{ "schemaVersion": "2.0.0", "title": "Independent proof records for serious candidates", "snapshotId": "lmfdb-parent-graph-2026-08-18", "statusVocabulary": { "verified": "Independent evidence establishes the stated item.", "not_verified": "The item is required or relevant but has not been established.", "not_applicable": "The item does not apply to this record.", "not_checked": "No candidate-specific independent check has been performed." }, "theoremReadyRule": "A record is theorem-ready only when G0--G8 are all green and G7/G8 contain separate evidence IDs.", "records": [ { "proof_record_id": "proof:mc72-216-13-h-canonical-to-576-e4:v1", "dossier_url": "https://github.com/georgeturcasubb/georgeturcasubb.github.io/blob/main/covers-of-elliptic-curves/docs/examples/cubic-72-216.md", "paper_featured": true, "candidate_id": "mc72-216-13-h-canonical-to-576-e4", "candidate_version": 1, "degree": 3, "status": "theorem_ready", "theorem_ready": true, "derivation": { "theorem_ready": true, "all_gates_green": true, "independent_signoff_present": true, "adversarial_signoff_present": true, "rule": "G0 through G8 must all be green, with nonempty independent-verifier and adversarial-referee evidence IDs.", "candidate_status_recorded": "theorem_ready" }, "source": { "label": "72.216.13.h.1", "genus": 13, "level": 72, "qbar_gonality_bounds": [ 4, 13 ] }, "target": { "label": "24.72.1.h.1", "genus": 1, "chosen_origin": "canonical cusp 2", "weierstrass_coefficients": [ 0, 0, 0, 0, 8 ], "weierstrass_equation": "y^2 = x^3 + 8", "elliptic_lmfdb_label": "576.e4", "q_isogeny_class": "576.e", "conductor": 576, "rank_bounds": [ 1, 1 ], "non_torsion_point": [ 1, 3 ] }, "map": { "construction": "subgroup_inclusion", "geometric_degree": 3, "base_field": "Q", "explicit_coordinate_map_available": false, "group_inclusion_available": true, "explicit_kummer_description": "w^3 = 2h", "note": "The Kummer description is the separately verified paper computation; LMFDB stores no coordinate map for this parent edge." }, "stage4": { "map_degree_verified": { "state": "verified", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1", "SYNTHESIS-PRE-REVIEW-20260721-V1" ], "checked_on": "2026-08-24", "note": "The common-level subgroup inclusion has exact coarse degree three." }, "target_model_verified": { "state": "verified", "evidence_ids": [ "TARGET-ID-20260721-V1", "TARGET-ID-SIMULTANEOUS-20260721-V1", "GROUP-20260721-V1", "RANK-576E4-SAGE-20260721-V1" ], "checked_on": "2026-08-24", "note": "The pointed target is independently identified with 576.e4: y^2=x^3+8." }, "target_point_verified": { "state": "verified", "evidence_ids": [ "TARGET-ID-20260721-V1", "RANK-576E4-20260721-V1", "RANK-576E4-SAGE-20260721-V1" ], "checked_on": "2026-08-24", "note": "P=(1,3) lies on the actual pointed target and is non-torsion." }, "rank_verified": { "state": "verified", "evidence_ids": [ "TARGET-ID-20260721-V1", "RANK-576E4-20260721-V1", "RANK-576E4-SAGE-20260721-V1" ], "checked_on": "2026-08-24", "note": "Two exact computations certify algebraic rank bounds [1,1]." }, "geometric_gonality_verified": { "state": "verified", "evidence_ids": [ "THEOREM-APPLICABILITY-20260721-V2" ], "checked_on": "2026-08-24", "note": "The verified genus-13 source satisfies the rational-fibre reduction threshold 13>10." }, "genus_hypothesis_verified": { "state": "verified", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1" ], "checked_on": "2026-08-24", "note": "The smooth projective source has independently reproduced genus 13." }, "discriminant_divisor_verified": { "state": "verified", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ], "checked_on": "2026-08-24", "note": "The trace discriminant is D_f=2 Br(f), of degree 