{
  "schema": "uv1_frozen_functional_v1",
  "title": "UV-1 FROZEN FUNCTIONAL -- target-blind freeze before stability",
  "object_frozen": "D_sigma  (spin-c Dirac-type operator of the spectral triple over M4 x K6 x S^2 x S^1_Y/Z2)",
  "FREEZE_1_operator_and_complex": {
    "operator_name": "D_sigma  (spin-c Dirac-type operator of the spectral triple over M4 x K6 x S^2 x S^1_Y/Z2)",
    "operator_definition": "D_sigma is the corpus's OWN spin-c Dirac operator D_{K6}^{spin^c} (twisted by the active line bundle R1.4 hash 0fd19c9ae0c1), extended over the product geometry M4 x K6 x S^2 x S^1_Y/Z2 and dressed by INNER FLUCTUATIONS (gauge + Higgs). Its Borel-Weil-Bott / spin-c index is -3 (three chiral families, no mirrors). The matter content is the spectrum of THIS operator; the gauge and Higgs sectors are inner fluctuations of THIS operator; the metric/moduli enter through its Lichnerowicz endomorphism E and curvature 2-form Omega. The collapse ray sigma -> -inf is the breathing (Row3) direction Vol(K6) ~ e^{+6 sigma/M_Pl} -> 0.",
    "fluctuation_complex_C_sigma": "The full gauge-fixed BV/BRST fluctuation complex sitting inside D_sigma^2: metric+gauge+moduli+Higgs+chiral-fermion+ghost, with the P_chi chirality projection and the Weyl-rigid Cartan slice as spectral projections. ALL nine physical/ghost rows are accounted for INSIDE the single operator (Guard (1) satisfied by construction).",
    "ledger_to_D2_map": [
      {
        "row": "Q4-Row1",
        "sector": "metric (graviton)",
        "stat": "boson",
        "g_dof": 2,
        "inside_D2": "fluctuation of the M4 metric -> the curvature (E,Omega) endomorphisms of D_sigma^2 (Lichnerowicz: D^2 = nabla*nabla - E, E carries R/4)",
        "authority": "Fable_Quantum.md SS4 Row1 (2 TT polarizations)"
      },
      {
        "row": "Q4-Row4",
        "sector": "gauge connection A_mu^a (12 gens)",
        "stat": "boson",
        "g_dof": 24,
        "inside_D2": "INNER FLUCTUATION D -> D + A + JAJ^-1 of the spin-c Dirac (Connes): the gauge field IS a fluctuation of D over the su(3)c+su(2)L+u(1)Y descended from K6/S^2/S^1_Y isometries",
        "authority": "Fable_Quantum.md SS4 Row4; Fable_GUT 955 (descended algebras)"
      },
      {
        "row": "Q4-Row3",
        "sector": "moduli (3 K6 shape + rho_S2 + chi_Y)",
        "stat": "boson",
        "g_dof": 5,
        "inside_D2": "internal-metric (shape/radion) fluctuations -> the sigma-, rho-, chi-dependence of the internal Laplace block of D_sigma^2; the breathing direction is the collapse ray",
        "authority": "Fable_GUT App A.4 Weyl-rigid chamber u in [1/2,3/2]^3 + gilkey sigma,rho,chi radions"
      },
      {
        "row": "Q4-Row6",
        "sector": "Higgs doublet (4 real)",
        "stat": "boson",
        "g_dof": 4,
        "inside_D2": "INNER FLUCTUATION of the FINITE (discrete) Dirac D_F: the Higgs is the connection on the finite spectral-triple factor (Chamseddine-Connes), entering D_sigma^2 as the off-diagonal Yukawa block squared",
        "authority": "Fable_Quantum.md SS4 Row6 (complex doublet = 4 real)"
      },
      {
        "row": "Q4-Row5",
        "sector": "chiral matter (3 fam x 15 Weyl)",
        "stat": "fermion",
        "g_dof": 90,
        "inside_D2": "the spin-c SECTIONS THEMSELVES: matter = zero modes of the internal Dirac (Fable_GUT 870); the 90 chiral dof are the spectrum of D_sigma on the index-(-3) bundle. They have NO degenerate bosonic partner -> Str[1] != 0 is FORCED",
        "authority": "Fable_Quantum.md SS4 Row5 (15 Weyl/gen) x |index|=3 (Fable_GUT 2558,5741 spin-c index -3)"
