{ "claims": [ { "id": "harmonic-obstruction", "branch": "cubature", "title": "Equal-weight harmonic obstruction", "statement_latex": "Every complete radial Class-I grid is a spherical five-design. The limiting equal-weight sampling measure differs from uniform spherical area first at harmonic degree six. Paper IV also gives finite-frequency computations and an asymptotic expansion for the degree-six moment.", "scope": { "frequency": "Paper IV: every complete radial Class-I grid (five-design); limiting equal-weight measure mu_rad (degree-six obstruction)", "node_set": "radial Class-I grids and their weak limit mu_rad", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "theorem claim", "evidence": [ "paper proof", "exact invariant evaluation" ], "machine_checkable": "where supplied", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "not a cross-language conformance claim" }, "source_ids": [ "paper-iv" ], "artifact_ids": [], "audit_href": "/cubature#harmonic-obstruction", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f1-f9-exactness", "branch": "cubature", "title": "Positive cubature exactness for 1 ≤ f ≤ 9", "statement_latex": "For each frequency 1 \\le f \\le 9 the frozen Class-I node set X_f admits a strictly positive H_3-orbit-constant cubature rule of degree t_f.", "scope": { "frequency": "1 to 9", "node_set": "fixed radial Class-I grids", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "theorem claim", "evidence": [ "paper proof", "exact moment system", "exact weights" ], "machine_checkable": "yes where artifact supplied", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "replay of frozen orbit_masses_f1_f9.csv inside G6_RELEASE/1.1", "implementation": "G6_RELEASE/1.1 Rust, Julia, and pinned TypeScript cubature replay", "release_id": "G6_RELEASE/1.1", "expected_output_ref": "/g6/data/orbit_masses_f1_f9.csv" }, "source_ids": [ "paper-vi" ], "artifact_ids": [ "orbit-csv", "f9-csv", "class-i-cubature-1-0" ], "audit_href": "/cubature#f1-f9", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f1-f9-positivity", "branch": "cubature", "title": "Positivity of orbit masses for 1 ≤ f ≤ 9", "statement_latex": "The exact orbit masses established for 1 \\le f \\le 9 are strictly positive.", "scope": { "frequency": "1 to 9", "node_set": "fixed radial Class-I grids", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "theorem claim", "evidence": [ "exact inequalities", "exact weights" ], "machine_checkable": "yes", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "sign checks on frozen orbit masses in G6_RELEASE/1.1", "implementation": "G6_RELEASE/1.1 Rust, Julia, and pinned TypeScript cubature replay", "release_id": "G6_RELEASE/1.1", "expected_output_ref": "/g6/data/orbit_masses_f1_f9.csv" }, "source_ids": [ "paper-vi" ], "artifact_ids": [ "orbit-csv" ], "audit_href": "/cubature#f1-f9", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f10-f21", "branch": "cubature", "title": "Target-degree exactness for 10 ≤ f ≤ 21", "statement_latex": "For each frequency 10 \\le f \\le 21 a strictly positive rule reaches the target degree t_f. Maximality is not claimed.", "scope": { "frequency": "10 to 21", "node_set": "fixed radial Class-I grids", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "hosted classI_exact_certificates_f2_f21.json (1,815,267 bytes, SHA-256 7b3945ad…)", "historical verifier verify_classI_f2_f21.py and recovery revision verify_classI_f2_f21.r2.py", "per-frequency derived records /cubature/frequencies/f10.json through f21.json", "this-archive Python 3.11.2 r2 ALL PASS: 383 exact identities, 404 strictly positive masses" ], "machine_checkable": "yes", "publication": { "kind": "Zenodo preprint-associated artifact", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "exact finite moment identities on recovered classI_exact_certificates_f2_f21.json; homogeneous harmonic generators; not Paper VI and not independent third-party reproduction", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py (historical verify_classI_f2_f21.py also PASS)", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier", "cubature-f2-f21-verifier-r2", "cubature-f2-f21-dossier-pdf", "cubature-this-archive-f2-f21-stdout", "cubature-this-archive-f2-f21-ledger", "cubature-f10-record", "cubature-f11-record", "cubature-f12-record", "cubature-f13-record", "cubature-f14-record", "cubature-f15-record", "cubature-f16-record", "cubature-f17-record", "cubature-f18-record", "cubature-f19-record", "cubature-f20-record", "cubature-f21-record" ], "audit_href": "/cubature#f10-f21", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f10", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-10 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{10} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 33. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 10, "node_set": "fixed radial Class-I grid X_10", "arithmetic": "Q(sqrt(5))", "target_degree": 33, "nodes": 1002, "orbits": 14, "moment_conditions": 14, "exact_minimum_orbit": [ 10, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.10", "derived export /cubature/frequencies/f10.json", "this-archive r2 ledger frequencies.10 (14 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 10 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f10-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f10", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f11", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-11 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{11} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 35. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 11, "node_set": "fixed radial Class-I grid X_11", "arithmetic": "Q(sqrt(5))", "target_degree": 35, "nodes": 1212, "orbits": 16, "moment_conditions": 15, "exact_minimum_orbit": [ 11, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.11", "derived export /cubature/frequencies/f11.json", "this-archive r2 ledger frequencies.11 (15 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 11 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f11-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f11", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f12", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-12 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{12} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 39. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 12, "node_set": "fixed radial Class-I grid X_12", "arithmetic": "Q(sqrt(5))", "target_degree": 39, "nodes": 1442, "orbits": 19, "moment_conditions": 18, "exact_minimum_orbit": [ 12, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.12", "derived export /cubature/frequencies/f12.json", "this-archive r2 ledger frequencies.12 (18 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 12 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f12-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f12", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f13", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-13 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{13} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 41. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 13, "node_set": "fixed radial Class-I grid X_13", "arithmetic": "Q(sqrt(5))", "target_degree": 41, "nodes": 1692, "orbits": 21, "moment_conditions": 20, "exact_minimum_orbit": [ 13, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.13", "derived export /cubature/frequencies/f13.json", "this-archive r2 ledger frequencies.13 (20 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 13 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f13-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f13", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f14", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-14 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{14} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 45. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 14, "node_set": "fixed radial Class-I grid X_14", "arithmetic": "Q(sqrt(5))", "target_degree": 45, "nodes": 1962, "orbits": 24, "moment_conditions": 23, "exact_minimum_orbit": [ 14, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.14", "derived export /cubature/frequencies/f14.json", "this-archive r2 ledger frequencies.14 (23 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 14 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f14-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f14", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f15", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-15 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{15} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 47. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 15, "node_set": "fixed radial Class-I grid X_15", "arithmetic": "Q(sqrt(5))", "target_degree": 47, "nodes": 2252, "orbits": 27, "moment_conditions": 25, "exact_minimum_orbit": [ 15, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.15", "derived export /cubature/frequencies/f15.json", "this-archive r2 ledger frequencies.15 (25 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 15 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f15-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f15", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f16", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-16 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{16} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 51. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 16, "node_set": "fixed radial Class-I grid X_16", "arithmetic": "Q(sqrt(5))", "target_degree": 51, "nodes": 2562, "orbits": 30, "moment_conditions": 29, "exact_minimum_orbit": [ 15, 1, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.16", "derived export /cubature/frequencies/f16.json", "this-archive r2 ledger frequencies.16 (29 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 16 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f16-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f16", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f17", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-17 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{17} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 53. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 17, "node_set": "fixed radial Class-I grid X_17", "arithmetic": "Q(sqrt(5))", "target_degree": 53, "nodes": 2892, "orbits": 33, "moment_conditions": 31, "exact_minimum_orbit": [ 17, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.17", "derived export /cubature/frequencies/f17.json", "this-archive r2 ledger frequencies.17 (31 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 17 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f17-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f17", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f18", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-18 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{18} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 57. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 18, "node_set": "fixed radial Class-I grid X_18", "arithmetic": "Q(sqrt(5))", "target_degree": 57, "nodes": 3242, "orbits": 37, "moment_conditions": 35, "exact_minimum_orbit": [ 18, 0, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.18", "derived export /cubature/frequencies/f18.json", "this-archive r2 ledger frequencies.18 (35 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 18 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f18-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f18", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f19", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-19 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{19} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 59. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 19, "node_set": "fixed radial Class-I grid X_19", "arithmetic": "Q(sqrt(5))", "target_degree": 59, "nodes": 3612, "orbits": 40, "moment_conditions": 37, "exact_minimum_orbit": [ 11, 8, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.19", "derived export /cubature/frequencies/f19.json", "this-archive r2 ledger frequencies.19 (37 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 19 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f19-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f19", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f20", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-20 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{20} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 63. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 20, "node_set": "fixed radial Class-I grid X_20", "arithmetic": "Q(sqrt(5))", "target_degree": 63, "nodes": 4002, "orbits": 44, "moment_conditions": 42, "exact_minimum_orbit": [ 17, 3, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.20", "derived export /cubature/frequencies/f20.json", "this-archive r2 ledger frequencies.20 (42 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 20 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f20-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f20", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f21", "branch": "cubature", "title": "Target-degree exactness on the fixed frequency-21 Class-I grid", "statement_latex": "The fixed radial Class-I grid X_{21} admits a strictly positive H_3-orbit-constant cubature rule exact through degree 65. Maximality is not claimed. This does not enlarge Paper VI.", "scope": { "frequency": 21, "node_set": "fixed radial Class-I grid X_21", "arithmetic": "Q(sqrt(5))", "target_degree": 65, "nodes": 4412, "orbits": 48, "moment_conditions": 44, "exact_minimum_orbit": [ 20, 1, 0 ] }, "mathematical_status": "certificate-backed post-publication computation; not part of Paper VI v1.3", "evidence": [ "shared combined certificate frequencies.21", "derived export /cubature/frequencies/f21.json", "this-archive r2 ledger frequencies.21 (44 rows)" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted recovered certificate", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "frequency 21 identities inside the combined f=2–21 certificate", "implementation": "Python 3.11.2 verify_classI_f2_f21.r2.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f2_f21.stdout.txt" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [ "cubature-f2-f21-exact-certificate", "cubature-f2-f21-verifier-r2", "cubature-f21-record", "cubature-this-archive-f2-f21-ledger" ], "audit_href": "/cubature#f21", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f22-degree69", "branch": "cubature", "title": "Positive degree-69 rule on the fixed frequency-22 grid", "statement_latex": "X_{22} supports a strictly positive H_3-orbit-constant cubature rule through algebraic degree 69.", "scope": { "frequency": 22, "node_set": "fixed radial Class-I grid", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "certificate-backed theorem claim", "evidence": [ "exact moment system", "rank certificate", "exact positive weights" ], "machine_checkable": "yes", "publication": { "kind": "Zenodo preprint-associated artifact", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "named frozen Python 3.12.13 and Julia 1.11.6 verifier outputs only", "implementation": "Omega_Naught_ClassI_f22_exact_verifier.py and OmegaNaughtClassIF22ExactVerifier.jl", "release_id": "CLASS_I_CUBATURE/1.1 corrected release candidate", "expected_output_ref": "/cubature/Omega_Naught_ClassI_f22_verification_output.txt" }, "source_ids": [ "paper-vi", "certificate-f22-69" ], "artifact_ids": [ "f22-python-verifier", "f22-python-verifier-r2", "f22-julia-verifier", "f22-degree69-canonical", "f22-degree69-csv", "f22-moment-spec", "f22-release-json", "f22-addendum-pdf" ], "audit_href": "/cubature/f22#degree-69", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f22-obstruction", "branch": "cubature", "title": "Degree-70/71 obstruction on the fixed frequency-22 grid", "statement_latex": "The unique invariant solution at degrees 70/71 has a negative mass on orbit (21,1,0). By symmetrization, no nonnegative weights on the same fixed nodes reach degree 70.", "scope": { "frequency": 22, "node_set": "fixed radial Class-I grid", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "obstruction", "evidence": [ "forced negative weight", "exact dual certificate" ], "machine_checkable": "yes", "publication": { "kind": "Zenodo preprint-associated artifact", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "named frozen Python 3.12.13 and Julia 1.11.6 verifier outputs only", "implementation": "Omega_Naught_ClassI_f22_exact_verifier.py and OmegaNaughtClassIF22ExactVerifier.jl", "release_id": "CLASS_I_CUBATURE/1.1 corrected release candidate", "expected_output_ref": "/cubature/Omega_Naught_ClassI_f22_verification_output.txt" }, "source_ids": [ "certificate-f22-70" ], "artifact_ids": [ "f22-python-verifier", "f22-python-verifier-r2", "f22-julia-verifier", "f22-degree71-canonical", "f22-degree71-csv", "f22-dual-canonical", "f22-negative-orbit", "f22-moment-spec", "f22-release-json", "f22-addendum-pdf" ], "audit_href": "/cubature/f22#obstruction", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f23-f24-existence", "branch": "cubature", "title": "Strictly positive H₃-invariant rules on X₂₃ and X₂₄", "statement_latex": "X_{23} admits strictly positive H_3-invariant cubature weights exact through degree 71. X_{24} admits strictly positive H_3-invariant cubature weights exact through degree 75. No maximality assertion is included.", "scope": { "frequency": "23 and 24", "node_set": "fixed radial Class-I grids X_23 and X_24", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "theorem claim", "evidence": [ "locally hosted existence proof", "hosted original certificate ClassI_Cubature_f23_f24_exact_certificate_v1.1.json (1,407,835 bytes, SHA-256 e886510e…)", "hosted original verifier verify_ClassI_f23_f24_exact_v1.1.py (6,818 bytes, SHA-256 41d83727…)", "hosted original 109-row ledger ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt (10,905 bytes, SHA-256 4d56dc2c…)", "guarded launcher replay_recovered.py (1,621 bytes, SHA-256 a3bf98e7…)", "this-archive Python 3.11.2 replay_recovered.py ALL PASS: 109 identities; python3 -O refused", "printed finite-audit extract remains a distinct artifact" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted theorem statement", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "exact finite identities, positivity, and modular rank witnesses on the recovered CLASS_I_CUBATURE_F23_F24_EXACT/1.1 certificate; not Paper VI and not independent third-party reproduction", "implementation": "Python 3.11.2 replay_recovered.py -> verify_ClassI_f23_f24_exact_v1.1.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f23_f24.stdout.txt" }, "source_ids": [], "artifact_ids": [ "cubature-f23-f24-theorem", "cubature-f23-f24-exact-certificate", "cubature-f23-f24-verifier", "cubature-f23-f24-ledger", "cubature-f23-f24-replay-launcher", "cubature-f23-f24-audit-extract", "typed-network-dossier-pdf", "cubature-this-archive-f23-f24-stdout", "cubature-f23-record", "cubature-f24-record" ], "audit_href": "/cubature/f23-f24#theorem", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f23-f24-rank", "branch": "cubature", "title": "Rank and affine dimension four on the f=23 and f=24 moment systems", "statement_latex": "\\operatorname{rank} A_{23}=52 with affine dimension 4; \\operatorname{rank} A_{24}=57 with affine dimension 4. Neither argument proves failure or impossibility at any higher degree.", "scope": { "frequency": "23 and 24", "node_set": "H_3-orbit moment systems on X_23 and X_24", "arithmetic": "Q(sqrt(5)); modular images in F_p, p=1000000000000000031" }, "mathematical_status": "corollary", "evidence": [ "two-sided modular determinants of chosen minors, as stated in the hosted proof and reprinted in the 5 September 2026 dossier" ], "machine_checkable": "yes; modular determinants re-executed by the hosted verifier on this archive", "publication": { "kind": "locally hosted theorem statement", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "two-sided modular determinants of chosen minors inside verify_ClassI_f23_f24_exact_v1.1.py on this archive", "implementation": "Python 3.11.2 replay_recovered.py", "release_id": "OMEGA_NAUGHT_RECOVERED_CUBATURE_EVIDENCE/2026-09-13", "expected_output_ref": "/cubature/evidence/2026-09-13-this-archive/f23_f24.stdout.txt" }, "source_ids": [], "artifact_ids": [ "cubature-f23-f24-theorem", "cubature-f23-f24-exact-certificate", "cubature-f23-f24-verifier", "cubature-f23-f24-audit-extract", "typed-network-dossier-pdf", "cubature-this-archive-f23-f24-stdout" ], "audit_href": "/cubature/f23-f24#rank", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "cubature-f23-conjecture", "branch": "cubature", "title": "Existence of strictly positive H_3-orbit-constant rules of target degree t_f for every f ≥ 25", "statement_latex": "Existence of strictly positive H_3-orbit-constant rules of target degree t_f for every f \\ge 25 is an open conjecture. No theorem in this archive covers those frequencies. Higher degrees at f=23 and f=24 are not shown impossible.", "scope": { "frequency": "25 and above", "node_set": "fixed radial Class-I grids", "arithmetic": "Q(sqrt(5))" }, "mathematical_status": "conjecture", "evidence": [ "none supplied" ], "machine_checkable": "no", "publication": { "kind": "Zenodo preprint-associated artifact", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "open conjecture" }, "source_ids": [ "class-i-cubature-1-1" ], "artifact_ids": [], "audit_href": "/cubature#f25", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "typed-network-interfaces", "branch": "network", "title": "Typed exact interfaces among Papers VII, V, I, II and frozen SRSD", "statement_latex": "K_{\\mathrm{coord}}\\cap K_\\Omega=\\mathbb{Q}(\\sqrt5). Distinguished scalar packages (\\varphi,\\varepsilon) and (\\tau_\\Omega,\\lambda) reconstruct one another. Restricted conjugacy T_h\\circ J_h=J_h\\circ F_q holds on D_{15}. Paper-II quotient and remainder equal the frozen SRSD encoding of Qd. Cubature theorems on X_{23} and X_{24} remain logically independent of \\Omega_0, the floor map, and SRSD.", "scope": { "frequency": "not a cubature theorem; D_15 for the I–II clause", "node_set": "typed interfaces between distinct theorem packages", "arithmetic": "exact real/quadratic identities; finite integer arithmetic on D_15" }, "mathematical_status": "theorem claim", "evidence": [ "locally hosted 5 September 2026 proof dossier", "cyclotomic algebraicity of sin(delta), cos(delta) and Lindemann for the field intersection", "exact enclosure of lambda and conjugacy on all 32768 states in D_15", "external specialization of the frozen SRSD encoder" ], "machine_checkable": "partial: D_15 conjugacy is a finite identity already audited separately on this site; cubature coefficient payload cited as LaTeX filecontents is not extractable from the hosted PDF; the original certificate, verifier, and ledger are hosted separately under /cubature/", "publication": { "kind": "locally hosted proof dossier", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not run", "scope": "the dossier's embedded network verifier is not hosted as a separate artifact; this is not a joining theorem and not a G6 replay" }, "source_ids": [ "paper-i", "paper-ii", "paper-iii", "paper-v", "paper-vii" ], "artifact_ids": [ "typed-network-dossier-pdf", "cubature-f23-f24-audit-extract" ], "audit_href": "/network#theorem", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_RUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is recorded for this claim. NOT_RUN here does not describe a hidden pass.