{
  "schema": "gap04_cloop_density_sign_result_v1",
  "purpose": "Derive (or precisely gate) the SIGN of c_loop's UNDERLYING heat-kernel density -- the density BEFORE the one-loop -(1/2) prefactor in the e^{-6 sigma} KK-Casimir wall c_loop = +1.3637e-5.",
  "density_sign_outcome": "GENUINELY-INDETERMINATE-gated-on-open-cloop",
  "c_loop_underlying_density_sign": "GATED-ON-OPEN-cloop-first-principles (UNBANKED)",
  "branch_selected": "still-owner-locked",
  "debit_reclassification": "STAYS BARE-COUNTED (the relative-one-loop-sign debit in gap04_reason_hunt.py is NOT promoted to DERIVED: we neither derived nor could derive the c_loop underlying-density sign data-blind)",
  "what_is_DERIVED_data_blind": {
    "D1_a4_cross_density_sign_pre_oneloop": "+",
    "D1_a4_cross_density_value": 1.193945858645035e-07,
    "D1_reason": "geometry-FORCED + convention-INVARIANT: (4pi)^{-D/2} > 0, master R^2 coeff +5/360 (cross 1/36) is the same in both Gilkey Laplacian conventions (R^2 even in curvature), and the single-signed Einstein registry R_K6=+30, R_S2=+2 is fixed by the frozen K6 x S2 geometry. All three factors positive.",
    "D2_global_one_loop_magnitude": -0.5,
    "D2_reason": "universal bosonic one-loop log-det: V^(1-loop) = -(1/2)(4pi)^{-D/2} sum_n a_n. GLOBAL prefactor MAGNITUDE only; a global prefactor does NOT force two coefficients same-SIGN.",
    "D3_c_a4_as_run_under_genuine_minus_half": -5.969729293225175e-08,
    "D3_c_a4_on_disk": -5.969729293225176e-08,
    "D3_matches_on_disk": true,
    "D3_reason": "global -(1/2) acting on a_4's OWN positive density gives c_a4 NEGATIVE = -5.97e-08 (byte-matches gilkey_a4_cross_terms.json)."
  },
  "what_is_NOT_derivable_and_why": {
    "decisive_unknown": "the SIGN of c_loop's UNDERLYING heat-kernel density (is c_loop a -(1/2) log-det of a NEGATIVE-density object -> BRANCH-STANDS, or a POSITIVE Casimir under a different bookkeeping -> BRANCH-RUNAWAY?).",
    "underlying_density_sign_banked": false,
    "banked_check_reason": "no underlying-density-sign slot anywhere; c_loop carried as a frozen positive WALL value only; z_renorm applies a Z-factor, no -(1/2) log-det",
    "first_principles_producer_present_on_disk": {
      "Final_physics_articles/scripts/gap_04\\src\\c_loop_NLO_match.py": false,
      "Final_physics_articles/scripts/gap_04\\src\\c_loop_NLO_match.json": false,
      "Final_physics_articles/scripts/gap_04\\src\\shell_projection_c_loop.py": false,
      "Final_physics_articles/scripts/gap_04\\src\\kk_casimir_c_loop.py": false,
      "Final_physics_articles/scripts/gap_04\\outputs\\c_loop_NLO_match.json": false
    },
    "any_first_principles_producer_present": false,
    "existing_producer_derives_a_density_sign": false,
    "existing_producer_note": "z_renormalized_c_loop.py reads c_loop as a FROZEN inherited number (C_LOOP_FRG2 = 1.3637877214788921e-05, 'Fable-Latest shell projection') and multiplies by prod_i Z_i^{g_i/2} ~0.99995. The g_i/2 is a wave-function power, NOT a one-loop -(1/2) log-det on the c_loop density; it derives NO density sign.",
    "named_open_object": "B-UQFC-14-FRG-2 (owner = Chris): 'UQFC SS14 FRG-2 truncation + FRG-6 trajectory closure feeding c_loop'. Fable_TOE.md records c_loop SCAFFOLD-BAND / FRG-2, positivity ASSUMED (P3 '-inf: V_loop wall (c_loop > 0)') and guarded by falsifier F6 -- the underlying-density sign is one layer BELOW even the assumed wall sign and is nowhere computed.",
    "nearest_casimir_machinery_disclaimer": "k6_zeta_casimir_insertion.py (the nearest first-principles Casimir machinery) explicitly states it 'does not claim to compute the full finite nonlocal determinant' -- it closes only the LOCAL Seeley-DeWitt/zeta coefficient, and for the Dirac/fermion bundle (a DIFFERENT object than the scalar KK-Casimir wall). It does NOT bank the scalar c_loop underlying-density sign."