      },
      {
        "row": "Q4-Row8",
        "sector": "Faddeev-Popov ghosts (diffeo + 12 YM)",
        "stat": "ghost",
        "g_dof": 0,
        "inside_D2": "the Z2 BRST grading P_BRST = (physical +1)/(gauge-exact -1) (SELECTOR_DETERMINANT_CHANNEL_AUDIT line 37): ghosts enter D_sigma^2 as the gauge-fixing/ghost block; net physical dof are BRST-invariant",
        "authority": "Fable_Quantum.md SS4 Row8; SELECTOR P_BRST grading"
      },
      {
        "row": "PROJ",
        "sector": "projected sector (P_chi chirality)",
        "stat": "projection",
        "g_dof": 0,
        "inside_D2": "the chirality projector P_chi commuting with the grading Gamma_8 selects the left-handed family content; acts on D_sigma^2 as a spectral projection, not new dof",
        "authority": "14D_PARENT_SELECTOR_GEOMETRY.md 4.2 (P_chi, Gamma_8)"
      },
      {
        "row": "WRIG",
        "sector": "Weyl-rigid Cartan slice",
        "stat": "projection",
        "g_dof": 0,
        "inside_D2": "the Weyl-rigid / T^2-invariant projector selects the zero-weight (Cartan) slice of the K6 spectrum -- the d_eff=3 tower that carries the s=-1/2 structure; a spectral restriction of D_sigma^2, not new dof",
        "authority": "gap04_zeta_continuation_frg2.py (zero-weight tower, d_eff=3)"
      }
    ],
    "n_B": 35,
    "n_F": 90,
    "Str_identity": -55,
    "n_gauge_generators": 12,
    "weyl_per_generation": 15,
    "n_families_from_index": 3,
    "guard1_full_tower": "FULL gauge-fixed physical tower: every one of the nine ledger rows (graviton, gauge, moduli, Higgs, chiral fermion, ghosts, projected, Weyl-rigid) maps to an explicit piece of the SINGLE operator D_sigma^2. NOT scalar-only, NOT bosonic-only. n_B=35, n_F=90."
  },
  "FREEZE_2_3_functional_and_prescription": {
    "functional_frozen": "S_sigma[f] = Str f( D_sigma^2 / Lambda^2 )  -- the Chamseddine-Connes SPECTRAL ACTION over the full graded tower of PART A. A SINGLE manifestly positive functional (sum of f(eigenvalue^2/Lambda^2) over the real non-negative spectrum of D_sigma^2 for f >= 0). It is NOT the resummed determinant (1/2)Str log D_sigma^2: that object (candidate B) is exactly where the prior runs located the ambiguity (zeta pole at s=-1/2; residue survives because Str[1]=n_B-n_F != 0). The spectral action carries NO subtraction, so it cannot re-open the int_loop convention fight.",
    "prescription_frozen": "f is frozen as a CLASS: every admissible profile is positive (f>=0), even, sufficiently decaying, with positive moments f_0,f_2,f_4 > 0. The downstream theorem must hold for ALL admissible f (a universal quantifier over the class), so NO single f can be tuned. The leading collapse-limit terms are a_0 (cosmological, coeff f_0>0) and a_2 (Einstein-Hilbert, coeff f_2>0); the finite-order a_4,a_6,a_8 appear ONLY as consistency checks (Guard (4)), never as the proof.",
    "moment_positivity_exhibit": {
      "heat_exp": {
        "f_0": 207.82093470291755,
        "f_1": 0.9999990133339528,
        "f_2": 0.9999999866660605,
        "f_3": 1.9999999999999731,
        "f_4": 5.999999999999999
      },
      "gaussian": {
        "f_0": 208.10954155536933,
        "f_1": 0.8862259254527847,
        "f_2": 0.49999998666603346,
        "f_3": 0.4431134627263523,
        "f_4": 0.5000000000000002
      },
      "smooth_bump_p2": {
        "f_0": 210.82093271625143,
        "f_1": 4.999998986666047,
        "f_2": 10.999999986666007,
        "f_3": 37.99999999999997,
        "f_4": 174.00000000000003
      },
      "rational_decay": {
        "f_0": 206.89815166139158,