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega0-definitions", "branch": "omega0", "title": "Ω₀ reference identities", "statement_latex": "\\Omega_0=\\phi+\\pi,\\ \\tau=\\Omega_0^{-1},\\ \\lambda=\\phi/\\Omega_0,\\ \\mu=1-\\lambda. Identities such as \\lambda+\\mu=1 follow from the definitions.", "scope": { "frequency": null, "node_set": "not applicable", "arithmetic": "exact real/quadratic identities" }, "mathematical_status": "definition", "evidence": [ "paper proof" ], "machine_checkable": "yes as definitional identities", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "OMEGA_NAUGHT/1.0 named manifest outputs", "implementation": "Python, Julia, Java, and Rust", "release_id": "OMEGA_NAUGHT/1.0", "expected_output_ref": "/g6/OMEGA_NAUGHT_1_0.txt" }, "source_ids": [ "paper-v" ], "artifact_ids": [ "omega-naught-1-0" ], "audit_href": "/omega0", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "g6-cross-language-replay", "branch": "integration", "title": "Cross-language replay of the frozen G₆ canonical record", "statement_latex": "Named Rust, Julia, and pinned TypeScript implementations reproduce the specified frozen G6_RELEASE/1.1 canonical record.", "scope": { "frequency": "cubature_frequency_max=9 inside the canonical record", "node_set": "not a cubature theorem", "arithmetic": "implementation property" }, "mathematical_status": "standard consequence", "evidence": [ "verifier supplied" ], "machine_checkable": "yes", "publication": { "kind": "other", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "G6_RELEASE/1.1 canonical record only; cubature_frequency_max=9", "implementation": "Rust, Julia, and the pinned TypeScript replay", "release_id": "G6_RELEASE/1.1", "expected_output_ref": "/releases/G6_RELEASE_1.1/RELEASE.json" }, "source_ids": [ "g6-release-1-1" ], "artifact_ids": [ "g6-release-json" ], "audit_href": "/integration", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-naught-1-0-jprv", "branch": "integration", "title": "OMEGA_NAUGHT/1.0 Python, Julia, Java, and Rust conformance", "statement_latex": "Named Python, Julia, Java, and Rust implementations reproduce the frozen OMEGA_NAUGHT/1.0 manifest outputs.", "scope": { "frequency": "not a cubature theorem", "node_set": "not a cubature theorem", "arithmetic": "implementation property" }, "mathematical_status": "standard consequence", "evidence": [ "verifier supplied" ], "machine_checkable": "yes", "publication": { "kind": "other", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "OMEGA_NAUGHT/1.0 named manifest outputs", "implementation": "Python, Julia, Java, and Rust", "release_id": "OMEGA_NAUGHT/1.0", "expected_output_ref": "/g6/OMEGA_NAUGHT_1_0.txt" }, "source_ids": [ "paper-v" ], "artifact_ids": [ "conformance-python", "conformance-julia", "conformance-rust", "conformance-java", "omega-naught-1-0" ], "audit_href": "/reproduce", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "floor-first-disagreement", "branch": "dynamics", "title": "First floor-map disagreement at 42189065", "statement_latex": "The rational floor map with q=339949771344778/10^{15} and the real floor map with the enclosed lambda first disagree at d^*=42189065. An independent unimodular Farey-neighbor pair 846547/2490212 and 13495616/39698853 proves no smaller positive denominator can sit strictly between q and lambda.", "scope": { "frequency": "positive integers; witness at 42189065", "node_set": "floor maps on nonnegative integers", "arithmetic": "exact rationals + frozen Machin/isqrt enclosure of lambda" }, "mathematical_status": "certificate-backed finite exact fact", "evidence": [ "unimodular Farey neighbors", "mediant witness 14342163/42189065", "this-archive Python 3.11.2 companion checker" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted review package", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "focused companion checker only; not the original synthesis suite; PDF not supplied", "implementation": "omega_synthesis_claim_check.py", "release_id": "review package, not a source paper", "expected_output_ref": "/synthesis/omega_synthesis_claim_check.stdout.txt" }, "source_ids": [ "paper-i", "paper-ii" ], "artifact_ids": [ "synthesis-claim-check-py", "synthesis-claim-check-stdout", "synthesis-claim-check-env", "synthesis-review-md" ], "audit_href": "/synthesis", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "successor-I15", "branch": "dynamics", "title": "Exact 15-bit admissible interval and constrained optima", "statement_latex": "I_{15}=[4196/12343, 7039/20706). The unique least-denominator member is 4196/12343 with A=45392. The signed-32 horizon maximizer under Q\\le 65538 is 49136/144539 with A=2490212. The least-denominator rational preserving A_0=42189065 is 846547/2490212.", "scope": { "frequency": "15-bit domain D_15", "node_set": "nonnegative integers n < 32768 for I_15; A(q) as stated", "arithmetic": "integers + frozen 60-decimal open enclosure of lambda" }, "mathematical_status": "certificate-backed finite exact facts", "evidence": [ "Farey-neighbor identities", "certified floor values from the frozen lambda enclosure", "exhaustive D_15 fibers" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted research addendum", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "named Python verifier outputs only; not part of G6_RELEASE/1.1", "implementation": "Omega_Naught_Successor_Search_Verifier.py", "release_id": "research addendum, not a successor contract", "expected_output_ref": "/dynamics/successor_search_certificate.json" }, "source_ids": [ "paper-i", "paper-ii" ], "artifact_ids": [ "successor-python-verifier", "successor-certificate-json", "successor-transcript" ], "audit_href": "/dynamics/successors", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "successor-not-accepted", "branch": "dynamics", "title": "No accepted successor contract", "statement_latex": "The listed rationals are constrained optima on distinct guarantee sets. None is an accepted replacement of OMEGA_NAUGHT/1.0.", "scope": { "frequency": "not applicable", "node_set": "contract/version boundary", "arithmetic": "not applicable" }, "mathematical_status": "status statement", "evidence": [ "acceptance rule in the source audit" ], "machine_checkable": "not applicable", "publication": { "kind": "locally hosted research addendum", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "no successor serialization or multi-language capsule replay is claimed" }, "source_ids": [ "paper-i", "paper-ii" ], "artifact_ids": [ "successor-python-verifier" ], "audit_href": "/dynamics/successors", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-alpha-15bit-kernel", "branch": "dynamics", "title": "Omega-Alpha finite integer kernel and codebooks", "statement_latex": "For the protocol (S,Q,N)=(10^{15},339949771344778,23), every d in {0,...,32767} reconstructs exactly from the ordered remainder ledger; the maximum steps to zero is 10; the maps T(b)=F^{23}((b+2)S) on {0,...,255} and {0,...,65535} are strictly increasing with the published hashes.", "scope": { "frequency": "15-bit domain D_15 and finite codebooks", "node_set": "nonnegative integers in the declared domains", "arithmetic": "integers after S, Q, N are fixed" }, "mathematical_status": "exhaustively verified finite facts in the locally hosted synthesis", "evidence": [ "exhaustive 15-bit enumeration", "byte and 16-bit codebook enumeration", "frozen JSON test report" ], "machine_checkable": "yes", "publication": { "kind": "locally hosted July 2026 synthesis", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "named JSON keys in the frozen Omega-Alpha test report; not part of G6_RELEASE/1.1", "implementation": "omega_alpha_reference.py", "release_id": "Omega-Alpha synthesis 1.0, locally hosted, not a source paper", "expected_output_ref": "/omega-alpha/reference_test_report.json" }, "source_ids": [ "paper-i", "paper-ii" ], "artifact_ids": [ "omega-alpha-python", "omega-alpha-report-json", "omega-alpha-pdf" ], "audit_href": "/dynamics/omega-alpha#kernel", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-alpha-not-source", "branch": "dynamics", "title": "Omega-Alpha is not a source paper", "statement_latex": "The July 2026 Omega-Alpha synthesis is a scoped historical consolidation. It does not replace Papers I–VII, has no DOI on this site, and does not establish physical, cryptographic, or generic-compression theorems.", "scope": { "frequency": "not applicable", "node_set": "publication and scope boundary", "arithmetic": "not applicable" }, "mathematical_status": "status statement", "evidence": [ "synthesis abstract and claim-verification matrix" ], "machine_checkable": "not applicable", "publication": { "kind": "locally hosted July 2026 synthesis", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "scope and publication status, not a verifier outcome" }, "source_ids": [ "paper-i", "paper-ii" ], "artifact_ids": [ "omega-alpha-pdf", "omega-alpha-claim-matrix", "omega-alpha-history" ], "audit_href": "/dynamics/omega-alpha#omega-alpha-status", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-local-closure", "branch": "geometry", "title": "Construction-to-OMEGA uniqueness of the marked kernel", "statement_latex": "Fix a>0, \\varepsilon=1'', the ordered seed (\\alpha,d,b), and the bilateral extension a then a\\varphi. The six marks, all pairwise distances and fifteen center angles, the normalized Gram matrix, and an oriented frame (u,v,n) in \\mathbb{R}^3 are uniquely determined up to orientation-preserving Euclidean isometry. Intrinsic rank is two.", "scope": { "frequency": "not applicable", "node_set": "local OMEGA marked kernel", "arithmetic": "oriented Euclidean three-space" }, "mathematical_status": "theorem claim", "evidence": [ "paper proof, Theorem 3.4 (Construction-to-OMEGA)" ], "machine_checkable": "closed formulas evaluated by the companion generator and a browser reconstruction check; not a theorem checker", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "reconstruction generator, not a certificate package" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "paper-vii-pdf-v16", "paper-vii-kernel-json", "paper-vii-kernel-py" ], "audit_href": "/geometry#closure", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-angular-fan", "branch": "geometry", "title": "Three fundamental bearings and the complete center-angular law", "statement_latex": "\\theta_0=\\varepsilon/2,\\ \\theta_1=\\pi/4+\\varepsilon/4,\\ \\theta_2=\\arctan((\\varphi^2+\\sin(\\varepsilon/2))/\\cos(\\varepsilon/2)). Every center angle is |\\beta_i-\\beta_j|, equivalently \\cos\\angle p_{j,\\sigma}\\alpha p_{k,\\tau}=(c^2+\\sigma\\tau x_j x_k)/(\\sqrt{c^2+x_j^2}\\sqrt{c^2+x_k^2}).", "scope": { "frequency": "not applicable", "node_set": "six center rays of the local kernel", "arithmetic": "oriented Euclidean plane" }, "mathematical_status": "theorem claim", "evidence": [ "paper proofs, Theorem 6.1" ], "machine_checkable": "float64 evaluation of the closed formulas; not a theorem checker", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "reconstruction generator, not a certificate package" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "paper-vii-pdf-v16", "paper-vii-kernel-json", "paper-vii-kernel-jl" ], "audit_href": "/geometry#fan", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-center-relative", "branch": "geometry", "title": "Exact 3D reconstruction and positive-homothety family", "statement_latex": "p_{j,\\sigma}=\\alpha+a(c u+\\sigma x_j v) with (u,v,n) from the ordered seed. The family \\{K_\\Omega(a):a>0\\} is a torsor for (\\mathbb{R}_{>0},\\times). SE(3) placement preserves distances, center angles, incidence, and handedness.", "scope": { "frequency": "not applicable", "node_set": "oriented Euclidean three-space", "arithmetic": "Euclidean rigid motions and positive homothety" }, "mathematical_status": "theorem claim", "evidence": [ "paper proofs, Proposition 9.1 and Section 10" ], "machine_checkable": "companion generator exports landmarks, JSON, and sampled OBJ; not a theorem checker", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "reconstruction generator, not a certificate package" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "paper-vii-pdf-v16", "paper-vii-kernel-py", "paper-vii-kernel-obj", "paper-vii-kernel-jl", "geometry-local-object-jl", "geometry-local-object-stdout", "geometry-local-object-diagram", "geometry-local-object-frame" ], "audit_href": "/geometry#r3", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-outer-scaffold", "branch": "geometry", "title": "Notebook outer scaffold is an adopted transcription; remaining completion is not claimed", "statement_latex": "Adopting the repeated a,a\\varphi annotations defines S,E,L with W-H=L_{chord} and \\|S-\\alpha\\|=a\\varphi+h_{sag}. The larger circle, additional curved links, and signed-depth completion are not required for Theorem 3.4 and Proposition 9.1 and are not claimed as unique.", "scope": { "frequency": "not applicable", "node_set": "notebook outer architecture", "arithmetic": "oriented Euclidean plane, optional signed depth" }, "mathematical_status": "adopted definition plus derived identities in the Zenodo preprint; remaining outer completion is a status statement", "evidence": [ "Paper VII v1.6 Definition 11.1, Proposition 11.3" ], "machine_checkable": "identities W-H=L_chord and \\|S-\\alpha\\| evaluated in the browser check; not a uniqueness certificate", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "scope and publication status, not a verifier outcome" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "paper-vii-pdf-v16", "paper-vii-kernel-jl" ], "audit_href": "/geometry#scaffold", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-drift-control", "branch": "geometry", "title": "Experimental 4D OMEGA rotation is not a privileged memory anchor", "statement_latex": "The experimental operator R(2\\pi\\lambda)\\oplus R(2\\pi\\mu) is a hypothesis layer, not a theorem of Paper VII. Across uniformly random 4D dynamics it performs like an ordinary fixed anchor; a matched-case reconstruction advantage is the generic advantage of a correct model.", "scope": { "frequency": "not applicable", "node_set": "synthetic 4D rotations and a 64D embedding proxy", "arithmetic": "IEEE binary32/binary16 simulation; not exact" }, "mathematical_status": "numerical experimental control; not a theorem", "evidence": [ "published results.json of OMEGA_DRIFT_EXPERIMENT_v0_1, seed 20260831" ], "machine_checkable": "Python/NumPy script; numerical reproducibility of the named run, not a theorem checker", "publication": { "kind": "locally hosted experimental control", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "experimental script, not a frozen certificate package" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "omega-drift-py", "omega-drift-results" ], "audit_href": "/geometry/drift#e4", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "The implementation field is not applicable. That is not a failed run.