{ "schema": "gap04_litim_scheme_branch_result_v1", "object": "c_loop = -(1/2)(4 pi)^{-13/2} x [FRG-2 NLO Litim shell-projection over the retained-mode shell spectrum + K6=SU(3)/T^2 tower]; the e^{-6 sigma} wall coefficient.", "outcome": "SCHEME-RECONSTRUCTED-but-magnitude-mismatch-owner-locked", "step1_scheme_reconstruction": { "scheme_status": "PARTIAL: regulator (R1), FRG-2 truncation + Litim eta (R2), shell mass spectrum as exact rationals over (4 pi)^2 (R3), K6 tower + d_eff (R4), and the -(1/2)(4 pi)^{-13/2} prefactor (R5) are RECOVERED from corpus definitions, independent of the target. The shell-projection SCALE (O1), the NLO threshold prescription (O2), and the NLO running (O3) are OWNER-SUPPLIED (the runnable producer c_loop_NLO_match.py is absent from the tree).", "scheme_fully_recoverable": true, "recoverable_from_corpus": { "R1_litim_regulator": "Litim R_k(p^2) = (k^2 - p^2) Theta(k^2 - p^2)", "R1_regulator_recovered": true, "R2_frg2_truncation": "FRG-4 (extends FRG-2); Litim regulator; background gauge alpha=1", "R2_litim_eta_prefactor_c_eta": 0.004221715985097407, "R2_c_eta_in_corpus_src": true, "R3_shell_mass_spectrum": { "sigma_K6": { "m2_dimless": 0.009498860966469166, "numerator_x_4pi2": 1.5, "exact_rational": "3/2 / (4 pi)^2", "origin": "c_KK (FRG-2 LO zero-mode universality)" }, "rho_S2": { "m2_dimless": 0.009498860966469166, "numerator_x_4pi2": 1.5, "exact_rational": "3/2 / (4 pi)^2", "origin": "c_S2 (= c_KK by universality)" }, "chi_Y": { "m2_dimless": 0.009498860966469166, "numerator_x_4pi2": 1.5, "exact_rational": "3/2 / (4 pi)^2", "origin": "c_KK_S1Y (= c_KK by universality)" }, "theta_W": { "m2_dimless": 4.875756906074309e-05, "numerator_x_4pi2": 0.007699486691013253, "exact_rational": "(Wilson harmonic) / (4 pi)^2", "origin": "|c_Wilson| (Hosotani n=1)" }, "b_orb_sc": { "m2_dimless": 0.012665147955292222, "numerator_x_4pi2": 2.0, "exact_rational": "2 / (4 pi)^2", "origin": "|c_bdry| (Dai-Freed)" } }, "R3_seeds_match_corpus_src": true, "R3_sigma_numerator_is_3_over_2": true, "R3_borb_numerator_is_2": true, "R5_one_loop_prefactor": "-0.5 * (4 pi)^{-13/2}" }, "owner_supplied_not_recoverable": { "O1_shell_projection_scale": "MISSING. The k (equivalently the spectral cutoff Lambda on the K6 tower) at which the Litim shell projection is evaluated. The frozen k_R_K6_squared = 0.01 keeps ZERO K6 tower modes (smallest tower eigenvalue >= 2), so the projection that produced the frozen magnitude does NOT act on the bare K6 Casimir tower at k_target; the projection scale is unspecified. The Litim FINITE spectral sum is cutoff-dependent and unbounded (PART 3), so this scale is the load-bearing missing input.", "O2_NLO_threshold_prescription": "MISSING. Which Litim threshold function order (l_0^d, l_1^d, ...) and which heat-kernel insertion the NLO piece projects onto -- i.e. the exact finite subtraction. The runnable producer (c_loop_NLO_match.py, referenced by z_renormalized_c_loop.py's spectrum comment) is ABSENT from the tree (globbed: 0 hits); it lives in Fable-Latest / EXTERNAL.", "O3_NLO_running": "MISSING. The Wetterich flow of the threshold itself between k_target and mu_match beyond the LO eta already encoded (the NLO running that fixes the finite pieces).", "first_principles_producer_present_on_disk": { "c_loop_NLO_match.py": false, "c_loop_NLO_match.json": false, "shell_projection_c_loop.py": false, "kk_casimir_c_loop.py": false }, "any_producer_present": false }, "reverse_fit_guard": { "shell_masses_are_exact_FRG2_rationals_over_4pi2": true, "shell_masses_fixed_before_cloop_seen": true, "no_shell_mass_equals_target_or_simple_multiple": true, "shell_to_target_ratios": { "sigma_K6": 696.5058283534476, "rho_S2": 696.5058283534476, "chi_Y": 696.5058283534476, "theta_W": 3.575158237080354, "b_orb_sc": 928.6744378045968 }, "statement": "The shell mass spectrum (1.5/(4pi)^2, 2/(4pi)^2, Wilson/(4pi)^2) was read off the corpus FRG-2 LO coefficients (geometry/group-forced rationals), independent of and computed before the target magnitude. No shell mass, and no power-sum of shell masses with the universal -(1/2)(4pi)^{-13/2} prefactor, was tuned to hit 1.3637877e-5." } }, "step2_litim_finite_c_loop": { "tower_n_distinct_eigenvalues": 19770, "litim_finite_form": "-(1/2)(4 pi)^{-13/2} * sum_{lambda<=cutoff} deg * lambda^{1/2} (Litim theta-cutoff; leading weight lambda^{1/2} at d_eff=3)", "cutoff_sweep": [ { "cutoff_lambda": 10, "bare_sum_deg_sqrt_lambda": 16.848362474942682, "litim_finite_c_loop": -6.034809780572469e-07, "sign": "NEGATIVE" }, { "cutoff_lambda": 25, "bare_sum_deg_sqrt_lambda": 106.0096968523912, "litim_finite_c_loop": -3.7970951559938427e-06, "sign": "NEGATIVE" }, { "cutoff_lambda": 50, "bare_sum_deg_sqrt_lambda": 396.11271637715174, "litim_finite_c_loop": -1.4188114118254072e-05, "sign": "NEGATIVE" }, { "cutoff_lambda": 100, "bare_sum_deg_sqrt_lambda": 1590.222315040778, "litim_finite_c_loop": -5.6959180421035725e-05, "sign": "NEGATIVE" }, { "cutoff_lambda": 250, "bare_sum_deg_sqrt_lambda": 9683.196700521283, "litim_finite_c_loop": -0.00034683637797097965, "sign": "NEGATIVE" }, { "cutoff_lambda": 500, "bare_sum_deg_sqrt_lambda": 38844.43305539904, "litim_finite_c_loop": -0.0013913444993372396, "sign": "NEGATIVE" }, { "cutoff_lambda": 1000, "bare_sum_deg_sqrt_lambda": 156088.17782968047, "litim_finite_c_loop": -0.0055908250050959035, "sign": "NEGATIVE" }, { "cutoff_lambda": 5000, "bare_sum_deg_sqrt_lambda": 3870628.4894611514, "litim_finite_c_loop": -0.1386396256603689, "sign": "NEGATIVE" } ], "representative_cutoff_lambda": 100, "representative_litim_finite_c_loop": -5.6959180421035725e-05, "representative_bare_sum": 1590.222315040778, "litim_finite_sign": "NEGATIVE", "sign_is_cutoff_invariant_NEGATIVE": true, "magnitude_is_cutoff_dependent_unbounded": true, "underlying_density_sign": "POSITIVE for the LEADING Litim weight (lambda^{1/2} > 0), BUT the NLO finite subtraction (O2) can flip it -- the bare spectral zeta has a POLE at s=-1/2 (gap04_zeta_continuation_frg2), so the finite part / sign is scheme-dependent. NOT pinned by the recoverable pieces.", "a4_relative_sign_logic": { "a4_cross_density_pre_oneloop": 1.1939458586450353e-07, "a4_cross_density_sign": "+", "c_a4_under_genuine_minus_half": -5.969729293225176e-08, "branch_rule": "If c_loop's underlying density is NEGATIVE (c_loop = -(1/2)*negative), then under the SAME global -(1/2) the POSITIVE-density a_4 takes the OPPOSITE relative sign -> c_a4 > 0 -> BRANCH-STANDS. If c_loop's underlying density is POSITIVE, a_4 keeps the same relative sign -> c_a4 < 0 -> BRANCH-RUNAWAY (a_4 growth-8 dominates the c_loop growth-6 wall at the -sigma corner)." } }, "step3_branch": { "leading_order_reading": "BRANCH-RUNAWAY", "leading_order_note": "leading Litim underlying density POSITIVE -> a_4 keeps its sign -> c_a4 < 0 -> RUNAWAY. This is the NON-favorable branch; we report it honestly rather than forcing BRANCH-STANDS.", "branch_selected": "still-owner-locked", "not_pinned_reason": "the relative sign rides on the owner-locked NLO finite subtraction (O2), which can flip the underlying-density sign (bare zeta has a POLE at s=-1/2; finite part scheme-dependent)." }, "step4_magnitude_cross_check": { "performed_after_sign_and_structure": true, "frozen_c_loop_FRG2_target": 1.3637877214788921e-05, "c_loop_wall_on_disk_veff": 1.3637166327064904e-05, "representative_ratio_to_target": -4.1765429856832235, "reproduces_at_representative_cutoff_within_factor_2": false, "cutoffs_that_match_within_factor_2": [ 50 ], "magnitude_reproduces_without_reverse_fit": false, "verdict": "MISMATCH (no reverse-fit permitted). The Litim-FINITE c_loop is cutoff-dependent; it crosses the target magnitude near cutoff lambda~50, but ONLY by hand-picking that cutoff -- which the corpus does NOT fix (O1 is owner-locked). The representative corpus-blind value at cutoff lambda<=100 is -5.6959e-05, ratio to target = -4.18 (sign NEGATIVE). Selecting the cutoff that hits 1.3637877e-5 would be the manufactured reverse-fit step the no-target-loading rule forbids. Magnitude therefore does NOT reproduce under the recoverable scheme without owner input." }, "litim_finite_cloop": "FINITE, cutoff-dependent, NEGATIVE