24." }, "derangement_verified": { "state": "verified", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ], "checked_on": "2026-08-24", "note": "The odd support is empty, so the derangement condition is vacuous." }, "rational_fibre_reduction_verified": { "state": "verified", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ], "checked_on": "2026-08-24", "note": "Theorem 3.5 / Song--Tucker applies because g(C)=13>3^2+1." }, "field_density_theorem_applicable": { "state": "verified", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ], "checked_on": "2026-08-24", "note": "The complete excluded-prime and empty-odd-support ledger verifies the density input." } }, "gates": { "G0": { "status": "green", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "SYNTHESIS-PRE-REVIEW-20260721-V1" ], "summary": "Stable new ID/version, native-level raw canonical generators, full-inverse adelic definitions, and direct common-level inclusion are explicit. The record is the sole canonical successor to rejected V1, not a second count of that failed metadata formulation; canonical labels separate it from the reserved i sibling." }, "G1": { "status": "green", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1" ], "summary": "The exact modular signature gives genus 13, independently reproduced. Full determinant and -I meet the cited moduli theorem, which supplies the smooth projective geometrically irreducible Q-curve; no affine or singular model is used." }, "G2": { "status": "green", "evidence_ids": [ "TARGET-ID-20260721-V1", "TARGET-ID-SIMULTANEOUS-20260721-V1", "GROUP-20260721-V1", "RANK-576E4-SAGE-20260721-V1" ], "summary": "The canonical genus-one target is pointed at rational cusp 2, and exact independently reviewed Q-descent identifies it with E:y^2=x^3+8 and cusp 2 with O=[0:1:0]. A targeted verifier exhibited 24 simultaneous 72-sheet branch/deck/Galois/pointing witnesses with zero failures, closing the adversarial descent objection. Sage independently certifies conductor 576 for that equation." }, "G3": { "status": "green", "evidence_ids": [ "TARGET-ID-20260721-V1", "RANK-576E4-20260721-V1", "RANK-576E4-SAGE-20260721-V1" ], "summary": "On the actual pointed target, P=(1,3) is non-torsion and exact algebraic rank bounds are [1,1], proved locally by two-isogeny descent and independently in proof-mode Sage." }, "G4": { "status": "green", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1", "TARGET-ID-20260721-V1", "TARGET-ID-SIMULTANEOUS-20260721-V1" ], "summary": "The direct inclusion H_can <= G_can^(72) of full-inverse canonical groups invokes the exact moduli construction over Q and extends uniquely to smooth projective compactifications. Composing with the verified pointed Q-isomorphism gives f:X_{H_can}->E over Q." }, "G5": { "status": "green", "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1", "SYNTHESIS-PRE-REVIEW-20260721-V1" ], "summary": "The cited moduli theorem makes the actual canonical morphism degree equal to [G_can^(72):H_can]=3 after all GL2, SL2, projective, coarse, determinant, -I, component, and descent conventions are reconciled. A separate Sage implementation reproduces the three cosets, S3 action, degree-three S/ST/T projection fibers, and Riemann-Hurwitz. Under the literal 'when possible' rule, this is sufficient although no genus-13 equation or homogeneous map exists; an equation-level function-field calculation is therefore unavailable, not claimed." }, "G6": { "status": "green", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ], "summary": "For the canonical cover, all twelve and only twelve branch fibers are cusps of type (3); D_f=2 times the reduced twelve-cusp divisor, B_f is empty, the fixed-point-free condition