{ "audit": "PC-001 ACTUAL NORMALIZATION AUDIT — Round 030 certificate C1–C10 against the actual PC-001 objects", "target": { "curve": "389.a1", "p": 11, "P": [ 0, 0 ], "Q": [ 1, 0 ], "a_11": -4, "w": 1 }, "calibrator": { "curve": "19.a1", "ainvs": [ 0, 1, 1, -769, -8470 ], "a_11": 3, "rank": 0 }, "verdict_rule": "NOT-APPLICABLE if the item does not apply; FAIL on a counterexample or nontrivial residual; UNMEASURED if the object has not been constructed by anyone; PASS only if the object exists and every measurement on it is green; never UNMEASURED→PASS; no majority vote", "items": { "C1": { "title": "Late common character", "requirement": "Target and calibrator use the same exact late character.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the projected late classes B̂_E(t), B̂_0(t) built from ONE auxiliary Eisenstein family E₂(1,χ₈)^crit with one frame and one quotient transport — equivalently the Coleman-family trace jets 𝒯_X(g) mod X² with a stated pseudocharacter identification — so that C_{π,E} and C_{π,0} exist as separate objects to compare; neither peer line has them (PC-001 §2: 仍要求真正構造 B_{E,2} 與 r_{0,1}; Attack 09: 未宣稱計算出 Coleman 族)", "what_exists": { "cited_theorem": "Loeffler–Rivero C1.9–C1.13 degeneration comparison and C1.11 uniqueness of the meromorphic Eichler–Shimura map given the auxiliary frame (PC-001 §1) — CITED_THEOREM", "hypotheses_measurable_here": { "both_partners_ordinary_at_11 (a11, unit root)": { "389.a1": [ -4, 7 ], "19.a1": [ 3, 3 ] }, "auxiliary_character_decent": { "chi8_even": true, "chi8_nontrivial": true, "chi8_of_11_not_1": true, "conductor_8": true }, "same_auxiliary_data_named_for_both": "E₂(1,χ₈), smoothing d = 5, χ₈(11) = −1" }, "hypotheses_not_instantiable": "that the SAME auxiliary quotient transport is applied to both partners — a property of a construction that has not been carried out" } }, "label": "CITED_THEOREM for the naturality argument; UNMEASURED for the actual equality C_{π,E} = C_{π,0}", "level": "E", "normalization": "PC-001 (1): one C_π = c_lead/q_f for both partners, Loeffler–Rivero's compatible Kato/Coleman/Gauss/adjoint conventions, the auxiliary frame fixed once", "dependencies": { "upstream": [ "the auxiliary family and its quotient transport (not constructed)" ], "downstream": [ "C2", "C4", "Round 030 nodes A–B", "PC-001 (2)–(3)" ] }, "gate": "src92 · pc001-actual-C1", "mutation": "split the late factor: C_{π,E} = u·C_{π,0} on the model — the identity (2) is off by exactly u", "notes": "" }, "C2": { "title": "Partner factors separated", "requirement": "No partner arithmetic is hidden in the common line.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the Petersson/period halves Π_E, Π_0 of D_i = L_p(Ad g_i)(0) in Loeffler–Rivero's normalisation (§B2.3, eq. (8)) — Attack 09 §4: 本輪沒有計算其值; PC-001 §3: 其完整 period 比值 D_0 沒有在本輪算出 — and the common line C_π j itself (C1) to inspect for hidden partner factors", "what_exists": { "lambda_E(0) mod 11": 5, "lambda_0(0) mod 11": 4, "S5(0)": 26, "S5_partner_independent": true, "E_adj_E mod 11": 3, "E_adj_0 mod 11": 7, "A_0(0)/A_E(0) measured part mod 11": 6, "A_0(0)/A_E(0) in full": "6 · (Π_E/Π_0) mod 11, Π_E/Π_0 unmeasured" } }, "label": "COMPUTED_HERE for λ_i(0), 𝒮₅(0), E_i; UNMEASURED for Π_E/Π_0 and for the common line", "level": "P", "normalization": "λ_i(0) at PC-001's scale (plus 0 ↦ 1, minus 4 ↦ 1 for 19.a1; index-3 minus line for 