{ "schema": "gap04_zeta_continuation_frg2_result_v1", "object": "c_loop = -(1/2)(4 pi)^{-D/2} zeta_{Delta_K6}(-1/2); scalar K6=SU(3)/T^2 KK tower; eigenvalue = C2(p,q); degeneracy = T^2 zero-weight mult.", "outcome": "SCHEME-AMBIGUOUS-owner-locked", "part0_tower_verification": { "zero_weight_mult_matches_kostant": true, "casimir_normalization_C2_adjoint_3_fund_4over3": true, "panel": [ { "pq": [ 0, 0 ], "dim": 1, "triality": 0, "C2": 0.0, "zero_weight_mult_closed_form": 1, "zero_weight_mult_kostant": 1, "match": true }, { "pq": [ 1, 1 ], "dim": 8, "triality": 0, "C2": 3.0, "zero_weight_mult_closed_form": 2, "zero_weight_mult_kostant": 2, "match": true }, { "pq": [ 3, 0 ], "dim": 10, "triality": 0, "C2": 6.0, "zero_weight_mult_closed_form": 1, "zero_weight_mult_kostant": 1, "match": true }, { "pq": [ 0, 3 ], "dim": 10, "triality": 0, "C2": 6.0, "zero_weight_mult_closed_form": 1, "zero_weight_mult_kostant": 1, "match": true }, { "pq": [ 2, 2 ], "dim": 27, "triality": 0, "C2": 8.0, "zero_weight_mult_closed_form": 3, "zero_weight_mult_kostant": 3, "match": true }, { "pq": [ 3, 3 ], "dim": 64, "triality": 0, "C2": 15.0, "zero_weight_mult_closed_form": 4, "zero_weight_mult_kostant": 4, "match": true }, { "pq": [ 4, 1 ], "dim": 35, "triality": 0, "C2": 12.0, "zero_weight_mult_closed_form": 2, "zero_weight_mult_kostant": 2, "match": true }, { "pq": [ 1, 4 ], "dim": 35, "triality": 0, "C2": 12.0, "zero_weight_mult_closed_form": 2, "zero_weight_mult_kostant": 2, "match": true }, { "pq": [ 6, 0 ], "dim": 28, "triality": 0, "C2": 18.0, "zero_weight_mult_closed_form": 1, "zero_weight_mult_kostant": 1, "match": true }, { "pq": [ 4, 4 ], "dim": 125, "triality": 0, "C2": 24.0, "zero_weight_mult_closed_form": 5, "zero_weight_mult_kostant": 5, "match": true }, { "pq": [ 2, 5 ], "dim": 81, "triality": 0, "C2": 20.0, "zero_weight_mult_closed_form": 3, "zero_weight_mult_kostant": 3, "match": true }, { "pq": [ 5, 2 ], "dim": 81, "triality": 0, "C2": 20.0, "zero_weight_mult_closed_form": 3, "zero_weight_mult_kostant": 3, "match": true } ] }, "part1_spectrum": { "coordinates": "shifted Eisenstein: a=p+1,b=q+1; lambda=(a^2+ab+b^2-3)/3; deg=min(a,b); a,b>=1; a==b mod3", "a_max": 900, "n_distinct_eigenvalues": 93790, "smallest_(lambda,deg)": [ [ 3.0, 2 ], [ 6.0, 2 ], [ 8.0, 3 ], [ 12.0, 4 ], [ 15.0, 4 ], [ 18.0, 2 ], [ 20.0, 6 ], [ 24.0, 5 ] ], "note": "eigenvalue = SU(3) Casimir of triality-0 rep; the smallest nonzero level is the (1,1) adjoint at C2=3 with deg=2." }, "part2_heat_trace_expansion": { "form": "Theta(t) ~ sum_k c_k t^{e_k}, half-integer e only", "coefficients": { "t^-1.5": 0.274199564987461, "t^-0.5": 0.41047907068834455, "t^+0.5": 0.28786411243788057, "t^+1.5": 0.13116487718919, "t^+2.5": 0.04498911804946684, "t^+3.5": 0.012820009022498139, "t^+4.5": 0.003508041503751712 }, "leading_c_minus_3half_fit": 0.274199564987461, "leading_c_minus_3half_continuum_pred_sqrt3_times_J": 0.27419972383114605, "continuum_J_int_min_ab": 0.1583092843656332, "leading_coeff_match": true, "effective_spectral_dimension_d_eff": 3, "validation_rel_err": [ { "t": 0.02857142857142857, "rel_err": 3.5974265633691775e-16 }, { "t": 0.01, "rel_err": 0.0 }, { "t": 0.005, "rel_err": 1.4549513940967977e-16 } ], "note": "d_eff=3 (Theta~t^{-3/2}), NOT 6: the zero-weight sublattice is a measure-zero slice of the full L^2(K6) spectrum." }, "split_machinery_validation": { "s0_convergent_test_point": 2.0, "direct_sum_extrapolated": 0.5235465016088529, "split_formula_value": 0.5235465948194805, "relative_agreement": 1.7803696018277585e-07, "PASS_machinery_validated": true }, "part3_continuation_to_minus_half": { "method": "Mellin / heat-kernel split (Epstein-Hurwitz): zeta(s) = 1/Gamma(s)[int_0^1 t^{s-1}(Theta-P_N) + sum_k c_k/(s+e_k) - n0/s + int_1^inf t^{s-1}(Theta-n0)]", "I0_int_tlo_1": -0.014533908701661756, "I0_cutoff_spread_5e-3_to_2e-2": 2.4474172688471185e-13, "Iinf_int_1_inf": 0.02425856958354306, "c_half_theta_coeff_t_plus_half": 0.28786411243788057, "POLE_at_minus_half": true, "residue_R_scheme_invariant": -0.08120496685733616, "residue_R_sign": "NEGATIVE", "finite_part_F_minimal_subtraction_mu_1": -0.4542999626860593, "finite_part_F_sign_at_mu_1": "NEGATIVE", "scheme_dependence": { "residue_sign_scheme_invariant": "NEGATIVE", "finite_part_sign_at_mu_1": "NEGATIVE", "ln_mu_star_where_finite_part_flips": -2.797242461068853, "mu_star_where_finite_part_flips": 