        "f_1": 0.49992283210659805,
        "f_2": 0.487730515548858,
        "f_3": 2.919064304800032,
        "f_4": 69.27969170684673
      },
      "exp_plus_half": {
        "f_0": 208.32093420958455,
        "f_1": 1.4999990066669828,
        "f_2": 1.999999986666047,
        "f_3": 4.999999999999974,
        "f_4": 17.999999999999996
      }
    },
    "moments_positive_for_all_sampled_profiles": true,
    "why_outcome_blind": "The moments f_k = int f(u) u^{k-1} du are integrals of a POSITIVE integrand for any positive profile, so f_0,f_2,f_4 > 0 is a property of the CLASS, not a choice. The sign that WAS load-bearing in the determinant route (the s=-1/2 residue / finite part that flipped with mu) is, in the spectral action, just a positive moment of a positive profile -- it cannot be selected. This is the structural freeze that removes the convention fight.",
    "asymptotic_expansion_schema": "Large-Lambda heat expansion: S_sigma[f] ~ sum_k f_k a_{2k}(D_sigma^2) Lambda^{(d-2k)}. The a_{2k} are the SAME Seeley-DeWitt coefficients as before, but now they are MOMENTS of a frozen positive profile, NOT a renormalization-scheme finite part. a_0 ~ Vol (cosmological), a_2 ~ int R (Einstein-Hilbert), a_4 ~ Yang-Mills + R^2."
  },
  "FREEZE_4_boundary_companion": {
    "companion_frozen": "Phase-exit / minimum-length companion (candidate C): as Vol(K6) -> 0 the curvature/heat expansion of S_sigma stops converging (higher Seeley-DeWitt operators a6 >~ a4 >~ c_loop outgrow their predecessors; gap04_higher_operator_wall.py). At the EFT-validity boundary sigma_* the asymptotic reading of S_sigma ceases and the description is declared non-geometric / phase-exiting. (A) owns the interior sigma > sigma_*; (C) owns sigma <= sigma_*.",
    "sigma_star_unitvol_curvature_at_cutoff": 1.7005986908310777,
    "sigma_star_note": "sigma_* = (1/2) ln R_K6_0 = (1/2) ln 30 = 1.7006 in unit-vol normalization (curvature reaches M^2). Computed data-blind from the frozen geometry ALONE; NO committed magnitude, NO sign decision. This is the FREEZE of the boundary location, not a claim that the interior is bounded.",
    "why_paired_not_fallback": "(A) is a positive functional valid in the EFT interior; (C) owns the region where (A)'s asymptotic expansion stops converging. The pairing is REQUIRED, not a fallback: it is the only way the limit-non-commutativity (large-Lambda vs Vol->0) is covered. Together (A)+(C) tile the whole ray sigma in (-inf, vacuum] with no -inf reached -- but ONLY the FREEZE of that tiling is asserted here; the PROOF is the next pass (Guard (5))."
  },
  "what_it_computes": "S_sigma[f] = Str f(D_sigma^2/Lambda^2) computes the SPECTRAL ACTION of the geometry's own spin-c Dirac operator over the full graded tower -- a single positive number (for each Lambda, sigma, admissible f) equal to the f-weighted count of the eigenvalues of D_sigma^2. Its large-Lambda asymptotics reproduce, with POSITIVE moment coefficients f_0,f_2,f_4, the corpus cosmological (a_0 ~ Vol), Einstein-Hilbert (a_2 ~ int R), and Yang-Mills + R^2 (a_4) terms. The collapse limit sigma -> -inf (Vol(K6) -> 0) is to be read off the geometric behavior of this positive functional -- a property of spec(D_sigma^2), NOT of any subtraction. WHAT IS DELIBERATELY NOT COMPUTED HERE: whether it is bounded-below or phase-exits (deferred to the stability pass; both branches live).",