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "geometry-construct-interface", "branch": "geometry", "title": "Kernel construct node log and presentation machine", "statement_latex": "OMEGA_KERNEL_NODES/1.0 exports the unit kernel at \\varepsilon=\\pi/648000 with exact-expression strings and binary64 coordinates, claim tags kernel|optional|not_implied, and a closed exclusion ledger. Display substitution \\varepsilon\\mapsto 12^\\circ is non-normative. Replay stops at stage 15 and does not enter the adopted scaffold.", "scope": { "frequency": "not applicable", "node_set": "canonical unit planar kernel plus optional S,E,L", "arithmetic": "Paper VII closed formulas; IEEE binary64 realization" }, "mathematical_status": "implementation interface around Paper VII; not a theorem and not a second evaluator", "evidence": [ "live paper-vii-kernel.ts landmarks", "canonical unit JSON", "browser construct machine", "exact scalar-interface Python certificate (scalar layer only)" ], "machine_checkable": "node-log validator against the live evaluator; overlay catalog closed; replay boundary; Python exact scalar certificate does not prove Euclidean claims", "publication": { "kind": "locally hosted technical release paper", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "canonical unit node log and replay boundary against paper-vii-kernel.ts", "implementation": "omega-naught Construct mode adapter v1.1.0", "release_id": "omega-kernel-construct/1.1.0", "expected_output_ref": "/geometry/construct/omega-kernel-nodes.unit.true.json" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "construct-paper-pdf", "construct-node-log", "construct-scalar-py" ], "audit_href": "/geometry#construct", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "typed-connection-audit", "branch": "network", "title": "De-branded executable audit of typed maps", "statement_latex": "v_L\\circ H_t=T_t\\circ v_L. On D=\\{0,\\ldots,32767\\}, \\lfloor\\lambda d\\rfloor=\\lfloor Qd/S\\rfloor. Finite F_q-orbits reconstruct from the terminal state and the ordered remainder ledger. Homothety of the six kernel marks scales squared Euclidean distances by t^{2}. Radial template identities are algebraic and carry no physical claim. Name identification of maps is not a joining theorem.", "scope": { "frequency": "not a cubature theorem; D={0,...,32767} for the floor clause", "node_set": "de-branded maps among affine contraction, rational floor, remainder ledger, external encoder, and Euclidean homothety of six kernel marks", "arithmetic": "SymPy symbolic identities; Python integers and fractions.Fraction; rational enclosures of sqrt(5) and pi; no binary64 in the D_15 certificate" }, "mathematical_status": "executable mathematical audit of de-branded maps; not a joining theorem and not a formal proof-assistant kernel", "evidence": [ "hosted typed_connection_proof.py, SHA-256 a0c32a3c37fbf217d8111aaa8dd5cc0050d0e302df3ae339b05d99556bb32281", "named Python 3.10.21 / SymPy 1.14.0 runtime", "exact rational enclosure of lambda via integer square-root bounds on sqrt(5) and Machin bounds on pi", "exhaustive floor equality on all 32768 states in D, with q0; one_step_inverse when A≠0. No hypothesis Q=3. Composition requires local correctness hypotheses.", "scope": { "frequency": null, "node_set": "marked scale family; rational floor map on nonnegative integers; twelve normalized icosahedral vertices; stated deformation family", "arithmetic": "exact integer/rational arithmetic; twelve-node moments in Q(sqrt(5)); analytic localization independent of the finite certificate" }, "mathematical_status": "written theorems in the 19 September 2026 Zenodo preprint 10.5281/zenodo.22848213; this-archive Python replay of the finite certificate is not a proof-assistant check of the analytic theorems 19 September 2026 localized-flow audit hosted at /papers/8/audit-01 is additional work covering Theorems 7.3–7.6 and 8.6; it is not peer review.", "evidence": [ "Zenodo record 22848213 retrieved 2026-09-19", "PDF SHA-256 9827d625…", "companion ZIP SHA-256 cbdf7a29…", "this-archive checker replay: 32768 rows, 300871 transitions, 56 monomials, certificate 32bcc2fe…" ], "machine_checkable": "omega_reference_checker.py and verify_external_bundle.py on this archive; PASS", "publication": { "kind": "Zenodo preprint", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "Not a G6 cross-language conformance claim. This-archive Python 3.10.21 replay of the paper's own checker is recorded in /papers/8/ENVIRONMENT.json: certificate 32bcc2fe…, 32768 rows, 300871 transitions. NumPy 2.2.6 not pinned 2.3.5. Not independent third-party reproduction." }, "source_ids": [ "paper-8" ], "artifact_ids": [ "paper-8-pdf", "paper-8-docx", "paper-8-zip", "paper-8-checker", "paper-8-certificate", "paper-8-bundle-reader", "paper-8-review-pdf", "paper-8-environment", "paper-8-audit01-pdf", "paper-8-audit01-md", "paper-8-audit01-py" ], "audit_href": "/papers/8", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "REPORTED_NOT_INSPECTED", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Reported evidence is named by the existing record. This audit did not inspect those logs and did not rerun them.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "paper-8-localized-flow-audit", "branch": "certification", "title": "paper 8 localized forced incompressible flow (Theorems 7.3–7.6 and 8.6)", "statement_latex": "For every admissible schedule (c>0,\\eta) and compact marked set K there exist a divergence-free velocity u and pressure p, independent of viscosity, such that for every \\nu>0 a compactly supported force f_\\nu realizes the volume-preserving deformation \\Phi^{(m)}_{c,\\eta} as a smooth finite-energy forced incompressible flow in every finite dimension m\\ge3, with exact marked trajectories in the cutoff plateau. Angular energy through degree five is exact on the twelve icosahedral vertices; the degree-six obstruction is Q(z^6)-\\int z^6 d\\sigma=-1/105.", "scope": { "frequency": "not a cubature 1–9 theorem; twelve-node degree-five exactness is the angular-energy connection only", "node_set": "arbitrary compact marked set K in the cutoff plateau; twelve normalized icosahedral vertices for Theorem 8.6", "arithmetic": "analytic localization; independent exact SymPy checks in dimensions 3 and 4; general-m index proof is in the written audit" }, "mathematical_status": "written theorems in the 19 September 2026 preprint; 19 September 2026 audit found no proof-breaking defect in Theorems 7.3–7.6 or 8.6. Audit is not institutional peer review and does not establish priority of the combined methods.", "evidence": [ "six-page proof PDF, 306508 bytes, SHA-256 e3985cb5…", "written audit OMEGA_Paper_VIII_Proof_Audit_01_Forced_Flow.md, 21406 bytes, SHA-256 4617fd3f…", "independent_checks.py, 9339 bytes, SHA-256 380c32bd…; no author modules imported", "supplied Python 3.12.14 results: 29 PASS groups, SHA-256 01b5ece2…", "this-archive Python 3.10.21 / SymPy 1.14.0: 29 PASS groups, identical check labels", "OMEGA_Localized_Flow_Complete_Proof.tex and original independent_checks.log were not supplied and are not hosted", "does not enlarge Paper VI; not a blowup theorem" ], "machine_checkable": "independent_checks.py on this archive; 29 groups PASS. Finite checks corroborate selected identities; they are not a formal verification of the complete theorem.", "publication": { "kind": "locally hosted audit of a Zenodo preprint section", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "This-archive replay of independent_checks.py only. Not a G6 conformance claim. Not a proof-assistant run.", "implementation": "Python 3.10.21 / SymPy 1.14.0", "release_id": "paper-8-proof-audit-01", "expected_output_ref": "/papers/8/audit-01/this_archive_independent_results.json" }, "source_ids": [ "paper-8" ], "artifact_ids": [ "paper-8-audit01-pdf", "paper-8-audit01-md", "paper-8-audit01-py", "paper-8-audit01-results", "paper-8-audit01-environment", "paper-8-audit01-this-archive-results", "paper-8-audit01-this-archive-log", "paper-8-audit01-hosting" ], "audit_href": "/papers/8/audit-01", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-object-exploration-02", "branch": "geometry", "title": "Six-mark projective recognition, landmark decoding, and baseline integration bounds", "statement_latex": "For the labeled marks p_i=\\alpha+a(cu+z_i v) with z=(-A,-B,-s,s,B,A), A=s+\\varphi^2, B=s+1, the outer cross-ratio R=1+\\varphi^2/(4AB) determines s=(\\varphi/2)(\\sqrt{R/(R-1)}-\\sqrt5) in the declared family. Affine records have minimum distance 5 (correct 2 errors or 4 erasures). Projective line records have minimum distance 4 (correct 1 error or 3 erasures). Uniform measure on [-A,A] admits a positive six-node rule of degree 3 and no nonnegative degree-4 rule for 01; twelve-node spherical 5-design plus one extra point", "node_set": "aligned icosahedral 12-set plus p_*=R^T(1,1,1)^T/\\sqrt3", "arithmetic": "Written polynomial and linear-algebra proofs; exact Fraction computation of the J-squared spherical moment; numerical NumPy corroboration for the remaining checker groups." }, "mathematical_status": "Written proofs in the 21 September 2026 PDF and the 22 September 2026 remarks. Not a theorem of Paper VII. Not a source paper.", "evidence": [ "PDF 344230 bytes SHA-256 a8c9e39e… (unchanged)", "restored TeX 27553 bytes SHA-256 fcaadf4b…; paste as received 7db0d779…", "REMARKS.md 7927 bytes SHA-256 9c217a5e…", "checker 2026-09-22.r2 10661 bytes SHA-256 ce3e5ea2…", "this-archive Python 3.10.21 / NumPy 2.2.6 captured 2026-09-22T12:35:49Z: TOTAL 30 PASS, stdout 60ad47fa… bound to checker ce3e5ea2…; four negative controls reject" ], "machine_checkable": "verify_boundary_reconstruction.py revision 2026-09-22.r2 on this archive; stdout must match this_archive_verify.stdout.txt and ENVIRONMENT.json must name checker SHA-256 ce3e5ea2…. Exact Fraction check of the J² spherical mean only. Remaining groups are numerical. Not a proof-assistant certificate.", "publication": { "kind": "locally hosted note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Implementation conformance for checker revision 2026-09-22.r2. Exact rational J-squared spherical moment; numerical NumPy corroboration for the other groups. Both normal and -O runs pass; both deliberate error controls are rejected in each mode. Not an exact certificate for all statements. Not G6. Not a proof-assistant run.", "implementation": "Python 3.10.21; NumPy 2.2.6", "release_id": "boundary-reconstruction/2026-09-22.r2", "expected_output_ref": "/geometry/boundary-reconstruction/this_archive_verify.stdout.txt" }, "source_ids": [ "boundary-note-2026-09-21", "boundary-remarks-2026-09-22" ], "background_source_ids": [ "paper-vii", "paper-iv" ], "artifact_ids": [ "boundary-pdf", "boundary-tex", "boundary-paste", "boundary-remarks", "boundary-py", "boundary-stdout", "boundary-environment", "boundary-hosting" ], "audit_href": "/geometry/boundary-reconstruction", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-ten-sample-reconstruction", "branch": "geometry", "title": "Explicit minimal ten-sample reconstruction of H≤2 ⊕ ℝJ", "statement_latex": "On F=H_{<=2}\\oplus R J, J=z(x^2+xy-y^2), the ordered samples at the physical points for a1,-a1,a2,-a2,a3,-a3,a4,a5,a6 and xi_*=(1,1,1)/sqrt(3) determine every field. The evaluation map S is an isomorphism onto R^{10}; the cardinal functions are S^{-1}. Ten scalar samples are minimal. Transport by the invariant matrix A satisfies S_A^{-1}=U_A S^{-1}. The worked field 2+3x+x^2-y^2+4J has beta=4 and value 721/300 at (1/5,-3/10,1/3).", "scope": { "frequency": "not a cubature theorem", "node_set": "nine specified aligned icosahedral nodes and one additional boundary point; not the browser tool's numeric labels", "arithmetic": "exact identities in Q(sqrt(5), r, sqrt(3)), r^2=(5+sqrt(5))/2" }, "mathematical_status": "Written supplementary note, 23 September 2026. Not a theorem of Paper VII and not an additional counted source paper. Ten is minimal for scalar point samples on this space. The thirteen-sample arrangement's three consistency equations are not an error-correction guarantee. General transport is the written argument; two rational executions are finite checks.", "evidence": [ "checker verify_ten_sample.py 12288 bytes SHA-256 cc850bda469cb33cc9d37c307dceede7e1b4e5ede002957c3ff7497c7a388c0f", "supplied verification.txt and verification_optimized.txt, each 1186 bytes, SHA-256 a6e04971faf30e665f00fc1b60a16488dddce315a83d715291b5298031268be0", "this-archive Python 3.10.21 at 2026-09-23T23:53:35Z: both python3 and python3 -O exited 0 and matched that transcript", "author record names Python 3.12.14; stdout bytes agree", "canonical Ten_Sample_Reconstruction.md hosted, 9973 bytes, SHA-256 967bc8afe442d6b0763f5d563b993a368161243d0f23b7274be46936d81639fe" ], "machine_checkable": "verify_ten_sample.py, standard library only. TOTAL 13 PASS. Exact algebraic comparisons. Not a proof-assistant certificate.", "publication": { "kind": "locally hosted supplementary note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "This-archive replay of the exact checker. The written note is the hosted Markdown. Not G6. Not a cubature rule.", "implementation": "Python 3.10.21", "release_id": "ten-sample-reconstruction", "expected_output_ref": "/geometry/boundary-reconstruction/ten-sample/verification.txt" }, "source_ids": [ "paper-vii" ], "background_source_ids": [ "boundary-note-2026-09-21" ], "artifact_ids": [ "ten-sample-note", "ten-sample-verifier", "ten-sample-supplied-normal", "ten-sample-supplied-optimized", "ten-sample-execution" ], "audit_href": "/geometry/boundary-reconstruction/ten-sample", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-exact-field-interface", "branch": "geometry", "title": "Exact field interface for the golden algebra", "statement_latex": "F=Q(sqrt(5))=Q(phi) is contained in K=Q(sqrt(2),sqrt(3),sqrt(5)), with [K:Q]=8 and [K:F]=4. Every element of K has unique coefficients on 1, sqrt(2), sqrt(3), sqrt(5), sqrt(6), sqrt(10), sqrt(15), sqrt(30). For x=u+v phi, N_{K/Q}(x)=(u^2+uv-v^2)^4. The Householder reflection of n=(2,3,0) lies in GL(3,Q) and sends n to -n. A positive cubature exact on products from a finite-dimensional continuous space has injective samples. On the existing degree-five rule this applies to H_{<=2}. J is invisible at the twelve nodes and the normalized integral of J^2 is 1/21.", "scope": { "field": "K=Q(sqrt(2),sqrt(3),sqrt(5)). Pi-dependent constants are not elements of K.", "cubature": "sufficient injectivity condition only; the degree-five application uses the hosted twelve-node rule and does not add a rule", "records": "eight reduced rational coefficients in schema omega-exact-coefficient/1; the JSON Schema checks shape and integer strings; the checker also rejects unreduced fractions and any zero other than 0/1; norms do not replace the coefficients" }, "mathematical_status": "Supplementary note, 25 September 2026. Established field arithmetic and its specified application to existing OMEGA objects. Not an additional source paper. The checker does not prove the Galois correspondence or the transcendence of pi.", "evidence": [ "Exact_Field_Interface.md 8938 bytes SHA-256 30b1c4c58f0baa470052a5d1aa3eaf98fed6cf117e11e57c4fe38cf925e0f0f4", "verify_exact_interface.py 17597 bytes SHA-256 d51b7595348d067163c08b88a7726b761662471a1f9b380201114d0112417bfd", "verification.txt and verification_optimized.txt, each 1895 bytes, SHA-256 345931205dac7acbf49422ca727e55ea333badc37ce3329f612a6ef7813d3977", "coefficient_record.schema.json 1588 bytes SHA-256 cd1b481dae752eae12f098d94317a1a2c17798a5b9eb9f669db9d110136f33e1", "this-archive Python 3.10.21 at 2026-09-25T02:39:45Z: python3 and python3 -O both exited 0, TOTAL 29 PASS" ], "machine_checkable": "verify_exact_interface.py, standard library only. TOTAL 29 PASS under python3 and python3 -O. The coefficient schema is applied by that checker. Not a proof-assistant certificate.", "publication": { "kind": "locally hosted supplementary note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Exact finite checks of the basis sign determinant, the norm identity, the rational reflection, an attained integer disagreement, a continuous infimum that is not a disagreement, one floor identity, the normalized integral of J^2, and the coefficient-record schema including rejection of missing, malformed, zero-denominator, and unreduced records. Not a new cubature rule.", "implementation": "Python 3.10.21", "release_id": "exact-field-interface", "expected_output_ref": "/geometry/exact-interface/verification.txt" }, "source_ids": [], "background_source_ids": [ "boundary-note-2026-09-21" ], "artifact_ids": [ "exact-interface-note", "exact-interface-verifier", "exact-interface-transcript", "exact-interface-transcript-optimized", "exact-interface-schema", "exact-interface-execution" ], "audit_href": "/geometry/exact-interface", "verification": { "srsd_applicability": "UNDETERMINED", "srsd_conformance": "NOT_REVIEWED", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Axes added 2026-10-07. They do not reclassify mathematical_status or implementation_conformance.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "formal_note": "No matching formalization was identified for this claim in this review. That is not a determination that none exists." } }, { "id": "omega-floor-kernel-399", "branch": "dynamics", "title": "399 rational-floor reference kernel", "statement_latex": "Q=339949771344778, S=10^{15}, F(d)=\\lfloor Qd/S\\rfloor, \\rho(d)=Qd-SF(d). For d=399, Qd=135\\cdot 10^{15}+639958766566422. The inverse d=(Sd'+r)/Q is an integer exactly when the record is valid. Q/S is not \\lambda.", "scope": { "domain": "nonnegative integers for the written argument; the local checkers cover seeds 0 through 32767 and one 81-digit example", "format": "unpadded extinction ledger, terminal zero, no zero-to-zero padding" }, "mathematical_status": "Explanatory supplement for the map already published on /dynamics and used by the reconstruction tool. Not a new theorem and not an additional source paper. A valid suffix is another seed.", "evidence": [ "PROOF_AND_GROK_HANDOFF.md 13304 bytes SHA-256 d47d2326f74f17ff2d763cba7df0e27f67be1ef401d1004ac4ce547260e67a7f", "floor_kernel.py 7737 bytes SHA-256 dc1b1633628fac20e0cc8fb5f80c37c54d888f496124417b6e71ddbd0b53b8a9", "verify_floor_kernel.mjs 4709 bytes SHA-256 be5d7a199d9f1ac7b092e7a6fa639d8407b911cc72294520c6cd0b9552d3bf04", "supplied site-reference transcripts: Python 806 bytes SHA-256 3a59103bb189ae68ef0aa4ef9f67525489f3b9243a912b69748a1718ebc1512d; Node 796 bytes SHA-256 c7b1ccb37dec27a23cb0aacabdfbe30ee6bc02aee1c2a36d7c6d940fa84664f3", "this archive Python 3.10.21 and Node v22.23.2 at 2026-09-24T00:50:51Z: all commands exit 0; site-reference stdout matches the supplied transcripts", "production module run reimported the 399 record to seed 399; a separate browser check of that control also recovered 399" ], "machine_checkable": "floor_kernel.py and verify_floor_kernel.mjs. RESULT=PASS. 32768 seeds, 300871 transitions, maximum depth 10. Not a proof-assistant certificate.", "publication": { "kind": "locally hosted explanatory supplement", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Finite-domain replay of the supplied reference checkers, plus one production-module 399 round trip. Not a new global equality with the irrational floor.", "implementation": "Python 3.10.21 and Node v22.23.2", "release_id": "floor-kernel-399/2026-09-24", "expected_output_ref": "/dynamics/floor-kernel/verification_python.txt" }, "source_ids": [ "paper-i", "paper-ii" ], "background_source_ids": [ "paper-8" ], "artifact_ids": [ "floor-kernel-note", "floor-kernel-py", "floor-kernel-mjs", "floor-kernel-node-verifier", "floor-kernel-python-transcript", "floor-kernel-node-transcript", "floor-kernel-sha256sums", "floor-kernel-execution", "floor-kernel-zip" ], "audit_href": "/dynamics/floor-kernel", "verification": { "srsd_applicability": "APPLICABLE", "srsd_reason": "The observation H(d)=Q*d with scale S uses the frozen encoder. The feedback d -> floor(Q*d/S) is application data, not part of the resolution definition.", "srsd_conformance": "NOT_REVIEWED", "part14_ref": "floor-transition-general", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "formal_note": "Floor.lean and Ledger.lean contain related division theorems at the canonical constants or for integer A,B,d. This review did not decide that they formalize the 399 ledger claim. This is not a determination that no formalization exists.", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "computational_note": "Historical execution evidence is recorded. This audit did not repeat that execution. The implementation_conformance field was not overwritten. The logs were not opened in this label correction.", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Specialization at the canonical Q and S. The statement text was not revised. Historical execution evidence is recorded. This audit did not repeat that execution.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED" } }, { "id": "floor-transition-general", "branch": "dynamics", "title": "General integer floor transition", "statement_latex": "Let Q and S be positive integers and let d be an integer. Set d_{next}=\\lfloor Qd/S\\rfloor and \\rho=Qd-Sd_{next}. Then 0\\le\\rhoS, residual-boundary rejections, and the canonical 399 product. Not a review of the 24 September ledger checker.", "implementation": "Python 3.10.21", "release_id": "floor-transition-general/2026-10-07", "expected_output_ref": "/dynamics/floor-kernel/evidence/2026-10-07/python-normal.txt" }, "source_ids": [ "paper-ii", "paper-iii" ], "artifact_ids": [ "floor-transition-general-py", "floor-transition-general-normal", "floor-transition-general-optimized", "floor-transition-general-execution", "floor-transition-general-execution-2026-10-08" ], "audit_href": "/dynamics/floor-kernel#general", "verification": { "srsd_applicability": "APPLICABLE", "srsd_reason": "The observation is H(d)=Qd with scale S. The decoder reconstructs Qd. Recovering d is a further inverse of H. The feedback map is application data. Q0: 0 ≤ remainder < B", "one_step_inverse when A≠0: (B*step + remainder)/A = d", "canonical_division when B>0, 0≤r0 the normalized nodes give an invertible Vandermonde sampler on polynomials of degree at most 5. The added bound ||V beta||_infty <= 1 defines K_Omega, with polar V^T C_1 and volume product 256/45. The map (beta,h) |-> (V beta, V^{-T} h) preserves the canonical symplectic form. A two-ball packing into the interior polar product is stated only if the external hypotheses A-087 and A-343 are assumed, and only for a strict capacity margin.", "scope": { "domain": "Normalized six marks with s>0, the model P_5, and the additional sample bound. The canonical sine seed is not evaluated by the finite checks.", "arithmetic": "exact arithmetic in Q(phi) for three rational seeds and the excluded s=0 boundary" }, "mathematical_status": "Direct identities for the declared sampler and sample bound. The packing statement is conditional on two external hypotheses that this note does not establish. Not a source paper and not an amendment of Paper VII.", "evidence": [ "NOTE.md 27115 bytes SHA-256 3e7f731acaaa05c4428ec4de3da5bb65873b3b581da51a52f20c668108ef7a66", "this archive Python 3.10.21 on 2026-10-08: four exact lines, exit 0 under python3 and python3 -O", "A-087 and A-343 were not executed" ], "machine_checkable": "Appendix A rational-seed checks only. Not the canonical sine seed and not the external symplectic statements.