.402, where the FIRST higher heat-kernel operator catches a lower one (a6 >~ a4 >~ c_loop). For sigma more negative than this the ordered expansion is INVALID."
  },
  "runaway_inside_or_outside_EFT": {
    "runaway_onset_sigma_a4_beats_cloop": -2.7156337438401494,
    "EFT_breakdown_onset_sigma": 1.4024289914042833,
    "runaway_inside_EFT_validity": false,
    "verdict": "The a4 cross-term can only OVERTURN the c_loop wall once it dominates c_loop -- at sigma < -2.716. But the a6 (and higher) operators catch a4 at essentially the SAME or LESS-negative sigma (~1.402), so the operator that 'wins' the -sigma runaway wins by VIOLATING the EFT ordering. The runaway onset is AT/BELOW the EFT-breakdown sigma: the perturbative runaway lies OUTSIDE the regime where the truncated heat-kernel expansion is trustworthy. A runaway that only appears where the EFT has broken down is NOT a trustworthy physical prediction."
  },
  "eft_validity": "The curvature/derivative (Seeley-DeWitt) expansion STOPS CONVERGING at the -sigma corner: the higher heat-kernel operators OUTGROW the lower ones (a6 e^{-12s} > a4 e^{-8s} > c_loop e^{-6s}), so the truncation is not ordered there. Operator-ratio crossings (data-blind, using on-disk c_loop=1.364e-05, c_a4=-5.970e-08 and a bracketed a6 magnitude) put the breakdown onset near sigma ~ 1.40; the a4-beats-c_loop runaway onset is at sigma ~ -2.72, i.e. AT/BELOW the breakdown. The perturbative runaway lies OUTSIDE EFT validity and is therefore NOT a trustworthy physical prediction. (Independently: the EFT cutoff is the KK scale set by R_K6; the unit-vol curvature already reaches M_13^2 near sigma ~ +1.70.)",
  "non_perturbative_sectors": [
    {
      "name": "V_NP (D-brane/NS5 instanton)",
      "exp_sigma": null,
      "exp_rho": null,
      "exp_chi": null,
      "growth_minus_sigma": "S_inst -> 0 at -sigma (no wall)",
      "order": "non-perturbative",
      "coeff_sign": "OPEN/OUTSIDE-CORPUS (requires brane stack / flux data)",
      "sign_status": "S_inst ~ Vol(cycle) GROWS at +sigma, SHRINKS at -sigma. e^{-S_inst} suppresses the +sigma corner (the named B-UQFC-14-NP-1 +modulus wall candidate) but at -sigma the action -> 0 and the dilute-instanton expansion BREAKS. No controlled forced-positive p>8 -sigma wall. Sign and magnitude OUTSIDE the frozen corpus."
    },
    {
      "name": "V_flux (flux quantization)",
      "exp_sigma": null,
      "exp_rho": null,
      "exp_chi": null,
      "growth_minus_sigma": "~ +Vol^{-2} (grows at -sigma)",
      "order": "non-perturbative (flux)",
      "coeff_sign": "POSITIVE by quantization, but power-law not p>8 exponential",
      "sign_status": "A quantized flux energy ~ n^2 / Vol(cycle) ~ e^{-c sigma} with POSITIVE coefficient (n^2>0) and GROWS at -sigma -- but it is a POWER-LAW (curvature-like, low growth rate, c ~ a few), NOT a p>8 exponential, so it does NOT outrun a6's e^{-12 sigma}. And the flux SECTOR (which cycles, which quanta) is OUTSIDE the frozen corpus."
    }
  ],
  "non_perturbative_forced_p_gt_8_wall": false,
  "nonperturbative_terms": "NO. The non-perturbative sectors (D-brane/NS5 instanton, flux quantization, wrapped-brane, KKLT uplift) have actions S_inst ~ Vol(cycle) that GROW at +sigma and SHRINK at -sigma -- so e^{-S_inst} suppresses the +sigma (decompactification) corner (the named B-UQFC-14-NP-1 +modulus wall candidate), NOT the -sigma corner. At the -sigma corner the instanton action -> 0 and the dilute-gas expansion BREAKS. A quantized-flux energy ~ n^2/Vol IS positive and grows at -sigma, but it is a LOW-growth power-law (curvature-like, not a p>8 exponential), so it does not outrun a6's e^{-12 sigma}; and its sector data is OUTSIDE the frozen corpus. There is NO derivable forced-positive p>8 non-perturbative wall that dominates BEFORE the perturbative runaway.",
  "well_verdict": "NO HIGHER-OPERATOR WALL IS FORCED, AND NO RUNAWAY IS TRUSTWORTHY -- the -sigma corner is PERTURBATIVELY UNDECIDABLE. (1) The fastest operator at the corner is a6 (e^{-12 sigma}), but its sign is NOT geometry-forced: only tr[E^3]=+1/6 is derived; the eight purely-gravitational cubic coefficients c1..c8 (a sign-mixed rational set, like a4's +5/-2/+2) and the K6-block cubic Riemann contractions K5..K8 are MISSING / MISSING_GEOMETRY (the true SU(3)/T^2 flag manifold has Weyl != 0). So a6 does NOT supply a forced-positive wall, and declaring it positive would violate the guard against convenient signs. (2) a8, a10, ... grow even faster and are larger open objects -- which is precisely