is vacuous, and the complete working excluded-prime set is {2,3}. The pointed Q-isomorphism transports these facts to f without changing degree or ramification. All literal Theorem 1.2/Corollary 3.8 conditions and the project rank requirement are met." }, "G7": { "status": "green", "evidence_ids": [ "G7-20260721-V1" ], "summary": "The independent verifier signed PASS against the exact frozen pre-promotion candidate SHA-256 2ff4e853d88fdb6cb0efe0c7fb99483cad19525ab73e7d0604fa2781d78757b1. A fresh standard-library implementation reproduced group inclusion, all degree conventions, genera, target Q-form, rank, ramification, trace discriminant, and integral spread; its 43-entry manifest verifies." }, "G8": { "status": "green", "evidence_ids": [ "G8-20260721-V1", "TARGET-ID-SIMULTANEOUS-20260721-V1" ], "summary": "The adversarial referee signed PASS against the same frozen pre-promotion candidate hash after attacking all mandated failure modes. It withheld sign-off for a target-descent presentation gap, obtained and independently reran the exact 24-witness simultaneous-conjugacy certificate, and concluded that no material objection remains; its 16-entry manifest verifies." } }, "evidence_ids": [ "GROUP-20260721-V1", "SOURCE-AUDIT-CONVENTIONS-20260721-V1", "P5-LEHNER-H-CANONICAL-SAGE-20260721-V1", "TARGET-ID-20260721-V1", "TARGET-ID-SIMULTANEOUS-20260721-V1", "RANK-576E4-20260721-V1", "RANK-576E4-SAGE-20260721-V1", "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "SYNTHESIS-PRE-REVIEW-20260721-V1", "G7-20260721-V1", "G8-20260721-V1" ], "stage5": { "branch_divisor": { "status": "verified", "notation": "Br(f)", "degree": 12, "galois_orbit_degrees": [ 1, 1, 2, 2, 2, 4 ], "ramified_fibres": 12, "fibre_partition": [ 3 ], "support_description": "the twelve cusps" }, "ramification": { "status": "verified", "degree": 3, "ramified_fibres": 12, "fibre_partition": [ 3 ], "ramification_index_at_each_point": 3, "total_riemann_hurwitz_contribution": 24, "support_description": "All ramification lies above the twelve target cusps.", "evidence_ids": [ "THEOREM-PRELIM-20260721-V1", "THEOREM-APPLICABILITY-20260721-V2", "TARGET-ID-20260721-V1" ] }, "trace_discriminant": { "status": "verified", "formula": "D_f = 2 Br(f)", "degree": 24, "coefficient_at_each_branch_point": 2 }, "odd_support": { "status": "verified", "formula": "B_f = empty", "empty": true, "splitting_field": "Q", "omega": [], "galois_group": "trivial" }, "derangement": { "status": "verified", "vacuous": true, "witness": "The identity acts without fixed points on the empty Galois set." }, "integral_spread": { "status": "verified", "excluded_primes": [ 2, 3 ] }, "rational_fibre_reduction": { "status": "verified", "method": "Theorem 3.5 / Song--Tucker genus inequality", "inequality": "g(C)=13 > 3^2+1=10" }, "field_density": { "status": "verified", "theorem": "Theorem 1.2 and Corollary 3.8 in the controlling draft", "note": "The fixed-point-free condition is vacuous because B_f is empty." } }, "caveats": [ "No genus-13 coordinate model or homogeneous coordinate formula is claimed.", "The exact abstract modular construction and independent common-level computation certify the Q-map and its degree." ], "provenance": { "candidate_path": "data/candidates/mc72-216-13-h-canonical-to-576-e4.json", "candidate_sha256": "4db273fe35abb0b6641ae333659909aca5d016253a9f3199102543270e807ec7", "evidence_kind": "certified_arithmetic_computation_and_exact_symbolic_computation" } }, { "proof_record_id": "proof:d5-l40-ck1-o2:2026-08-24", "dossier_url": "https://github.com/georgeturcasubb/georgeturcasubb.github.io/blob/main/covers-of-elliptic-curves/docs/examples/quintic-40-480.md", "paper_featured": false, "candidate_id": "d5-l40-ck1-o2-v3", "candidate_version": 