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Determinant factorization, both inverses, dual pairing, volume product 256/45, and the excluded s=0 boundary, on the three rational seeds in Appendix A. Not A-087 and not A-343.", "implementation": "Python 3.10.21", "release_id": "sampling-polar/2026-10-08", "expected_output_ref": "/geometry/sampling-polar/evidence/2026-10-08/python-normal.txt" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "sampling-polar-note", "sampling-polar-readme", "sampling-polar-py", "sampling-polar-sha256sums", "sampling-polar-normal", "sampling-polar-optimized", "sampling-polar-execution" ], "audit_href": "/geometry/sampling-polar", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This note is a sampler, a polar, and a conditional symplectic packing. It is not an SRSD resolution record.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/geometry/sampling-polar/evidence/2026-10-08/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The finite checks were run on this archive. They do not review the written symplectic argument and do not establish A-087 or A-343. An AI reading is not independent human review.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-08", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "The supplied note reports Python 3.12.14. That run was not reproduced. This audit executed Appendix A on Python 3.10.21. The two logs are byte-identical.", "formal_note": "No matching formalization of this note was identified in this review. That is not a determination that none exists." } }, { "id": "transition-invariants", "branch": "geometry", "title": "Transition invariants: symbolic identities and a finite ledger audit", "statement_latex": "The symbolic checks use free variables. Under the assumptions printed in the source, the zero normal forms are identities, not checks of one example: five normalized gaps a(phi, 1, 2s, 1, phi); selected-path ratio phi/(phi+pi); projective cross-ratio and the specialization (A+B)^2=4AB+phi^2; six-coefficient recovery after a reference-length change; both characteristic invariants for arbitrary degree-at-most-five data; the one-step identity Qd=Sy+rho and its induction step; and chi/m^2 for one solution at two lengths. Separately, the finite ledger audit covers seeds 0 through 32767 and horizons 1 through 23, which is 753664 prefix invariants. In that domain every seed reaches 0, and the longest time to 0 is step 10.", "scope": { "domain": "Free-variable identities in the declared polynomial ring, and a separate finite audit of seeds 0 through 32767 at horizons 1 through 23. The canonical sine is not evaluated.", "arithmetic": "exact rational arithmetic and integer division. Standard library only." }, "mathematical_status": "An exact symbolic-algebra checker with a finite canonical-ledger audit. Both executions reproduce the declared checks. Neither establishes physical unification or independent mathematical review.", "evidence": [ "OMEGA_Transition_Invariants_Proof.py 12112 bytes SHA-256 e94400b6e136bed009388c5d17e4f1c698235b7b938f75c266bbaf363a15a0ca", "supplied record Python 3.12.14 at 2026-10-11T23:19:35Z names that source hash; a matching hash checks byte integrity and does not itself verify that the recorded execution occurred", "this archive Python 3.10.21 on 2026-10-11 executed the hosted bytes: exit 0 under python3 and python3 -O; pass lines agree; 753664 prefix invariants on seeds 0-32767 and horizons 1-23; maximum extinction step 10" ], "machine_checkable": "Symbolic identities for the free variables named in the source, plus 753664 prefix checks on seeds 0-32767 and horizons 1-23. Not the canonical sine and not a physical calibration.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "The declared symbolic identities and the finite ledger audit in the hosted checker. Not a review of a written paper and not an execution of the canonical sine.", "implementation": "Python 3.10.21", "release_id": "transition-invariants/2026-10-11", "expected_output_ref": "/geometry/transition-invariants/evidence/2026-10-11/python-normal.txt" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "transition-invariants-py", "transition-invariants-readme", "transition-invariants-sha256sums", "transition-invariants-normal", "transition-invariants-optimized", "transition-invariants-execution", "transition-invariants-supplied-record" ], "audit_href": "/geometry/transition-invariants", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "These checks are polynomial identities and a finite integer ledger. They are not an SRSD resolution record, and the result does not forbid a later SRSD application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/geometry/transition-invariants/evidence/2026-10-11/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "This audit reran the hosted checker. The hosted logs have recorded SHA-256 digests. Matching those digests checks byte integrity. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. The supplied record names the same source hash, and the pass lines agree. An AI reading is not independent human review.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-11", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "The supplied record reports Python 3.12.14 and names the hosted source hash. This audit reran those bytes on Python 3.10.21. Matching pass lines supports execution agreement. Recorded digests check the logs' byte integrity. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "formal_note": "No matching formalization of this checker was identified in this review. That is not a determination that none exists." } }, { "id": "core-theorem-note", "branch": "geometry", "title": "Self-similarity, numerical difference, and exact recovery", "statement_latex": "The 13 October 2026 note states: the marked-kernel evolution by G/(G+C) is conjugate to Omega(v)=1+lambda(v-1) on v>1; the integer step is a quantized rational homothety, and a quantized lambda-homothety on the stated floor-agreement domain; one-step and finite-horizon differences separate into approximation and quantization; a valid trajectory is bijective with its terminal state and ordered remainders, while a global identity alone is not validity; a validated record with initial scale delta*d_0 determines the ideal scales. The finite checker audit covers seeds 0 through 32767 at horizon 23.", "scope": { "domain": "The note's fixed kernel, ratio, and parameters Q, S, delta, N. Symbolic checker identities use free s or an indeterminate X. The finite audit is seeds 0 through 32767 and horizon 23. The disagreement threshold 42189065 is certified, not enumerated.", "arithmetic": "exact rational arithmetic, integer division, and symbolic coefficients in Q(phi)[s]. Standard library only. No floating-point values in the checker." }, "mathematical_status": "The hosted note states five conclusions and written arguments for them. This archive has not reviewed those arguments as a proof. The checker rerun reproduces the declared symbolic checks, rational certificates, and finite ledger audit. Not a source paper and not an amendment of Paper VII.", "evidence": [ "omega_geometry_recovery_verify.py 16815 bytes SHA-256 e044133aceefb442bfa226bdf61359a58bca413e12196f090ff19eef3ef84ab5", "supplied record Python 3.12.14 on 13 October 2026 names that source hash", "this archive Python 3.10.21 on 2026-10-13 reran both commands: exit 0, logs byte-identical, stdout matches the supplied record body", "finite audit: 32768 states, horizon 23, 753664 prefix invariants, maximum extinction step 10" ], "machine_checkable": "The eight declared checker groups. Not a review of the written proofs, not the canonical sine, and not an enumeration of every integer below 42189065.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "The declared symbolic identities, rational certificates, rejection tests, and finite ledger audit in the hosted checker. Not a review of the written proofs.", "implementation": "Python 3.10.21", "release_id": "core-theorem/2026-10-13", "expected_output_ref": "/geometry/core-theorem/evidence/2026-10-13/python-normal.txt" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "core-theorem-pdf", "core-theorem-txt", "core-theorem-py", "core-theorem-readme", "core-theorem-sha256sums", "core-theorem-normal", "core-theorem-optimized", "core-theorem-execution", "core-theorem-supplied-record" ], "audit_href": "/geometry/core-theorem", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This note is a geometric scaling family and an integer witness encoding. It is not an SRSD resolution record, and the result does not forbid a later SRSD application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/geometry/core-theorem/evidence/2026-10-13/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "This audit reran the hosted checker. The hosted logs have recorded SHA-256 digests. Matching those digests checks byte integrity. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. The written arguments were not reviewed. An AI reading is not independent human review.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-13", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "The supplied record reports Python 3.12.14 and names the hosted source hash. This audit reran those bytes on Python 3.10.21. The logs match the supplied transcript body. Matching pass lines supports execution agreement. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "formal_note": "No matching formalization of this note was identified in this review. That is not a determination that none exists." } }, { "id": "integration-proofs", "branch": "geometry", "title": "Integration proof bundle, 14 October 2026", "statement_latex": "The supplied ledger states 34 theorem records under explicit hypotheses, and 12 excluded or unestablished claims. The Python checker reports 2314 algebra checks in Q(phi). An eq check on free variables is a polynomial identity for that program. The finite audit covers seeds 0 through 32767 at horizon 23. The floor-agreement threshold 42189065 is certified, not enumerated. The auxiliary seed 1/2 is not the canonical sine.", "scope": { "domain": "The files supplied in this upload: the ledger, the certificate DAG, the Python checker, the readable proof text, and the supplied status. Julia, Rust, adversarial, SymPy, and provenance files were not supplied.", "arithmetic": "exact Q(phi) polynomial arithmetic and integer division. Python standard library only. No floating-point JSON numbers in the certificate." }, "mathematical_status": "The hosted ledger states written arguments. This archive has not reviewed them as a proof. The Python checker was rerun and reproduced the declared algebra checks and finite audit. Not a source paper, not an amendment of Paper VII, and not a certificate of an external mathematics release.", "evidence": [ "certificates.json 1722817 bytes SHA-256 daaee23b3c27bbb320d0f8db2c8d00a4a077f08e5fe0056864323873ddf71b19", "verify.py 18214 bytes SHA-256 423b426276850abf4b3a35b36d54918870eaa400941dc1e916ef8261cd82326d", "supplied python_normal.json 2820 bytes SHA-256 eb4dab39fc196a2f1f09b9cac36f94ec0d28fae6595b07a349f28fc02dfbed91, Python 3.13.5", "this archive Python 3.10.21 on 2026-10-15: exit 0 under python3 and python3 -O; logs byte-identical; mathematical fields match the supplied report except the runtime string; 2314 algebra checks; 32768 seeds; horizon 23; 753664 transitions; maximum extinction step 10" ], "machine_checkable": "The 2314 algebra checks and the finite audits executed by the hosted verify.py. Not the written proofs. Not Julia, Rust, the adversarial script, or the SymPy script.