{ "schema": "gap04_full_supertrace_residue_result_v1", "object": "Str_pole = Res_{s=-1/2} sum_species (-1)^F g_species zeta_species = r0 * (n_B - n_F); the full-spectrum supertrace of the s=-1/2 pole residue (a local Seeley-DeWitt coefficient) on K6^{W-rig} x S^2 x S^1_Y/Z2.", "outcome": "RESIDUE-SURVIVES-scheme-dependent-Lambda-hard", "partA_per_dof_pole_residue": { "method": "Mellin/heat-kernel split on the SU(3)/T^2 zero-weight (Weyl-rigid/T^2-invariant) tower; r0 = c_{+1/2}/Gamma(-1/2)", "t_plus_half_heat_coeff_c_half": 0.28786411243788057, "t_minus_three_half_leading_c_lead": 0.274199564987461, "per_real_scalar_dof_residue_r0": -0.08120496685733616, "matches_prior_R_minus_0_0812": true, "single_dof_slice_d_eff": 3 }, "partB_field_content_ledger": { "authority": "Fable_Quantum.md SS4 nine-row quantum-object ledger (graviton, KK vectors, moduli, gauge, chiral fermions, Higgs, KK tower, ghosts, antifields); gauge algebra su(3)+su(2)+u(1); chiral family index -3 (Fable_GUT C2/E); Weyl-rigid chamber (Fable_GUT App A.4).", "bosons_physical": [ { "row": "Q4-Row1", "name": "graviton h_mn (physical)", "g_dof": 2, "F": 0, "authority": "Fable_Quantum.md SS4 Row1: 2 TT polarizations" }, { "row": "Q4-Row4", "name": "gauge A_mu^a (physical, 12 gens x2)", "g_dof": 24, "F": 0, "authority": "Fable_Quantum.md SS4 Row4: su(3)+su(2)+u(1)=12 generators, 2 transverse pol each" }, { "row": "Q4-Row3", "name": "moduli (3 K6 shape + rho_S2 + chi_Y)", "g_dof": 5, "F": 0, "authority": "Fable_GUT App A.4 Weyl-rigid chamber u in [1/2,3/2]^3 (3 shape) + gilkey sigma,rho,chi radions" }, { "row": "Q4-Row6", "name": "Higgs doublet (4 real scalars)", "g_dof": 4, "F": 0, "authority": "Fable_Quantum.md SS4 Row6: complex doublet = 4 real (1 physical h + 3 Goldstone)" } ], "faddeev_popov_ghosts_covariant": [ { "row": "Q4-Row8", "name": "diffeo ghost c^mu", "removes_dof": 2, "authority": "Fable_Quantum.md SS4 Row8: diffeo ghost" }, { "row": "Q4-Row8", "name": "YM ghosts c^a (12)", "removes_dof": 24, "authority": "Fable_Quantum.md SS4 Row8: 12 YM ghosts" } ], "fermions": [ { "row": "Q4-Row5", "name": "chiral matter (3 generations x 15 Weyl)", "g_dof": 90, "F": 1, "n_families_from_index": 3, "weyl_per_generation": 15, "authority": "Fable_Quantum.md SS4 Row5 (15 Weyl/gen) x family index |-3|=3 (Fable_GUT C2/E, Borel-Weil-Bott spin-c index)" } ], "weyl_per_generation": 15, "n_families_from_index": 3, "supertrace_weights": { "scheme_P_physical": { "n_B": 35, "n_F": 90, "Str1": -55 }, "scheme_C_covariant_minus_ghosts": { "graviton_net_10_minus_8": 2, "gauge_net_48_minus_24": 24, "moduli": 5, "higgs": 4, "n_B": 35, "n_F": 90, "Str1": -55 }, "n_B_scheme_invariant": true, "Str1": -55 }, "note_W_rigidity": "The Weyl-rigidity projector restricts the K6 metric moduli to the Weyl-rigid chamber u in [1/2,3/2]^3 (3 Cartan-shape breathing directions), i.e. the T^2-invariant zero-weight slice -- which is exactly the d_eff=3 tower that produces the s=-1/2 pole. The projector does NOT pair bosons with fermions; it only selects the Cartan-invariant bosonic moduli." }, "partC_supertrace_residue": { "per_species_residues": [ { "species": "graviton h_mn (physical)", "statistics": "boson(+)", "g_dof": 2, "signed_weight": 2, "residue_at_minus_half": -0.16240993371467233, "authority": "Fable_Quantum.md SS4 Row1: 2 TT polarizations" }, { "species": "gauge A_mu^a (physical, 12 gens x2)", "statistics": "boson(+)", "g_dof": 24, "signed_weight": 24, "residue_at_minus_half": -1.948919204576068, "authority": "Fable_Quantum.md SS4 Row4: su(3)+su(2)+u(1)=12 generators, 2 transverse pol each" }, { "species": "moduli (3 K6 shape + rho_S2 + chi_Y)", "statistics": "boson(+)", "g_dof": 5, "signed_weight": 5, "residue_at_minus_half": -0.4060248342866808, "authority": "Fable_GUT App A.4 Weyl-rigid chamber u in [1/2,3/2]^3 (3 shape) + gilkey sigma,rho,chi