0.06097798015346915, "finite_part_sign_is_scheme_dependent": true } }, "zeta_minus_half_value": "POLE at s=-1/2 (no scheme-free finite value): zeta(s) = R/(s+1/2) + F + O(s+1/2), R=-0.081205 (scheme-invariant), F=-0.4543 (mu=1/R_K6; scheme-dep)", "zeta_minus_half_sign": "residue R NEGATIVE (scheme-invariant); finite part F sign SCHEME-DEPENDENT (flips at mu*=0.06098)", "branch_selected": "still-owner-locked", "cross_check_against_frozen_cloop": { "performed_after_sign_computed": true, "frozen_c_loop_FRG2_target": 1.3637877214788921e-05, "c_loop_wall_on_disk_veff": 1.3637166327064904e-05, "minus_half_one_loop_prefactor": -0.5, "factor_4pi_to_minus_13_over_2": 7.163675151870211e-08, "c_loop_from_finite_part_F": 1.6272286770948435e-08, "ratio_to_frozen_target": 0.0011931685932252535, "reproduces_frozen_magnitude_within_factor_2": false, "verdict": "MISMATCH: the minimal-subtraction scalar-zeta finite part gives |c_loop| ~ 1.627e-08, about 838x smaller than the frozen FRG-2 magnitude 1.3638e-05. The frozen value is set by the owner's FRG-2 NLO Litim shell-projection scheme (specific finite subtraction + full 13D field content + mode-by-mode shell mass spectrum), which is MORE than the bare scalar K6 zeta(-1/2). Honest non-reproduction." }, "scheme": "Regularization = Epstein-Hurwitz / Mellin heat-kernel zeta on the SU(3)-Casimir (Eisenstein-form) lattice. Because zeta has a POLE at s=-1/2, a finite value requires a subtraction scheme. Minimal subtraction (mu=1/R_K6) gives F=-0.45430<0; the corpus c_loop uses FRG-2 NLO Litim shell-projection, a DIFFERENT finite scheme. FRG-2 Litim vs minimal-subtraction MS-bar give different finite parts (and the finite-part SIGN is mu-dependent), so the result is SCHEME-AMBIGUOUS-owner-locked.", "scheme_ambiguity": "The pole at s=-1/2 means the finite part is NOT scheme-free. Only the residue (R<0) is scheme-invariant. The finite-part sign flips at the renormalization scale mu*=0.06098, so without the owner's named FRG-2 Litim shell-projection scheme the BRANCH sign is genuinely ambiguous. This is the honest endpoint -- NOT a manufactured favorable branch.", "well_verdict": "OWNER-LOCKED / CONDITIONAL -- but the residual is now SHARPER. The continuation is COMPLETE (not deferred): zeta_{Delta_K6}(s) has a SIMPLE POLE at s=-1/2 (Theta(t) carries a nonzero t^{+1/2} term, coeff=+0.28786), so the one-loop vacuum energy is UV-divergent and its finite part is scheme-dependent. The scheme-INVARIANT residue is NEGATIVE (R=-0.08120), which is the sign BRANCH-STANDS would want -- but it is the residue of a DIVERGENCE, not the finite wall coefficient. The finite part F that actually sets the wall is scheme-dependent (its sign flips with the renormalization scale mu), so the FAVORABLE BRANCH-STANDS is NOT forced. The well stays gated on the owner's FRG-2 NLO Litim shell-projection SCHEME -- now a single named scheme choice (a smaller residual), not a whole missing computation.", "what_requires_chris": "(1) The named FRG-2 NLO Litim shell-projection SCHEME definition that fixes the finite subtraction at s=-1/2 (which renormalization scale mu, which shell-by-shell mass spectrum) -- this is what pins the finite-part sign and reproduces the frozen 1.3637877e-5 magnitude; the bare scalar K6 zeta(-1/2) is a POLE and gives only a scheme-invariant NEGATIVE residue, not a scheme-free finite wall coefficient. (2) Confirmation that the wall coefficient is the FINITE PART under that scheme (not the residue), and that the full 13D field content (the divergence's coefficient) is renormalized consistently.", "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 tower and the entire continuation are built from dim-6 geometry and SU(3) group theory ALONE. The frozen c_loop magnitude (1.3637877e-5) was used ONLY at the final cross-check, strictly AFTER the sign/structure were computed, and ONLY as a geometry-side comparison -- never as an input to the continuation. The favorable BRANCH-STANDS was NOT forced: the residue is reported NEGATIVE (scheme-invariant) but the wall-setting finite part is reported SCHEME-DEPENDENT and the magnitude cross-check is reported as an honest MISMATCH.", "provenance": { "veff_coefficients_frg4.yaml_sha256": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac", "mpmath_used": true }, "non_promotion": "no gate flipped; no status word emitted for any gate; countersign-ready continuation + scheme-ambiguity analysis only." }