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "The hosted Python checker against the hosted certificate file. Not a review of the written proofs, and not the engines or scripts that were absent from this upload.", "implementation": "Python 3.10.21", "release_id": "integration-proofs/2026-10-15", "expected_output_ref": "/geometry/integration-proofs/evidence/2026-10-15/python-normal.json" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "integration-certificates", "integration-proofs-json", "integration-verify-py", "integration-proofs-text", "integration-bundle-readme", "integration-theorem-index", "integration-readme", "integration-sha256sums", "integration-normal", "integration-optimized", "integration-execution", "integration-supplied-normal", "integration-supplied-status" ], "audit_href": "/geometry/integration-proofs", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This ledger is a collection of geometric, arithmetic, and interface arguments. It is not an SRSD resolution record, and the result does not forbid a later SRSD application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/geometry/integration-proofs/evidence/2026-10-15/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "This audit reran the hosted Python checker. The hosted logs have recorded SHA-256 digests. Matching those digests checks byte integrity. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. The written arguments were not reviewed. Julia and Rust were not run. An AI reading is not independent human review.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-15", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "The supplied record reports Python 3.13.5. This audit reran the same certificate bytes on Python 3.10.21. The reports match except the runtime string. Matching mathematical fields supports execution agreement. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "formal_note": "No proof-assistant run of this ledger was identified in this review. That is not a determination that none exists." } }, { "id": "relative-representation", "branch": "geometry", "title": "Relative representation and recoverable information, 15 October 2026", "statement_latex": "The thesis states a compositional reconstruction theorem: an injective measurement, an injective serialization, and an ordered-remainder encoding have an inverse on the valid image. The printed checker separates 28 free-variable polynomial identities from a finite ledger audit of seeds 0 through 32767 at horizon 23. The auxiliary seed 1/2 is not the canonical sine. The floor-agreement threshold 42189065 is certified, not enumerated.", "scope": { "domain": "The supplied thesis PDF, and a transcription of the checker printed in Appendix B. The companion source, the thesis Python 3.13.5 logs, the companion audit log, the mutation tests, and the manifest were not supplied.", "arithmetic": "exact rational polynomial arithmetic, Q(phi), and integer division. Python standard library only. No floating-point constants in the transcribed checker." }, "mathematical_status": "The hosted thesis states written arguments. This archive has not reviewed them as a proof. A transcription of the printed checker was rerun and reproduced the declared symbolic checks and finite audits. Not a source paper, not an amendment of Paper VII, and not a certificate of the unseen companion bytes.", "evidence": [ "thesis PDF 769556 bytes SHA-256 1a302b7e51340db486a3ff8fc2bfbdcb753c4c12dcfdeabaaa90abe42fd0982e", "transcribed verify_omega_thesis.py 18211 bytes SHA-256 e6330f8327f3ce39c61ca666864171a3470044117147b20a3237ba1931e7d506; this is the digest the program prints for its own bytes, not an unseen companion digest", "this archive Python 3.10.21 on 2026-10-15: exit 0 under python3 and python3 -O; logs byte-identical; 28 identities; 32768 seeds; horizon 23; 753664 prefix identities; maximum extinction step 10; payload 62 bytes", "thesis-reported Python 3.13.5 JSON logs were not supplied and were not compared" ], "machine_checkable": "The declared groups executed by the transcribed checker. Not the written proofs. Not the companion audit log, mutation tests, or manifest.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "The transcribed Appendix B checker. Not a review of the written proofs, and not the unsupplied 3.13.5 logs.", "implementation": "Python 3.10.21", "release_id": "relative-representation/2026-10-15", "expected_output_ref": "/geometry/relative-representation/evidence/2026-10-15/python-normal.json" }, "source_ids": [ "paper-vii" ], "artifact_ids": [ "relative-representation-pdf", "relative-representation-py", "relative-representation-readme", "relative-representation-sha256sums", "relative-representation-normal", "relative-representation-optimized", "relative-representation-execution" ], "audit_href": "/geometry/relative-representation", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This thesis is a geometric and arithmetic reconstruction argument. It is not an SRSD resolution record, and the result does not forbid a later SRSD application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/geometry/relative-representation/evidence/2026-10-15/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "This audit reran a transcription of the printed checker. The hosted logs have recorded SHA-256 digests. Matching those digests checks byte integrity. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. The written arguments were not reviewed. Julia, Rust, and Lean were not run. An AI reading is not independent human review.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-15", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "The thesis reports Python 3.13.5. Those JSON logs were not supplied. This audit ran the transcribed listing on Python 3.10.21. The two modes match each other. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "formal_note": "No proof-assistant run of this thesis was identified in this review. That is not a determination that none exists." } }, { "id": "supplement-1-workflows", "branch": "geometry", "title": "Supplementary workflows: canonical readouts, integer witnesses, and degree-2 transfer", "statement_latex": "An affine field on the six canonical marks is recovered from the outer readings. A rank-losing integer layer is recovered from the quotient, the remainder, and the trailing Smith coordinates. The twelve-node rule transfers samples exactly through harmonic degree 2 between a grid and two rotations of that grid.", "scope": { "domain": "CANONICAL_AFFINE_READOUT/1, INTEGER_LAYER_WITNESS/1, and the unnormalized twelve-node rule at degree at most 2. Not the 399 ledger, not the 23-step byte ledger, and not a higher subdivision frequency.", "arithmetic": "exact rational arithmetic in Q(phi) and integer floor division. Python standard library only. No floating-point values. The canonical sine is named and is not evaluated." }, "mathematical_status": "The checker proves the inverse formulas as linear forms, the Smith identity, and the Gram and transfer identities on the declared grids. The classical cyclotomic degree formula is cited and is not reproved. Not a reviewed proof of that formula, not a proof-assistant certificate, not peer review, and not an amendment of Paper VII.", "evidence": [ "verify_supplement_1.py 25845 bytes SHA-256 09d2fa0e6b5ac34a50344177c67aa4a3b0159565c8ac3aeccf728db4437c51e3", "this archive Python 3.10.21 on 2026-10-15: exit 0 under python3 and python3 -O; logs byte-identical; outer readings 5+2phi+2s and 1-2phi-2s; witnesses 7 and 6; serialized witness 77 bytes, not claimed optimal; three grids; nine basis recoveries", "classical degree [Q(cos(2*pi/1296000)):Q] = 172800 cited, not reproved; sine not evaluated; SageMath not called; Lean and SMT not generated" ], "machine_checkable": "The declared canonical, Smith, and transfer checks. Not a reproof of the cyclotomic degree formula. Not items 4, 5, or 6 of the implementation directive.", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Python 3.10.21 execution of verify_supplement_1.py. The browser tools repeat those exact checks and are not a separate certificate.", "implementation": "Python 3.10.21", "release_id": "supplement-1/2026-10-15", "expected_output_ref": "/tools/supplement-1/evidence/2026-10-15/python-normal.json" }, "source_ids": [], "artifact_ids": [ "supplement-1-py", "supplement-1-readme", "supplement-1-sha256sums", "supplement-1-example-readout", "supplement-1-example-witness", "supplement-1-normal", "supplement-1-optimized", "supplement-1-execution" ], "audit_href": "/tools/witness-lab", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "These are exact readout, witness, and transfer checks. They are not an SRSD resolution record.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/tools/supplement-1/evidence/2026-10-15/execution.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The hosted logs have recorded SHA-256 digests. Matching those digests checks the logs’ byte integrity. The logs document the archive’s reported executions. Rerunning the checker tests whether those results are reproducible. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. A hash identifies bytes; it does not certify their mathematical correctness.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-15", "historical_execution": "NOT_APPLICABLE", "current_audit_execution": "EXECUTED", "computational_note": "Python 3.10.21, both modes, exit 0, logs byte-identical. Formal execution was not run.", "formal_note": "No Lean or SMT package was generated. Formal execution was not run. That is not a determination that a formalization exists elsewhere." } }, { "id": "geometry-scale-compatibility", "branch": "dynamics", "title": "Selected scale compatibility", "statement_latex": "G(a)=(a\\\\varphi+a\\\\varphi)/2=a\\\\varphi and C_{\\\\mathrm{circ}}(a)=a\\\\pi give \\\\lambda=\\\\varphi/(\\\\varphi+\\\\pi). Scale evolution and the scalar affine map are conjugate on positive scales. The planar curve does not select the 600-cell coupling.", "scope": { "domain": "positive scales; the 600-cell coupling is a separate declaration", "arithmetic": "real identities under the stated definitions" }, "mathematical_status": "Written note. Not reviewed as a proof. Not a source paper.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "unification-note", "unification-proofs" ], "audit_href": "/geometry/unification", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "24 October supplementary note. Not an additional counted source work.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "reference-recovery-note", "branch": "dynamics", "title": "Observation fibers and recoverable relationships", "statement_latex": "A property factors through an observation exactly when it is constant on the observation fibers. Exact nonzero scaling is injective. Integer quantization, not that scaling, can have several predecessors.", "scope": { "domain": "the stated observation maps; singular mixing can hide kernel information", "arithmetic": "exact statements in the hosted note" }, "mathematical_status": "Written note. Not reviewed as a proof. Not a source paper.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "reference-recovery-proofs" ], "audit_href": "/geometry/reference-recovery", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "24 October supplementary note. The checker row is separate.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "reference-recovery-checker", "branch": "geometry", "title": "Reference-recovery checker, site run of 24 October", "statement_latex": "The restored checker’s declared groups exit 0. That execution is not a review of the written arguments.", "scope": { "domain": "the groups named by omega_reference_verify.py", "arithmetic": "the checker’s own arithmetic; one group is numerical" }, "mathematical_status": "Implementation record of a named script. Not a proof review and not peer review.", "evidence": [ "hosted checker", "this archive execution file dated 2026-10-24" ], "machine_checkable": "the script’s declared groups", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "python3 and python3 -O of the hosted checker on 24 October 2026. Not rerun for the 27 October supplement.", "implementation": "Python 3.10.21", "release_id": "reference-recovery/2026-10-24", "expected_output_ref": "/geometry/reference-recovery/evidence/2026-10-24-site/normal.stdout.txt" }, "source_ids": [], "artifact_ids": [ "reference-recovery-checker", "reference-recovery-site-execution" ], "audit_href": "/geometry/reference-recovery", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "A checker transcript is not an SRSD resolution record.