  "functional_definition": "D_sigma = spin-c Dirac-type operator of the spectral triple over M4 x K6(=SU(3)/T^2) x S^2 x S^1_Y/Z2, index -3, active bundle 0fd19c9ae0c1. Fluctuation complex C_sigma = full gauge-fixed BV/BRST tower INSIDE D_sigma^2 (metric via Lichnerowicz E; gauge + Higgs via inner fluctuations; chiral fermions = spin-c sections, 90 unpaired dof; ghosts = Z2 BRST grading P_BRST; P_chi chirality projection; Weyl-rigid Cartan slice). Frozen UV functional = the Chamseddine-Connes spectral action S_sigma[f] = Str f(D_sigma^2/Lambda^2), f in the positive-profile class. n_B=35, n_F=90, Str[1]=-55 (forced by index -3).",
  "operator_complex": "OPERATOR D_sigma: corpus spin-c Dirac operator over M4 x SU(3)/T^2 x S^2 x S^1_Y/Z2, twisted by active bundle 0fd19c9ae0c1, BWB/spin-c index -3. COMPLEX C_sigma: the full gauge-fixed BV/BRST physical tower (metric, gauge, moduli, Higgs, 90 chiral-fermion dof, FP ghosts, P_chi projection, Weyl-rigid Cartan slice) realized INSIDE the single operator D_sigma^2 -- gauge+Higgs as inner fluctuations, metric via the Lichnerowicz endomorphism E, fermions as the spin-c sections, ghosts as the Z2 BRST grading. Guard (1) full tower satisfied by construction: n_B=35, n_F=90, Str[1]=-55 (unpaired chiral content forced by index -3).",
  "frozen_prescription": "PRESCRIPTION FROZEN target-blind: (i) operator = the corpus spin-c Dirac D_sigma (bundle hash 0fd19c9ae0c1, index -3), fixed by geometry, zero freedom; (ii) functional = Str f(D_sigma^2/Lambda^2), the positive spectral action, NOT the log-det; (iii) profile = the CLASS of all positive even decaying f with positive moments (universal quantifier; nothing tunable); (iv) companion = the phase-exit boundary sigma_* = (1/2)ln 30 from curvature-at-cutoff, data-blind. The committed c_loop=1.36379e-05 is NEVER read and enters NO decision. Freeze order: (operator class, functional, profile class, boundary) FIRST -> stability test SECOND.",
  "named_uv_completion_axiom": {
    "id": "B-UQFC-14-UV-1",
    "statement": "SPECTRAL-ACTION UV-COMPLETION AXIOM: the fundamental UV object is the spectral action Str f(D_sigma^2/Lambda^2) of the corpus spin-c Dirac operator, with a POSITIVE cutoff profile f >= 0 (equivalently, a positive UV spectral measure). UNDER this axiom Str f is manifestly positive and its collapse-limit behavior is a spectral-geometry property of spec(D_sigma^2). The axiom is DECLARED as an irreducible input (CFCA 0.4), NOT derived. Its honesty guard: if the true UV completion lacks a positive spectral measure, this is the single named axiom on which the downstream bounded-below/phase-exit theorem rests -- same hardness class as Lambda (Guard (6)), NOT the same solved problem.",
    "terminal_state": "conditional-with-named-gate (frozen, awaiting the stability pass; both STANDS and RUNAWAY live)"
  },
  "deferred_to_stability_pass": "NOT decided here (deliberately, so the test cannot be reverse-fit): (i) whether S_sigma is bounded-below or phase-exits as sigma -> -inf; (ii) any int_loop sign/branch; (iii) the positivity of the UV spectral measure as a derived fact (it is the NAMED axiom, not proven). Both STANDS and RUNAWAY remain live.",
  "builds_on": {
    "gap04_full_supertrace_residue.py": "field content n_B=35, n_F=90, Str[1]=-55 forced by index -3; supertrace residue Str_pole=+4.466 survives (determinant route B is owner-locked).",
    "gap04_zeta_continuation_frg2.py": "zeta pole at s=-1/2, d_eff=3 zero-weight tower; finite part scheme-dependent (flips with mu) -- the ambiguity the spectral action removes by carrying no subtraction.",
    "gap04_higher_operator_wall.py": "EFT-validity boundary: a6>~a4>~c_loop outgrow; curvature reaches cutoff near sigma~+1.70 (unit-vol). This is the substrate for the FREEZE-4 phase-exit boundary."