the diagnostic that the curvature expansion has STOPPED CONVERGING at the corner. (3) Because higher operators outgrow lower ones there, the a4-cross-term runaway only 'wins' by violating the EFT ordering: it turns on at/below the EFT-breakdown sigma, so the runaway is OUTSIDE EFT validity and is NOT a trustworthy prediction. (4) No forced-positive p>8 non-perturbative -sigma wall exists in the frozen corpus (instanton/brane actions wall the +sigma corner, not -sigma; flux energy is positive but a low-growth power-law and its sector is outside the corpus). HONEST ENDPOINT: the no-runaway claim is neither confirmed (no forced wall) nor refuted (no trustworthy runaway); it is UNDECIDABLE at perturbative order because the EFT breaks down at exactly the corner in question. What resolves it is a RESUMMED / UV-complete treatment (the full Gilkey-Avramidi a6 with FORCED c1..c8 and the K6-block Riemann data, or the non-perturbative sector) -- owner-must-rule (Chris). The favorable STANDS was NOT forced; the unfavorable RUNAWAY was NOT forced either.",
  "cross_check_frozen_magnitude_post_hoc": {
    "performed_after_structure_and_signs": true,
    "frozen_c_loop_FRG2_target": 1.3637877214788921e-05,
    "c_loop_on_disk": 1.3637166327064904e-05,
    "c_a4_on_disk": -5.969729293225176e-08,
    "note": "the frozen magnitudes are used ONLY in the EFT operator-ratio test and this cross-check; they never select a sign or a branch."
  },
  "integrity_attest": "Guards honored. (a) No sign was chosen because it gives STANDS: a6's sign is reported OPEN, not positive; the endpoint is undecidable, not favorable. (b) The committed c_loop=1.3637877e-5 was read ONLY for the EFT operator-ratio test and the final magnitude cross-check, NEVER to pick a sign. (c) No anthropic 'we exist' move: the runaway is dismissed ONLY by the EFT-breakdown computation, not by existence. (e) No gap-specific stabilizing epicycle was introduced; the only 'stabilizer' considered (a6) is reported as sign-OPEN and the non-perturbative sectors as outside-corpus. a6 was NOT declared positive for convenience. A sign would be FORCED only by the missing geometry/group computation (Gilkey-Avramidi order-6 recursion + SU(3)/T^2 Riemann data), which is not performed here and is owner-must-rule. No observed value entered on any input side.",
  "what_requires_chris": "(A6 SIGN -- the load-bearing item) The eight universal purely-gravitational a6 cubic coefficients c1..c8 (for R^3, R|Ric|^2, R|Riem|^2, tr Ric^3, Ric.Ric.Riem, Ric.Riem.Riem, Riem^3 I1, Riem^3 I2) from the full Gilkey 1995 Th.4.8.16 / Avramidi 2000 Ch.4 order-6 recursion, the mixed E/Omega coefficients, the graviton-bundle endomorphism E-matrix on Sym^2 TM, AND the index-resolved Riemann/curvature-2-form of the true SU(3)/T^2 flag manifold (its structure constants -> K5..K8 on the K6 block; the corpus has only scalar |Riem|^2 and the manifold has Weyl != 0). Only with these is the SIGN of a6's e^{-12 sigma} coefficient forced. (EFT) Whether the -sigma corner admits a RESUMMED / UV-complete treatment (or is intrinsically strong-curvature) -- since the truncated curvature expansion provably does not converge there. (NON-PERTURBATIVE) The B-UQFC-14-NP-1 object: which cycles/quanta/brane stack supply a forced-sign non-perturbative term -- all outside the frozen corpus.",
  "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 operator tower and growth rates are built from the frozen SU(3)/T^2 x S^2 x S^1_Y geometry + Gilkey heat-kernel structure ALONE. a6's sign is reported OPEN (not the convenient +); the perturbative runaway is dismissed ONLY by the EFT-breakdown computation, not by any desire for stability and not by 'we exist'. The committed c_loop magnitude was read strictly post-structure, for the EFT-ratio test and cross-check only.",
  "provenance": {
    "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86",
    "veff_coefficients_frg4.yaml": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac",
    "gilkey_a4_cross_terms.py": "6e58698a4658ff1f49847617d8a236be8321f2c97227f570829111ba3208212d",
    "a6_recursion_scaffold.json": "e79df30272495d23d819b6faca8e4165c5fda2c5ee7246b0bb1eee30dbf2de3d",
    "curvature_contractions_d13.json": "60e354e68ed8168b6f3c2af4c7e6f99e36a49d999c5d66d01e965966c4ada058"
  },
  "non_promotion": "no gate flipped; no status word emitted for any gate. STAGED higher-operator / EFT-validity instrument. NOT a sign-derivation and NOT promotable as one: a6's sign is open and the corner is outside EFT validity."
}