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "execution_ref": "/geometry/reference-recovery/evidence/2026-10-24-site/EXECUTION.txt", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The 27 October audit did not repeat this run. A matching hash checks the log’s bytes. It does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Recorded site run: Python 3.10.21, both modes, exit 0. Not a Lean proof.", "formal_note": "No matching formalization of this checker was identified in this review." } }, { "id": "omega-language-v0-1", "branch": "dynamics", "title": "Scalar reference language v0.1", "statement_latex": "Scalar ranks and a 120-coordinate residue witness have different domains, framing, and decoders. The scalar parser does not accept vector records.", "scope": { "domain": "scalar OMEGA-LANGUAGE records; not Z^120", "arithmetic": "the scalar codec’s stated integers" }, "mathematical_status": "Specification note. Not a proof that the decoder authenticates a sender.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "omega-language-spec" ], "audit_href": "/language", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Structural validity is not sender authentication.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-real-dynamics", "branch": "dynamics", "title": "600-cell real signal model", "statement_latex": "\\\\mathcal F(v)=\\\\mathbf 1+\\\\lambda P(v-\\\\mathbf 1) with P=M/25 and M=13I+A. The update is a contraction in the infinity norm, fixes only the constant vector 1, preserves the mean by scalar OMEGA, and is equivariant under adjacency-preserving permutations.", "scope": { "domain": "real 120-vectors on the declared 600-cell graph", "arithmetic": "real arithmetic under the stated definitions" }, "mathematical_status": "Written all-domain statement under the definitions. Not reviewed as a proof. The planar geometry does not select this coupling.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "cell600-checker" ], "audit_href": "/dynamics/600cell", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Hosting the checker does not promote its ancillary numerical audit as this theorem.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-full-remainder", "branch": "dynamics", "title": "Full-remainder inverse on the integer network", "statement_latex": "For y=\\\\lfloor C_{\\\\mathrm{net}} d\\\\rfloor and r=Bd-Dy, the certificate MN=4948372000 I implements the inverse. Every finite ordered remainder history recovers earlier states by the telescoping identity.", "scope": { "domain": "integer vectors in the declared domain, every finite horizon", "arithmetic": "exact integer arithmetic" }, "mathematical_status": "Written all-integer statement. Not reviewed as a proof. Quotient agreement is not remainder equality.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "cell600-checker", "cell600-test-vector" ], "audit_href": "/dynamics/600cell/witnesses", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The checker’s local name C = 25 is the scalar denominator factor, not the circular readout.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-mod8-witness", "branch": "dynamics", "title": "Modulo-8 joint witness", "statement_latex": "On \\\\mathbb Z^{120}, the pair (T(d),(d_i \\\\bmod 8)) determines d. The residue witness is 360 bits, or 45 bytes. The next state and the graph order are stored or shared separately.", "scope": { "domain": "all integer 120-vectors; not a claim of global optimality or polynomial-time decoding", "arithmetic": "exact integer arithmetic" }, "mathematical_status": "Written theorem plus a historical checker record. The written argument was not reviewed. The 431-bit midpoint method remains correct in its restricted sharpness. 306 bits is not a necessity on the canonical cube.", "evidence": [ "26 October 2026 assessment stdout", "hosted source hash fbcf5bd8…" ], "machine_checkable": "the checker’s declared joint tests, not an unrestricted browser inverse", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "Historical full checker stdout. Not a site rerun and not a polynomial-time claim.", "implementation": "Python 3.12.14 historical record", "release_id": "600cell-joint-witness/2026-10-26", "expected_output_ref": "/dynamics/600cell/evidence/2026-10-26-assessment/normal.stdout" }, "source_ids": [], "artifact_ids": [ "cell600-checker", "cell600-graph-fixture", "cell600-test-vector", "cell600-joint-execution", "cell600-joint-stdout" ], "audit_href": "/dynamics/600cell/witnesses", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "A network witness is not an SRSD resolution record.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "HISTORICAL_RECORDED_NOT_RERUN", "execution_ref": "/dynamics/600cell/evidence/2026-10-26-assessment/EXECUTION.json", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The hosted logs have recorded SHA-256 digests. Matching those digests checks the logs’ byte integrity. The logs document the archive’s reported executions. Rerunning the checker tests whether those results are reproducible. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "REPORTED", "current_audit_execution": "NOT_REPEATED", "computational_note": "Historical Python 3.12.14. This archive’s Python is 3.10.21 and SciPy is absent. A site stdout, if stored later, is a different run. Not a Lean proof.", "formal_note": "No matching formalization was identified. This row does not inherit 147 Lean declarations." } }, { "id": "cell600-306-bit-bound", "branch": "dynamics", "title": "306-bit lower bound on the unbounded domain", "statement_latex": "2^{305} Q^{120} \\\\det M < D^{120} < 2^{306} Q^{120} \\\\det M, so some output fiber needs at least 306 bits of fixed-width side information on the unbounded domain. This is not a 306-bit necessity on the canonical cube.", "scope": { "domain": "unbounded integer domain, including a nonnegative translate of a finite demanding fiber", "arithmetic": "exact determinant and counting" }, "mathematical_status": "Written counting argument. Not reviewed. Not a codec optimality theorem.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "cell600-checker" ], "audit_href": "/dynamics/600cell/witnesses", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Separate from the 360-bit construction.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-ten-step-extinction", "branch": "dynamics", "title": "Ten-step extinction on the canonical cube", "statement_latex": "For d in \\\\{0,\\\\ldots,32767\\\\}^{120}, 32767 Q^{10} < S^{10}, so every such input reaches zero by ten steps. The uniform maximum attains 32767, 11139, 3786, 1287, 437, 148, 50, 16, 5, 1, 0. Horizon 23 is an audit convention.", "scope": { "domain": "the nonnegative canonical cube, proved by the power comparison, not by enumerating the cube", "arithmetic": "exact integer comparison" }, "mathematical_status": "Written domain statement. Not an enumeration and not a validity limit at horizon 23.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "cell600-checker" ], "audit_href": "/dynamics/600cell/witnesses", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Not a browser inverse.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-parameter-interval", "branch": "dynamics", "title": "Canonical network parameter interval", "statement_latex": "I_{\\\\mathrm{net}}=[211975/623548,\\\\,268150/788793), width 25/491850297564. The scalar interval I_{15} stays the scalar certificate. 4196/12343 disagrees at local z=127813 and 22813643/2^{26} disagrees at z=165245.", "scope": { "domain": "local integers 0 through 819175, each realizable on the canonical cube", "arithmetic": "exact integer and rational arithmetic" }, "mathematical_status": "Exact finite certificate plus the local-attainability argument. The written necessity argument was not reviewed. This does not amend the scalar successor contract.", "evidence": [ "network_parameter_audit.py", "this archive Python 3.10.21, both modes, JSON identical to the assessment file" ], "machine_checkable": "819176 local floors and 32768 scalar states in the script", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "pass", "scope": "python3 and python3 -O produced the assessment JSON. SciPy was not used.", "implementation": "Python 3.10.21, standard library", "release_id": "600cell-parameter/2026-10-27", "expected_output_ref": "/dynamics/600cell/evidence/2026-10-27-assessment/network_parameter_audit.json" }, "source_ids": [], "artifact_ids": [ "cell600-parameter-audit", "cell600-parameter-json", "cell600-parameter-site-execution" ], "audit_href": "/dynamics/600cell/parameters", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "A rational floor interval is not an SRSD resolution record.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "PASSED_DECLARED_CHECKS", "execution_ref": "/dynamics/600cell/evidence/2026-10-27-site/EXECUTION.txt", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "The hosted logs have recorded SHA-256 digests. Matching those digests checks the logs’ byte integrity. The logs document the archive’s reported executions. Rerunning the checker tests whether those results are reproducible. A matching hash does not itself verify that the recorded execution occurred or that the checker is mathematically correct. A hash identifies bytes; it does not certify their mathematical correctness.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "REPORTED", "current_audit_execution": "EXECUTED", "computational_note": "Python 3.10.21, normal and optimized, exit 0, JSON byte-identical. Not the joint-witness suite and not a Lean proof.", "formal_note": "No matching formalization was identified in this review." } }, { "id": "cell600-positive-simplex", "branch": "dynamics", "title": "Positive-simplex corollary", "statement_latex": "\\\\mathcal G(p)=\\\\operatorname{softmax}(\\\\lambda P \\\\operatorname{clr}(p)) on strictly positive 120-vectors. It is invertible on the zero-mean chart. It is not six-mark membership and not a finite codec.", "scope": { "domain": "strictly positive real probability vectors in dimension 120", "arithmetic": "exact real arithmetic" }, "mathematical_status": "Written derived result. No execution status and no formal status are inherited.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [], "audit_href": "/dynamics/600cell#positive-simplex", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Linked from TGA as a separate derivation.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } }, { "id": "cell600-decoder-budget", "branch": "dynamics", "title": "Decoder cost is not a polynomial-time claim", "statement_latex": "The inverse-norm bound is below 16, so each coordinate has at most two candidates, but the search may examine up to 2^{120} leaves before pruning. No unrestricted browser inverse is provided.", "scope": { "domain": "the stated exact search; exhaustion is unresolved, not invalid", "arithmetic": "exact forward inequalities" }, "mathematical_status": "Finite observations plus an exponential worst-case bound. Not a polynomial-time theorem.", "evidence": [ "hosted website supplement", "written argument not reviewed as a proof" ], "machine_checkable": "not this written statement", "publication": { "kind": "locally hosted derived note", "peer_reviewed": false }, "independent_reproduction": { "status": "none-recorded", "records": [] }, "implementation_conformance": { "status": "not applicable", "scope": "written statement; no checker is claimed for this row" }, "source_ids": [], "artifact_ids": [ "cell600-checker" ], "audit_href": "/dynamics/600cell/witnesses", "verification": { "srsd_applicability": "NOT_APPLICABLE", "srsd_reason": "This row is a geometric or network identity. It is not an SRSD resolution record. That does not forbid a later application.", "srsd_conformance": "NOT_APPLICABLE", "written_proof_review": "NOT_REVIEWED", "formal_evidence": "NOT_ASSESSED", "computational_evidence": "NOT_APPLICABLE", "language_review": "NOT_REVIEWED", "ste_source": "SOURCE_UNAVAILABLE", "note": "Arithmetic consistency is not sender authentication.", "schema": "OMEGA_VERIFICATION_AXES/2", "snapshot": "2026-10-27", "historical_execution": "NONE_RECORDED", "current_audit_execution": "NOT_REPEATED", "computational_note": "No execution is claimed for this written row. A Python PASS on another row is not this statement.", "formal_note": "No matching formalization was identified in this review. This row does not inherit the Lean package’s 147 declarations." } } ] }