389.a1), the ω⁻¹ branch at x⁻¹; E_i = (1 − β_i/α_i)(1 − β_i/(11α_i)) mod 11; Π_i in Loeffler–Rivero §B2.3 (8)", "dependencies": { "upstream": [ "C1", "Attack 09 (14)–(15)", "RUN-077/080/081 numbers" ], "downstream": [ "PC-001 (2)–(3)", "Round 021 §20 partner ratio", "Attack B of the handoff" ] }, "gate": "src92 · pc001-actual-C2", "mutation": "tag λ_i as common: the partner ratio read off the tags no longer equals the model's A_0(0)/A_E(0)", "notes": "RUN-088 supplied λ_0(0)/λ_E(0) ≡ 3; with the adjoint Euler multipliers the measured part of the partner ratio is 3·(3/7) ≡ 6 (mod 11); the whole remaining uncertainty is the Petersson/period ratio Π_E/Π_0. PC-002's cup factor 3 is not it (RUN-081)." }, "C3": { "title": "Coordinate factors separated", "requirement": "No analytic coordinate weight is hidden in exact normalization.", "verdict": "PASS", "evidence": { "object_constructed": true, "measured_ok": true, "counterexample": false, "applicable": true, "object_tested": "the cyclotomic coordinate of the four actual 11-adic series L_E, λ_E, L_0, λ_0 (t¹²⁰, layer 11³) and of PC-001's j(t) = log(1+t)/log₁₁(12): one exponent map e(u) = log_γ⟨u⟩, γ = 12, for both partners; PC-001's identity (2) as a series identity", "scope": "the PASS is on the objects that exist — the series and the identity; the coordinate law of the unconstructed B̂_i is (1)'s statement, forced once they live in the same Λ = Z₁₁[[Γ]]; if they are later built in another coordinate, reopen under R6 (a convention boundary changed)" }, "label": "COMPUTED_HERE", "level": "G (convention) — of the handoff's three, E: an exact statement, neither partner- nor lattice-dependent", "normalization": "cyclotomic coordinate t with σ_t(γ) = 1 + t, γ = 12 for the target and 12 for the calibrator; ⟨u⟩ = u·ω(u)⁻¹; PC-001's j(t) = log(1+t)/log₁₁(12)", "dependencies": { "upstream": [ "RUN-071/081 series", "PC-001 (1)–(2)", "Round 009 u^{−r}", "Round 021 j-tensor" ], "downstream": [ "C4", "Round 030 node A" ] }, "gate": "src92 · pc001-actual-C3", "mutation": "apply the jet weight as u^{+r}, or build the calibrator's series with generator 23 while the target keeps 12", "notes": "" }, "C4": { "title": "Pre-leading source identified", "requirement": "The auxiliary leading tensor or equivalent q-blind source exists.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "Θ_aux ∈ Sym^m(T_X^*) ⊗ 𝒞_aux — the auxiliary leading tensor in the weight direction X before quotient transport (Rounds 024–025) — or the identification of the Kummer unit's Eisenstein extension class with the meromorphic Eichler–Shimura leading period of Attack 09 (16); Attack 09 §6: 尚未建立它與式 (16) 的精確比較", "what_exists": { "candidate": "the Kummer unit ε = 1 + √2 and its 11-adic logarithm — an explicit nonzero Eisenstein extension class", "log_over_11_sqrt2_mod_11": 5, "q_blind_by_construction": true, "why_q_blind": "computed from ε, 11 and the precision alone; no quotient transport enters" } }, "label": "COMPUTED_HERE for the candidate's value and q-blindness; UNMEASURED for the tensor and its identification", "level": "E", "normalization": "ε = 1 + √2 in Z[√2]/121, log₁₁ ε = log(ε²⁴)/24, the value log ε/(11√2) mod 11; the frame (c_lead, q_f) of Round 021 for the reframe", "dependencies": { "upstream": [ "C1", "C3", "Attack 09 §2, §6" ], "downstream": [ "C5", "C7", "Round 030 nodes B–C" ] }, "gate": "src92 · pc001-actual-C4", "mutation": "let the candidate source be built with q_f: it then moves under q_f ↦ u q_f", "notes": "" }, "C5": { "title": "Auxiliary-to-arithmetic defect classified", "requirement": "Every defect factor is tagged E, L, G, or P.