  },
  "two_live_branches": {
    "BRANCH_STANDS": {
      "condition": "owner/first-principles rules c_loop's UNDERLYING density NEGATIVE (c_loop = -(1/2) x negative); under the SAME global -(1/2) the positive-density a_4 takes the OPPOSITE relative sign -> c_a4 = +5.97e-08, a positive wall",
      "c_a4": 5.969729293225175e-08,
      "effect": "the -sigma FRG-4 well STANDS",
      "named_falsifier": "F1_ext (true interior critical point exists)"
    },
    "BRANCH_RUNAWAY": {
      "condition": "c_loop's underlying density POSITIVE (positive Casimir under a different bookkeeping); a_4 keeps the SAME relative sign as its own positive density under -(1/2) -> c_a4 = -5.97e-08",
      "c_a4": -5.969729293225175e-08,
      "effect": "a_4 cross-term (growth 8 at -sigma) DOMINATES the c_loop wall (growth 6); a negative dominant term -> V -> -inf, a REAL separate 3-modulus -sigma runaway threatening F1",
      "named_falsifier": "F1 (any unbounded V_eff -> -inf direction)"
    }
  },
  "disjointness_dominance_at_neg_sigma": {
    "growth_a4_cross_term": 8.0,
    "growth_c_loop_wall": 6.0,
    "a4_dominates_neg_sigma_corner": true
  },
  "well_verdict": "OWNER-LOCKED / CONDITIONAL: the -sigma FRG-4 3-modulus well's survival is gated on the c_loop underlying-density sign, which is the named OPEN first-principles object (B-UQFC-14-FRG-2). It is NOT forced to stand (the genuine global -(1/2) on a_4's own positive density gives c_a4 NEGATIVE -> runaway) and NOT forced to fail (the relative sign vs c_loop is unbanked). Both branches carry named falsifiers; the owner must compute c_loop from first principles to decide.",
  "what_requires_chris": "Compute c_loop from first principles: run the regularized KK-tower Casimir / shell-projection (zero-point / heat-kernel) integral that produces the e^{-6 sigma} coefficient, and READ OFF the sign of its underlying density BEFORE the one-loop -(1/2) prefactor. This is the named open FRG-2 object B-UQFC-14-FRG-2. Until it exists, the density sign is GATED-ON-OPEN and the Gap-04 well stays owner-locked.",
  "provenance_hashes": {
    "frozen_inputs.yaml": "8397469ebf982c0c31e7aadef0181337e2fe1b9c7160ca25c000a0cbb0cbac86",
    "veff_coefficients_frg4.yaml": "9bf1d45871adcf67d754a3908c67eb6ad9d36780d4ef29336ea932c8705fa0ac",
    "gilkey_a4_cross_terms.json": "86ea94dff5efdb295432dfcd494bae18aebc0b0bf44db181d090ddb29fe10845",
    "z_renormalized_c_loop.json": "64736eb9ca556e64a8cf02e321d59c773e166ff0e27ff784222b78940f499e81",
    "z_renormalized_c_loop.py": "2d00436d4367d6f4581be5fd36ca574d70c823bd224cba93e2ab3d90e7cce2c0"
  },
  "frozen_inputs_used": {
    "R_K6_0": 30.0,
    "R_S2_0": 2.0,
    "a4_R2_master_coeff_5_over_360": 0.013888888888888888,
    "a4_cross_prefactor_1_over_36": 0.027777777777777776,
    "D_bulk": 13,
    "factor_4pi_to_minus_D_over_2": 7.163675151870211e-08,
    "global_one_loop_prefactor": -0.5,
    "c_loop_Z_frozen_wall_value": 1.3637166327064904e-05,
    "c_loop_FRG2_shell_input": 1.3637877214788921e-05
  },
  "no_target_loading": true,
  "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). Every DERIVED line (D1/D2/D3) is licensed by the committed computation in this passage and byte-checked against on-disk artifacts. The favorable branch was NOT forced: the genuine global -(1/2) on a_4's own positive density is reported NEGATIVE, and the well is left owner-locked because the deciding c_loop underlying-density sign is the named open object.",
  "non_promotion": "no gate flipped; no status word emitted for any gate; countersign-ready gated-on-open analysis only."
}