Status
Publication status: seven Zenodo preprints and one Zenodo report. Paper V (OMEGA NAUGHT Ω₀) is the report, DOI 10.5281/zenodo.22048540. Papers I–IV, VI, VII, and paper 8 are preprints. Paper VII is version 1.6 at DOI 10.5281/zenodo.22233573. paper 8 is DOI 10.5281/zenodo.22848213 (site label paper 8; manuscript heading Paper VIII). Not peer reviewed unless a specific work is later linked to a peer-reviewed publication.
Computational reproducibility: exact verifiers and/or finite certificates are supplied where listed.
Independent third-party reproduction: none recorded on this site unless a named reproduction and its public evidence are linked.
The author-reported DOI 10.5281/zenodo.23688676 is reserved and unpublished. It is not a ninth source work, not peer reviewed, and has no publication date on this site. The 600-cell pages are a website supplement to that pending manuscript.
| Claim / object | Paper / source | Mathematical status | Computational status | Conformance status |
|---|---|---|---|---|
| Paper I — quantized contraction | I | Source preprint; exact identities | Named historical G₆ replay (implementation property) | Not a theorem |
| Paper II — rational floor maps | II | Source preprint; remainder ledgers | Named historical G₆ replay (implementation property) | Not a theorem |
| Paper III — SRSD external audit instrument | III | Frozen v1.0. Foundational closure means internal sufficiency, not a new foundation for mathematics. | Condensed listing supplied; not a theorem checker | Not a theorem |
| Paper IV — harmonic obstruction | IV | 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. | Where supplied in cubature artifacts | Not a theorem |
| Paper V — Ω₀ reference identities | V | Source report; many identities definitional | Exact notebooks and OMEGA_NAUGHT/1.0 replay | Not a theorem |
| Paper VI — cubature 1 ≤ f ≤ 9 | VI | Source preprint; exact moment systems. Independent of Ω₀ and SRSD. | Frozen CSV replayed by historical G₆ | Not a theorem |
| Paper VII — marked kernel | VII | Construction-to-curve theorem: unique marked kernel from the one-arcsecond seed and sequential bilateral extensions. Complete center-angular fan; Gram rank two; exact R³ realization. Metric outer scaffold adopted; remaining completion not claimed. | Companion generator evaluates closed formulas. Zenodo preprint v1.6. | Not a theorem |
| Paper VII — exact 3D reconstruction | VII | p_{j,σ}=α+a(c u+σ x_j v); SE(3) placement; (R>0,×) torsor on the scaled family. | Companion JSON/OBJ generator. Not a theorem checker. | Not a theorem |
| paper 8 — exact arithmetic certification | 8 | Zenodo preprint 10.5281/zenodo.22848213. Remainder certificates, first disagreement, fibers, homothety/Hausdorff identity, twelve-node degree-five cubature with exact degree-six obstruction, localized forced realization. Not a blowup theorem. Does not enlarge Paper VI. | This-archive Python 3.10.21: bundle PASS; regenerated certificate 32bcc2fe…; interface/model/boundary tests. NumPy 2.2.6, not pinned 2.3.5. | Named runtime PASS; not independent reproduction |
| paper 8 Proof Audit 01 — localized forced flow | audit of paper 8 | Theorems 7.3–7.6 and 8.6 valid under stated hypotheses. Force constructed after the motion. Priority of combined methods not established. Not a source paper. | This-archive Python 3.10.21 / SymPy 1.14.0: independent_checks.py 29 groups PASS. Original TEX and original log not supplied. | Named runtime PASS; not independent reproduction |
| OMEGA object exploration 02 — six-mark recognition and baseline integration | local exploration | Cross-ratio recovers s in the declared family; affine distance 5 / projective distance 4; positive degree-three and no nonnegative degree-four on the canonical six nodes; 36° Gauss–Lobatto is a related family member. Canonical seed remains one arcsecond. | This-archive Python 3.10.21 / SymPy 1.14.0: 21 check groups PASS. Decoder rejects truncated, empty, scalar, and out-of-budget records. No author OMEGA modules imported. | Named runtime PASS; not independent reproduction |