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the comparison defect δ = a_D q_D η_D between the auxiliary source (C4) and the arithmetic pre-leading line — both sides must exist before a defect between them can be tagged", "what_exists": { "factors_of_PC001_(1)–(3)_and_Attack_09_(16)_(20)_with_roles": { "C_pi": "E", "c_lead": "E", "q_f": "E", "j": "G", "log11_gamma": "G", "S5": "L", "lambda_i(0)": "P", "E_adj_i": "P", "Pi_i": "P", "kappa_E": "P", "ell_0": "P", "16 = e11/theta": "L", "416 = 16*26": "L", "Gauss(chi_8)": "G", "plus_index_2": "L" }, "all_tagged": true, "untagged": [] } }, "label": "DERIVED_FROM_VERIFIED_INPUTS for the ledger of existing factors; UNMEASURED for δ", "level": "mixed (E/L/G/P per factor)", "normalization": "Round 026's categories E / L / G / P on the factors of PC-001 (1)–(3) and Attack 09 (16), (20)", "dependencies": { "upstream": [ "C4" ], "downstream": [ "C6", "C7" ] }, "gate": "src92 · pc001-actual-C5", "mutation": "leave one factor of the ledger without a role", "notes": "" }, "C6": { "title": "Period--de Rham exact status resolved", "requirement": "Either canonical cancellation or one explicit residual pairing scalar.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "η_pdR: the pairing scalar between the crystalline-quotient period frame of the auxiliary side (Attack 09 §5) and the arithmetic period frame — either a canonical-pairing theorem (route A, η = 1) or one explicit scalar with its frame (route B); neither exists", "what_exists": { "period_side_alignments": "λ_0⁺ = 1 (RUN-080) and λ⁻ = 6 (RUN-076): modular-symbol frames vs Néron periods — period-side data, not the period–de Rham pairing", "opposite_weights_give_invariance_not_one": true } }, "label": "UNMEASURED", "level": "E", "normalization": "Round 027's coupled frame mutation p ↦ up with the weights (+1, −1) on (δ_per, δ_dR)", "dependencies": { "upstream": [ "C4", "C5" ], "downstream": [ "C7", "Round 030 node D" ] }, "gate": "src92 · pc001-actual-C6", "mutation": "swap the period / de Rham exponent signs: the product stops being invariant", "notes": "" }, "C7": { "title": "Determinant exact status resolved", "requirement": "Either canonical cohomological equality or explicit $\\kappa_{\\det}$.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the auxiliary→arithmetic determinant comparison map (or the cohomological map it induces) for PC-001; Attack 06's H (RUN-073) is the descent pairing matrix, not this comparison; none exists", "what_exists": { "abstract_K1_identity_on_explicit_complexes": true, "actual_map": null } }, "label": "UNMEASURED (the abstract identity is COMPUTED_HERE and in RUN-089)", "level": "E", "normalization": "κ_det(h) = Π det(H^i(h))^{(−1)^i} on two-term complexes, cohomology recomputed in random bases (Round 028)", "dependencies": { "upstream": [ "C4", "C6" ], "downstream": [ "Round 030 node E", "Round 031–032 determinant comparisons" ] }, "gate": "src92 · pc001-actual-C7", "mutation": "hand the audit a cohomologically nontrivial automorphism as the actual map: κ_det ≠ 1 is a FAIL (R3)", "notes": "" }, "C8": { "title": "Lattice / finite data separated", "requirement": "No exact unit ambiguity is attributed to a length / index.