radions" }, { "species": "Higgs doublet (4 real scalars)", "statistics": "boson(+)", "g_dof": 4, "signed_weight": 4, "residue_at_minus_half": -0.32481986742934466, "authority": "Fable_Quantum.md SS4 Row6: complex doublet = 4 real (1 physical h + 3 Goldstone)" }, { "species": "chiral matter (3 generations x 15 Weyl)", "statistics": "fermion(-)", "g_dof": 90, "signed_weight": -90, "residue_at_minus_half": 7.308447017160255, "authority": "Fable_Quantum.md SS4 Row5 (15 Weyl/gen) x family index |-3|=3 (Fable_GUT C2/E, Borel-Weil-Bott spin-c index)" } ], "Str_identity_n_B_minus_n_F": -55, "per_dof_residue_r0": -0.08120496685733616, "supertrace_pole_residue_Str_pole": 4.466273177153489, "cancels": false, "cancellation_test_is_integer_Str1": "cancellation requires Str[1] == 0 EXACTLY (an integer dof balance); Str[1] = -55, so the test is immune to float noise in r0." }, "supertrace_residue": "Str_pole = r0 * (n_B - n_F) = (-0.081205)(-55) = 4.466273 -- NONZERO (SURVIVES). n_B=35, n_F=90.", "per_species_residues_summary": { "graviton h_mn (physical)": -0.16240993371467233, "gauge A_mu^a (physical, 12 gens x2)": -1.948919204576068, "moduli (3 K6 shape + rho_S2 + chi_Y)": -0.4060248342866808, "Higgs doublet (4 real scalars)": -0.32481986742934466, "chiral matter (3 generations x 15 Weyl)": 7.308447017160255 }, "sign_if_invariant": "NOT fixed: Str_pole != 0, so the s=-1/2 pole SURVIVES, the finite part stays scheme-dependent, and no basis/regulator-independent int_loop sign exists. The residue is a genuine local heat-kernel (Seeley-DeWitt) obstruction.", "forced_branch": "NONE forced -- scheme-dependent / Lambda-hard. Both BRANCH-STANDS (int_loop<0) and BRANCH-RUNAWAY (int_loop>0) remain live; neither is excluded by a banked theorem (guard (f)).", "int_a4_sign": "POSITIVE", "partD_branch_propagation": { "if_cancels": "Str_pole=0 -> finite part scheme-INVARIANT -> data-blind minimal-subtraction F<0 -> int_loop<0 -> c_a4>0 (a4 density +(1/36)R_K6 R_S2>0, growth-8>6) -> BRANCH-STANDS", "if_survives": "Str_pole!=0 -> pole SURVIVES -> finite part scheme-DEPENDENT -> no forced sign -> Lambda-hard; BOTH branches live", "actual": "NONE forced -- scheme-dependent / Lambda-hard. Both BRANCH-STANDS (int_loop<0) and BRANCH-RUNAWAY (int_loop>0) remain live; neither is excluded by a banked theorem (guard (f))." }, "partE_banked_invariant_linkage": "TIED to the BANKED three-family index (-3). The chiral spectrum has n_F = 90 fermionic dof (45 Weyl = 3 families x 15 Weyl x 2) with NO degenerate bosonic partners; n_B = 35. n_B - n_F = -55 != 0. The index -3 is the banked topological integer (Fable_GUT C2/E, Borel-Weil-Bott spin-c) that FORCES three unpaired chiral generations -- so the very invariant that delivers three families is what makes Str[1] != 0 and BLOCKS the supertrace cancellation. The (non)cancellation is therefore NOT a fresh convention: it is forced by a banked invariant. But it pays AGAINST the favorable sign -- it forbids the SUSY-type cancellation that would have made int_loop scheme-invariant. (Consilience holds; it just does not rescue STANDS.)", "determinant_line_route": "The canonical spin-c DETERMINANT LINE of the index bundle (Quillen metric; banked via the |Index|=3 construction) carries a forced sign for the FERMION-determinant global phase (anomaly cancellation, Fable_GUT E), but it acts on the chiral fermion functional determinant, NOT on the BOSONIC scalar K6-Casimir finite part that sets c_loop. With Str[1] = -55 != 0 the bosonic s=-1/2 pole is not removed, so the determinant line does NOT force int_loop's sign. (It WOULD be the route IF the residue cancelled and a residual global phase remained -- but it does not cancel.)", "d_eff_after_quotienting": "Per-species relevant slice is d_eff=3 (ODD; half-integer heat powers, verified r0 from t^{+1/2}). Full quotienting (Weyl-rigid projection) + ghost subtraction change the COEFFICIENT n_B, not the per-mode d_eff parity. Boson/fermion pairing would even-ize the effective slice ONLY if Str[1]=0; since Str[1]=-55 != 0, the NET residue rides on a d_eff=3 (ODD) slice and the s=-1/2 pole SURVIVES. d_eff parity is DECIDED BY the integer Str[1]: ODD (3) -- pole survives.", "well_verdict": "OWNER-LOCKED / Lambda-hard -- the favorable STANDS is NOT forced. The full-spectrum supertrace residue at s=-1/2 does NOT cancel: Str[1] = n_B - n_F = 35 - 90 = -55 != 0 (a banked INTEGER, immune to numerical noise), so Str_pole = r0 * Str[1] = (-0.08120)(-55) = 4.46627 != 0. Because the pole RESIDUE is a LOCAL Seeley-DeWitt coefficient that does NOT cancel across the complete (graviton + KK vectors + moduli + gauge + ghost-subtracted + chiral-fermion + Higgs) content, the int_loop finite part remains genuinely SCHEME-DEPENDENT: no basis-independent, regulator-independent sign exists. The three-family index (-3) -- the banked invariant that forces three UNPAIRED chiral generations -- is precisely what makes n_B != n_F, so the non-cancellation is FORCED by a banked theorem (not a fresh convention), and it forces AGAINST the favorable cancellation. d_eff stays ODD (3); the pole survives. This is the honest Lambda-hard endpoint: the Gap-04 well stays owner-locked, with BOTH STANDS and RUNAWAY live, neither excluded by a prior theorem (guards (c),(f) respected).", "integrity_guards_attest": { "a_sign_not_chosen_for_STANDS": "The supertrace says SURVIVES (non-cancellation); we report the UNFAVORABLE Lambda-hard endpoint, not STANDS.", "b_committed_c_loop_never_read": "The committed c_loop magnitude (1.3637877e-5) was NEVER opened and entered NO input/sign decision; no finite subtraction tuned to any target. (veff_coefficients file deliberately not in the authority list.)", "c_we_exist_not_used": "Runaway is NOT excluded by anthropics; it stays live.", "d_full_geometry_not_scalar_only": "Used the full nine-row content (graviton, KK/gauge vectors, moduli, ghosts, chiral fermions, Higgs), NOT a scalar-only tower.", "e_no_single_gap_stabilizer": "No stabilizing term was added; only the existing field content and its banked invariants were used.", "f_runaway_not_excluded_without_theorem": "No prior banked theorem excludes the runaway chamber; it remains live. We do NOT exclude it." }, "what_requires_chris": "(1) The CURVATURE-WEIGHTED per-species residue refinement: the leading supertrace weight is n_B - n_F (computed here, NONZERO), but the full a_{d+1} Seeley-DeWitt coefficient also carries species-specific E-endomorphism and Omega-bundle terms (graviton/gauge/fermion curvature couplings). Those refine the per-species residue at the SAME pole; since the LEADING dof weight already does not cancel (and cannot, by the index -3), the refinement cannot rescue cancellation, but the exact per-species a_{d+1} densities on K6 x S^2 x S^1_Y are the owner-signed heat-kernel computation. (2) Equivalently: the named FRG-2 NLO Litim shell-projection SCHEME that fixes the (now confirmed scheme-DEPENDENT) finite part -- B-UQFC-14-FRG-2, the unfilled CONVENTION_FREEZE slot. The supertrace test CONFIRMS the finite part is genuinely scheme-dependent (the pole does not cancel), so the sign is owner-must-rule, NOT derivable data-blind.", "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 field content is from Fable_Quantum.md SS4 + the banked index -3; the per-dof residue r0 is re-derived from SU(3)/T^2 group theory + frozen geometry ALONE. The cancellation test is the INTEGER Str[1] = n_B - n_F, which is target-blind and favorable-outcome-blind. The favorable STANDS was NOT forced: the supertrace SURVIVES and we report Lambda-hard.", "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 full-spectrum supertrace residue analysis only." }