3, "degree": 5, "status": "branch_verified", "theorem_ready": false, "derivation": { "theorem_ready": false, "all_gates_green": false, "independent_signoff_present": true, "adversarial_signoff_present": true, "rule": "G0 through G8 must all be green, with nonempty independent-verifier and adversarial-referee evidence IDs.", "candidate_status_recorded": "candidate" }, "source": { "label": "40.480.33.ck.1", "genus": 33, "level": 40, "qbar_gonality_bounds": [ 8, 10 ] }, "target": { "label": "40.96.1.o.2", "genus": 1, "chosen_origin": "cusp 1/8", "origin_kind": "cusp", "origin_parameter": "1/8", "weierstrass_coefficients": [ 0, 0, 0, 100, 0 ], "weierstrass_equation": "y^2 = x^3 + 100*x", "elliptic_lmfdb_label": "800.d4", "cremona_label": "800a4", "q_isogeny_class": "800.d", "conductor": 800, "rank_bounds": [ 1, 1 ], "non_torsion_point": [ 5, 25 ], "torsion_order": 2, "plane_model": { "coordinates": [ "u", "v", "w" ], "equation": "-5*u^3 + 10*u^2*v - 10*u*v^2 - u*w^2 + 2*v*w^2 = 0", "origin": [ 0, 1, 0 ] }, "pointed_isomorphism": { "forward": [ "-10*u*w + 20*v*w", "100*u^2 - 200*u*v + 200*v^2", "u*w" ], "inverse": [ "2*Y*Z", "X*Y/10 + Y*Z", "X^2 + 100*Z^2" ], "degree": 1, "base_field": "Q", "presentation_note": "These are rational coordinate triples with base loci. Exact inverse identities in the function fields give the unique Q-isomorphism between the smooth projective curves; they are not globally base-point-free triples." } }, "map": { "construction": "subgroup_inclusion", "geometric_degree": 5, "base_field": "Q", "explicit_coordinate_map_available": false, "group_inclusion_available": true, "note": "No isogeny is inserted into the degree-five modular cover.", "pointed_model_map_note": "The forward and inverse coordinate triples are rational formulas verified as inverse in the function fields; they extend uniquely between the smooth projective curves. They are not claimed to be base-point-free coordinate presentations (the forward triple has two geometric base points and the inverse triple has three, including O)." }, "stage4": { "map_degree_verified": { "state": "verified", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-G7-FULL-CONJUNCTION-20260818-IV1" ], "checked_on": "2026-08-24", "note": "Independent common-level group computations certify coarse degree five." }, "target_model_verified": { "state": "verified", "evidence_ids": [ "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "checked_on": "2026-08-24", "note": "Producer, independent verifier, and referee identify the actual pointed target as 800.d4." }, "target_point_verified": { "state": "verified", "evidence_ids": [ "D5-RANK-40-96-1-O-2-20260818-V2", "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "checked_on": "2026-08-24", "note": "P=(5,25) lies on 800.d4 and is certified non-torsion." }, "rank_verified": { "state": "verified", "evidence_ids": [ "D5-RANK-40-96-1-O-2-20260818-V2", "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "checked_on": "2026-08-24", "note": "The algebraic rank of the actual target is certified to be one." }, "geometric_gonality_verified": { "state": "verified", "evidence_ids": [ "D5-G7-FULL-CONJUNCTION-20260818-IV1", "D5-SOURCE-AUDIT-ADDENDUM-20260818T103533Z" ], "checked_on": "2026-08-24", "note": "Castelnuovo--Severi independently proves the exact lower bound needed here (at least six)." }, "genus_hypothesis_verified": { "state": "verified", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-SOURCE-AUDIT-20260818T102220Z" ], "checked_on": "2026-08-24", "note": "Two exact group/orbit implementations give source genus 33." }, "discriminant_divisor_verified": { "state": "verified", "evidence_ids": [ "D5-THEOREM-ADDENDUM-20260818T100903Z", "P5-THEOREM-ADDENDUM-20260824" ], "checked_on": "2026-08-24", "note": "The trace discriminant is D_f=4 Br(f), of degree 64." }, "derangement_verified": { "state": "verified", "evidence_ids": [ "D5-THEOREM-ADDENDUM-20260818T100903Z", "P5-THEOREM-ADDENDUM-20260824" ], "checked_on": "2026-08-24", "note": "The odd support is empty, so the derangement condition is vacuous." }, "rational_fibre_reduction_verified": { "state": "verified", "evidence_ids": [ "D5-THEOREM-ADDENDUM-20260818T100903Z", "P5-THEOREM-ADDENDUM-20260824" ], "checked_on": "2026-08-24", "note": "Theorem 3.5 / Song--Tucker applies because g(C)=33>5^2+1." }, "field_density_theorem_applicable": { "state": "not_verified", "evidence_ids": [ "P5-THEOREM-ADDENDUM-20260824" ], "checked_on": "2026-08-24", "note": "The complete integral-spread and excluded-prime ledger is still missing." } }, "gates": { "G0": { "status": "green", "evidence_ids": [ "D5-SQL-20260818T100140Z", "D5-GROUP-VERIFY-20260818-CK1-O2-V1" ], "summary": "Stable V3 successor identity, exact raw level-40 generators, parent conjugator, and the fine/coarse materialization anomaly are recorded explicitly; V2 remains frozen as its predecessor." }, "G1": { "status": "green", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-SOURCE-AUDIT-20260818T102220Z" ], "summary": "The modular construction and two exact group/orbit implementations prove a smooth projective geometrically integral Q-source of genus 33." }, "G2": { "status": "green", "evidence_ids": [ "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "summary": "Producer, independent Sage, and adversarial review identify the actual pointed target over Q as LMFDB 800.d4 (Cremona 800a4), y^2=x^3+100x, with cusp 1/8 sent to the origin." }, "G3": { "status": "green", "evidence_ids": [ "D5-RANK-40-96-1-O-2-20260818-V2", "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "summary": "The point (5,25) lies on the actual target and is non-torsion; the independent database-disabled Sage computation certifies rank 1 and torsion order 2." }, "G4": { "status": "green", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-SOURCE-AUDIT-20260818T102220Z" ], "summary": "The exact subgroup inclusion and primary modular-function-field construction give the finite Q-map on smooth projective curves." }, "G5": { "status": "green", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-G7-FULL-CONJUNCTION-20260818-IV1" ], "summary": "Two exact group/coset implementations give relative GL2, SL2, projective, and coarse degree exactly 5, with all determinant, -I, and component conventions checked." }, "G6": { "status": "yellow", "evidence_ids": [ "D5-THEOREM-ADDENDUM-20260818T100903Z" ], "summary": "Ramification, empty odd support, and the no-pencil conclusion are exact, but an explicit integral spread and complete excluded-prime ledger remain missing for theorem application." }, "G7": { "status": "green", "evidence_ids": [ "D5-G7-FULL-CONJUNCTION-20260818-IV1", "P5-D5-POINTED-MODEL-IV-20260824" ], "summary": "Independent verifiers signed the original map/degree/genus/rank/no-pencil conjunction and separately checked the retained pointed-target formulas, degree-one function-field identities, target-coordinate point, and exact rank; the modular forms themselves were not regenerated in the Sage review." }, "G8": { "status": "green", "evidence_ids": [ "D5-REFEREE-20260818-CK1-O2-V2", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "summary": "Adversarial reviews passed the cover/rank/no-pencil conjunction and the successor pointed-model claims after resolving the Cremona/LMFDB spelling and coordinate-triple base-locus traps; G6 remains yellow." } }, "evidence_ids": [ "D5-SQL-20260818T100140Z", "D5-GRAPH-20260818T095432Z", "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-SOURCE-AUDIT-20260818T102220Z", "D5-TARGET-ID-40-96-1-O-2-20260818-IV1", "D5-LOCAL-RANK-20260818T095822Z", "D5-RANK-40-96-1-O-2-20260818-V2", "D5-THEOREM-ADDENDUM-20260818T100903Z", "D5-SOURCE-AUDIT-ADDENDUM-20260818T103533Z", "D5-G7-FULL-CONJUNCTION-20260818-IV1", "D5-REFEREE-20260818-CK1-O2-V2", "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824", "P5-THEOREM-ADDENDUM-20260824", "D5-POINTED-TARGET-MODEL-20260824-V1", "P5-D5-POINTED-MODEL-IV-20260824", "P5-D5-POINTED-MODEL-REFEREE-20260824" ], "stage5": { "branch_divisor": { "status": "verified", "notation": "Br(f)", "degree": 16, "galois_orbit_degrees": [ 1, 1, 2, 2, 2, 8 ], "ramified_fibres": 16, "fibre_partition": [ 5 ], "support_description": "the sixteen cusps" }, "ramification": { "status": "verified", "degree": 5, "ramified_fibres": 16, "fibre_partition": [ 5 ], "ramification_index_at_each_point": 5, "total_riemann_hurwitz_contribution": 64, "support_description": "All ramification lies above the sixteen target cusps.", "evidence_ids": [ "D5-GROUP-VERIFY-20260818-CK1-O2-V1", "D5-THEOREM-ADDENDUM-20260818T100903Z", "P5-THEOREM-ADDENDUM-20260824" ] }, "trace_discriminant": { "status": "verified", "formula": "D_f = 4 Br(f)", "degree": 64, "coefficient_at_each_branch_point": 4 }, "odd_support": { "status": "verified", "formula": "B_f = empty", "empty": true, "splitting_field": "Q", "omega": [], "galois_group": "trivial" }, "derangement": { "status": "verified", "vacuous": true, "witness": "The identity acts without fixed points on the empty Galois set." }, "integral_spread": { "status": "not_verified", "tracked_primes": [ 2, 5 ], "excluded_primes_complete": false }, "rational_fibre_reduction": { "status": "verified", "method": "Theorem 3.5 / Song--Tucker genus inequality", "inequality": "g(C)=33 > 5^2+1=26", "irreducible_fibre_claim": false }, "field_density": { "status": "not_verified", "note": "The complete integral-spread ledger and theorem-applicability certificate are not yet available." } }, "caveats": [ "This proof record consumes the immutable version-3 successor candidate; earlier versions remain historical provenance and are not silently mutated.", "The pointed-model formulas have finite base loci as coordinate triples; the certified claim is the function-field birational isomorphism and its unique extension on smooth projective curves.", "The integral-spread and excluded-prime ledger is incomplete, so this record is not theorem-ready." ], "provenance": { "candidate_path": "data/candidates/d5-l40-ck1-o2-v3.json", "candidate_sha256": "14c75a7edc9827f578d1d779b804f2e2254a629ca9d63aa5ef9c19de2406e913", "evidence_kind": "certified_arithmetic_computation_and_exact_symbolic_computation", "pointed_model_review_path": "evidence/part5-lmfdb-companion/P5-D5-POINTED-MODEL-IV-20260824/review.json", "pointed_model_review_sha256": "b78bfd7b3005d1e8e138615e20a1fb8e5721a41ea1d0273ceef1223ea4973a76", "pointed_model_referee_path": "evidence/part5-lmfdb-companion/P5-D5-POINTED-MODEL-REFEREE-20260824/review.json", "pointed_model_referee_sha256": "a71ec20b9800f72af84c09250c44a8a0e18ea34554def70d0f434267d7a73328", "successor_independent_binding_path": "evidence/part5-lmfdb-companion/P5-D5-POINTED-MODEL-IV-20260824/successor-binding-v3-final.json", "successor_independent_binding_sha256": "d98e02d18690e010ee22b2d4a23e72b50e563dbc379096c449641d3482441cf1", "successor_referee_binding_path": "evidence/part5-lmfdb-companion/P5-D5-POINTED-MODEL-REFEREE-20260824/successor-binding-addendum-final2.md", "successor_referee_binding_sha256": "0eaac286ebbd198de6dd9aaa197e94076ceab2fd11fed9e510f5790f39cce5f8" } } ] }