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the local integral maps Φ_ℓ of an actual Selmer complex / mapping cone for PC-001, whose v_ℓ(det Φ_ℓ) would be compared with Smith and cone Euler lengths (Rounds 007, 033–034) — no such complex is constructed; and the prime-to-11 saturation of the target's rational modular-symbol line", "what_exists": { "19.a1": { "plus_primitive_content": 1, "minus_primitive_content": 1, "plus_normalisation_index": 2, "plus_index_valuations": { "2": 1, "3": 0, "5": 0, "7": 0, "11": 0 }, "minus_normalisation_index": 1, "global_Z_primitivity_measured": true, "role": "L: the index 2 is a length between the Z-primitive vector and PC-001's normalised one, v_2 = 1, v_11 = 0" }, "389.a1": { "plus_line": "mod 11 only, normalised at its first nonzero coordinate (index 5)", "minus_line": "mod 11 only, normalised at index 3; a unit multiple λ⁻ = 6 of the Ω⁻-normalised symbol on 14 twists (RUN-076)", "first_nonzero_indices": [ 5, 3 ], "global_Z_primitivity_measured": false, "role": "the 11-adic normalisation is measured; the prime-to-11 saturation of the target's rational line is not" }, "Gauss(chi_8)": { "role": "G: Ω_F⁻ = Ω_E⁻/G(χ₈) (Attack 09 §3.2), a convention; G(χ₈)² = 8, an 11-unit" } } }, "label": "COMPUTED_HERE for the calibrator's lattice data; UNMEASURED for the Selmer/cone lengths and the target's global saturation", "level": "L", "normalization": "Z-primitive vectors of the rational eigenlines at level 19 (content 1); PC-001's normalised vectors (plus 0 ↦ 1, minus 4 ↦ 1); mod-11 lines at level 389 normalised at their first nonzero coordinate", "dependencies": { "upstream": [ "RUN-080 lines", "RUN-076 alignment" ], "downstream": [ "Round 030 node F", "Rounds 033–034" ] }, "gate": "src92 · pc001-actual-C8", "mutation": "record the normalisation index 2 as a unit (valuation 0)", "notes": "" }, "C9": { "title": "Residual line flatness", "requirement": "No residual holonomy.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "object_needed": "the residual comparison local system on the actual comparison objects — vertices: the auxiliary source, the arithmetic source, the period and de Rham frames, the determinant routes; edges: the actual transports with their scalars (C4–C7) — none of the objects exists, so the graph has no vertices", "what_exists": { "actual_transport_graph": { "vertices": 0, "edges": 0, "tree": [], "cotree": [], "cycle_products": [], "all_trivial": null } } }, "label": "UNMEASURED", "level": "E", "normalization": "multiplicative transports on directed edges, BFS spanning tree, cotree cycle products (Rounds 016, 029)", "dependencies": { "upstream": [ "C4", "C5", "C6", "C7" ], "downstream": [ "C10", "Round 030 node G" ] }, "gate": "src92 · pc001-actual-C9", "mutation": "insert one actual cotree edge with holonomy ≠ 1: a hard FAIL (R4)", "notes": "" }, "C10": { "title": "Residual normalization fixed", "requirement": "Canonical section or one exact anchor.", "verdict": "UNMEASURED", "evidence": { "object_constructed": false, "measured_ok": null, "counterexample": false, "applicable": true, "blocked_by": "C9", "object_needed": "a theorem-level canonical trivialisation of the residual line (route A) or one exact arithmetic anchor in the connected flat component (route B) — attemptable only after C9 is green, and C9 is unmeasured", "what_exists": { "formal_one_anchor_fixes_the_section": true, "numerical_candidate": { "candidate": "q_fin = 1 from L''(E,1)/2 = Ω·Reg·q_fin", "is_arithmetic_anchor": false, "why_not": "a numerical normalisation chosen for convenience is not an arithmetic anchor (handoff C10); and it is a consistency check, not a proof bridge (§14)" } } }, "label": "UNMEASURED (blocked by C9)", "level": "E", "normalization": "sections of a flat residual system as a K^×-torsor; one anchor s(v₀) = 1 (Round 029)", "dependencies": { "upstream": [ "C9" ], "downstream": [ "Round 030 node G", "NORMALIZATION-CLOSED" ] }, "gate": "src92 · pc001-actual-C10", "mutation": "remove the anchor from the formal torsor, or accept the numerical q_fin = 1 as an anchor with C9 unmeasured", "notes": "" } }, "first_blocker": "C1", "normalization_status": "NORMALIZATION-UNMEASURED", "reopening_trigger_if_any": { "R1_frame_mutation_changed_an_invariant": false, "R2_common_factor_shown_partner_dependent": false, "R3_nontrivial_K1_automorphism": false, "R4_residual_holonomy": false, "R5_new_independent_character": false, "R6_convention_boundary_changed": false, "note": "none fired; R2 cannot be excluded while C1 is unmeasured" }, "round030_dependency_impact": "no FAIL: Round 030's closure theorem (42.1) stays a conditional theorem whose hypotheses C1–C10 are, for the actual PC-001, one PASS (C3) and nine UNMEASURED with the earliest at C1; no node of the post-normalisation DAG A–G is cut; the certificate is not frozen", "round031_035_dependency_impact": "no dependency broken by this audit: Rounds 031–035 (in the tree, not yet verified here) rest on the target-side determinant architecture, which none of the nine UNMEASURED items contradicts; they are held pending their own verification round", "dependency_cones": { "cone_if_C1_fails_later": [ "C2", "C4", "PC-001 (2)–(3)", "Round 021 §20", "Round 030 nodes A–B" ], "cone_if_C7_fails_later": [ "Round 030 node E", "Rounds 031–032 determinant comparisons" ], "cone_if_C9_fails_later": [ "C10", "Round 030 node G" ] }, "consistency_check": { "L2_over_2_this_tree": 0.7593165002884076, "omega_this_tree": 4.98042512171011, "regulator_this_tree": 0.15246017794314376, "heights": { "h_P": 0.32700077365160496, "h_Q": 0.47671165934373955, "pairing": 0.058522674844878914 }, "ratio": 0.9999999999999747, "ratio_rounded_15": 0.999999999999975, "q_fin": 1, "within_1e_13": true, "role": "CONSISTENCY CHECK, not PROOF BRIDGE (handoff §14); evidence for no C-item" }, "attack_order_used": [ "C1", "C2", "C3", "C4", "C6", "C7", "C5", "C8", "C9", "C10" ], "claim_labels_legend": [ "COMPUTED_HERE", "CITED_THEOREM", "DERIVED_FROM_VERIFIED_INPUTS", "UNMEASURED", "FAILED" ], "bsd": "open; a numerically exact BSD-shaped identity is not a proof of BSD (handoff §0, §14; this tree since RUN-001)", "next_objects_ranked": [ "1. the projected late classes B̂_E, B̂_0 from one auxiliary Eisenstein family (C1) — equivalently the Coleman-family trace jets 𝒯_X(g) mod X²", "2. the Petersson/period halves Π_E, Π_0 of the adjoint values D_i = L_p(Ad g_i)(0) in Loeffler–Rivero's normalisation (C2; Attack B)", "3. the auxiliary leading tensor Θ_aux, or the identification of the Kummer unit's class with the ES leading period (C4; Attack D)", "4. the auxiliary→arithmetic comparison map and its K1 residual (C7; Attack E)", "5. the period–de Rham pairing scalar η_pdR (C6)", "6. the residual transport graph (C9; Attack F), then the anchor (C10)" ] }