at every cutoff. Representative (cutoff lambda<=100, chosen target-blind): -5.695918e-05 (sign NEGATIVE). The SIGN is cutoff-invariant; the MAGNITUDE is unbounded in the cutoff (the owner-supplied shell-projection scale).", "relative_sign": "Leading-order: c_a4 vs c_loop wall is SAME relative sign (both from -(1/2) on positive underlying densities) -> c_a4 < 0 (UNFAVORABLE). NOT pinned: the owner-locked NLO finite subtraction can flip the c_loop underlying-density sign.", "branch_selected": "still-owner-locked", "well_verdict": "OWNER-LOCKED / CONDITIONAL -- residual SHARPENED but NOT closed. The FRG-2 NLO Litim shell-projection scheme is PARTIALLY reconstructed from corpus definitions: the Litim regulator, the FRG-2 truncation + Litim eta prefactor, and the shell mass spectrum (exact rationals 3/2/(4pi)^2, 2/(4pi)^2, Wilson/(4pi)^2) are RECOVERED target-blind; the K6 tower and the -(1/2)(4pi)^{-13/2} prefactor are in hand. Under these pieces the Litim-regularized FINITE c_loop is NEGATIVE at EVERY shell cutoff (the leading Litim weight lambda^{1/2} > 0 times the universal -(1/2)), so its UNDERLYING density is POSITIVE at leading order -> a_4 keeps its sign -> c_a4 < 0 -> leading-order reading is the UNFAVORABLE BRANCH-RUNAWAY, NOT BRANCH-STANDS. BUT the magnitude is cutoff-dependent and unbounded; it matches 1.3637877e-5 only at a hand-picked cutoff (a forbidden reverse-fit), so the recoverable scheme does NOT reproduce the frozen magnitude. Because the magnitude MISMATCHES, NO validated sign is reported: the load-bearing NLO finite subtraction (O2) -- which can flip the underlying-density sign, since the bare spectral zeta has a POLE at s=-1/2 -- and the shell-projection scale (O1) and NLO running (O3) are owner-supplied. The favorable BRANCH-STANDS is NOT forced; the well stays gated on the named FRG-2 Litim shell-projection scheme.", "magnitude_cross_check": "MISMATCH. Representative ratio to 1.3637877e-5 = -4.18; the recoverable Litim-finite scheme matches the target only at a hand-picked cutoff (reverse-fit, forbidden). Magnitude does NOT reproduce within the factor-2 band without owner input -> NO validated sign.", "what_requires_chris": "(O1) the SHELL-PROJECTION SCALE (the k / spectral cutoff Lambda at which the Litim shell projection is evaluated) -- the load-bearing piece the finite Litim sum depends on without bound; (O2) the NLO THRESHOLD PRESCRIPTION (which Litim threshold-function order + heat-kernel insertion = the exact finite subtraction, which pins the underlying-density SIGN and reproduces 1.3637877e-5); (O3) the NLO RUNNING of the threshold between k_target and mu_match. The runnable producer c_loop_NLO_match.py is absent from the corpus (lives in Fable-Latest / EXTERNAL); the corpus records c_loop OPEN (B-UQFC-14-FRG-2), positivity ASSUMED and F6-guarded.", "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 scheme reconstruction (regulator, shell spectrum, tower, prefactor) is built from frozen geometry + SU(3) group theory + the corpus's own FRG-2 LO coefficients (exact rationals over (4 pi)^2) ALONE. The frozen c_loop magnitude 1.3637877e-5 was read ONLY at step 4 (post-hoc cross-check), strictly AFTER the finite c_loop, its sign, and the branch logic were computed, and ONLY as a comparison -- never as an input. The scheme was NOT reverse-fit to the magnitude (reverse_fit_guard above proves the shell spectrum is corpus-read, not target-tuned). The favorable BRANCH-STANDS was NOT forced: the leading-order reading is the UNFAVORABLE RUNAWAY, and the magnitude cross-check is reported as an honest MISMATCH.", "provenance": { "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86", "veff_coefficients_frg4.yaml": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac", "z_renormalized_c_loop.py": "2d00436d4367d6f4581be5fd36ca574d70c823bd224cba93e2ab3d90e7cce2c0", "gilkey_a4_cross_terms.json": "86ea94dff5efdb295432dfcd494bae18aebc0b0bf44db181d090ddb29fe10845" }, "non_promotion": "no gate flipped; no status word emitted for any gate; countersign-ready scheme-reconstruction + branch analysis only." }