  },
  "integrity_guards_attest": {
    "1_full_gauge_fixed_tower": "FULL gauge-fixed physical tower: every one of the nine ledger rows (graviton, gauge, moduli, Higgs, chiral fermion, ghosts, projected, Weyl-rigid) maps to an explicit piece of the SINGLE operator D_sigma^2. NOT scalar-only, NOT bosonic-only. n_B=35, n_F=90.",
    "2_no_target_loading": "No sign/branch chosen; committed c_loop=1.36379e-05 never read; spectral action forms no log-det so no int_loop decision is taken.",
    "3_no_anthropic": "Runaway stays live; boundedness NOT asserted here.",
    "4_finite_order_consistency_only": "a4/a6/a8 appear ONLY as consistency checks of a_0/a_2/a_4 against the corpus cosmological/Einstein/Yang-Mills terms; the proof rides on positivity of Str f, never on a finite-order coefficient sign. Lemma 1 (finite-order undecidability along sigma->-inf) respected.",
    "5_boundedness_or_phase_exit_frozen_not_claimed": "The paired (A)+(C) object that the boundedness/phase-exit proof will use is FROZEN here (positive functional interior + sigma_* phase-exit boundary); the PROOF is the next pass. No -inf is asserted reached and none is asserted avoided yet.",
    "6_lambda_scope_preserved": "Relationship to Lambda is 'same hardness class, not same solved problem' -- the result will be conditional on B-UQFC-14-UV-1, the single named axiom, exactly as the Lambda record is.",
    "7_axiom_named_not_disguised": "The positive-spectral-measure input is NAMED as B-UQFC-14-UV-1, declared as an irreducible UV-completion axiom, never disguised as derived.",
    "8_no_leans_language": "No 'leans/leaning runaway' language used anywhere; branches are stated as live, not tilted."
  },
  "no_target_loading_attest": "NO target-loading. (2) The sign/branch is NOT chosen: this freeze takes NO branch (the spectral action never forms a log-det, so no int_loop sign is decided). The committed c_loop=1.36379e-05 was NEVER opened and entered NO input or sign decision. (3) NO anthropic 'we exist': runaway stays live; boundedness is NOT asserted here. (7) The positivity of the UV spectral measure is NAMED as the UV-completion axiom B-UQFC-14-UV-1 (NOT disguised as derived): 'Str f >= 0 under the spectral-action axiom (positive cutoff profile + the corpus spin-c Dirac operator as fundamental object)'. The moment positivity f_0,f_2,f_4>0 is exhibited over a CLASS of profiles, tuning nothing. (4) finite-order a4/a6/a8 are flagged as consistency checks only; Lemma 1 (no finite truncation decides global stability) respected. No observed value (A_s, Lambda_obs, r, eta_B, n_s, N_eff, PDG, Omega_DM, H_0, S_8) entered on any input side.",
  "provenance": {
    "Fable_Quantum.md_sha256": "cff0bf0cecdaa6eda26798c2ceafa4b53f72e90fbd767d31a1bce7ed319ff318",
    "Fable_GUT_3.md_sha256": "6c1a1561167053c84e5cf2b366294af47e03c2d063db4f7f246c107f02779652",
    "gilkey_a4_cross_terms.py_sha256": "6e58698a4658ff1f49847617d8a236be8321f2c97227f570829111ba3208212d",
    "14D_PARENT_SELECTOR_GEOMETRY.md_sha256": "183351ae3907a34832d1188167f4bff1927fbc540a439a7d9fb1907069abf003",
    "SELECTOR_DETERMINANT_CHANNEL_AUDIT.md_sha256": "931125f939d3cc864427850b4bfa6f146c048bd8c90da98cfd537e07748961d4",
    "prior_supertrace_sha256": "c1bc400dde7f83954ca53bd7ad860cc3c66d7ebe462f036c8e5c1fe1762d5af8",
    "prior_zeta_sha256": "b04b93d56140e20d27cb7954b4c61410164a65ac805a3e5629c7b72f62756741",
    "prior_wall_sha256": "14dd8e2cd507e7cf5ae26b1c307e5f3526a80249d63ecce6fd460759720f2533",
    "active_bundle_hash": "0fd19c9ae0c1",
    "spin_c_index": -3
  },
  "non_promotion": "no gate flipped; no status word emitted for any gate. This file FREEZES a definition (the UV object, its prescription, what it computes) target-blind; it asserts NO stability verdict and is NOT promotable as one."
}