| Affine dynamics, sampling, and quadrature | local note | d_+(s)=3 for 0<s<s_* and 5 for s_*≤s<1; signed max 5; R≤R_* criterion; sharp amplification. Canonical seed remains one arcsecond. Not a source paper. | This-archive Python 3.10.21 and OpenJDK 17.0.20: four executions byte-identical to expected.txt, TOTAL 21 PASS. Julia/Rust sources not received. | Named runtime PASS; not independent reproduction |
| Boundary reconstruction supplementary note | local note | Boundary reconstruction supplementary note: the original 21 September 2026 PDF is preserved. The corrected checker computes the spherical mean of J² exactly using rational arithmetic; the remaining groups provide numerical corroboration. Normal and optimized runs pass, and deliberate wrong-value and wrong-coefficient controls are rejected. The written proofs are in the original note and the supplementary remarks. Reconstruction is restricted to the declared aligned field spaces. | This-archive Python 3.10.21 / NumPy 2.2.6 captured 2026-09-22T12:35:49Z. Checker 2026-09-22.r2 SHA-256 ce3e5ea2…. Stdout 60ad47fa… bound to that program. Four negative controls reject. | Named runtime PASS; not independent reproduction |
| Explicit ten-sample reconstruction | local supplementary note | Nine specified icosahedral samples and one extra boundary value determine H≤2 ⊕ ℝJ. Cardinal functions invert the evaluation map. Ten is minimal. S_A^{-1}=U_A S^{-1}. Worked value 721/300. Not an additional source paper. | This-archive Python 3.10.21 at 2026-09-23T23:53:35Z: normal and -O both TOTAL 13 PASS, stdout a6e04971…, checker cc850bda…. Supplied transcripts name Python 3.12.14 and match those bytes. The written note is hosted: 9973 bytes, SHA-256 967bc8af…. | Named checker PASS; written note hosted; not independent reproduction |
| Exact field interface | local supplementary note | F=Q(√5) sits inside K=Q(√2,√3,√5), degree 8 over Q and degree 4 over F. Eight ordered rational coefficients reconstruct each element. N_{K/Q}(u+vφ)=(N_{F/Q})^4. The reflection of n=(2,3,0) is rational. Positive cubature exact on products gives injective samples; the existing degree-five rule is injective on H≤2. J vanishes on the twelve nodes and the normalized integral of J² is 1/21. Field recovery and record recovery are separate. Not an additional source paper. | This-archive Python 3.10.21 at 2026-09-25T02:39:45Z: normal and -O both TOTAL 29 PASS, stdout 34593120…, checker d51b7595…. The coefficient schema rejects a bad count, a malformed integer string, and a zero denominator. The checker also rejects an unreduced fraction and any zero other than 0/1. The Galois correspondence and the transcendence of π are cited, not reproved. | Named checker PASS; written note hosted; not independent reproduction |
| 399 rational-floor reference kernel | local explanatory supplement | The existing map F(d)=floor(Qd/S), rho(d)=Qd-SF(d), with Q=339949771344778 and S=10^15. The 399 product is 135*10^15+639958766566422. A terminal state and ordered remainders recover the trajectory when the inverse is an integer. Q/S is not λ. Not a new theorem and not a source paper. | This archive, Python 3.10.21 and Node v22.23.2, 2026-09-24T00:50:51Z: 32768 seeds, 300871 unpadded transitions, depth 10, domain checksum 804ac6ba…. Site-reference stdout matches the supplied transcripts. A module run and a separate browser check both recovered 399 from the production record. | Named checker PASS; not independent reproduction |
| OMEGA–Navier–Stokes geometric extension 3.0 | local extension | Family Φ_{c,η} scales N by c² and preserves J=zN(w); volume-preserving. Forced smooth realizations. Not a source paper. Not a blowup result. | Python 3.11.2 / SymPy 1.14.0: 62 exact checks; auditor and mutation harness PASS. | Named runtime PASS; not independent reproduction |
| Floor-map fibers and 32-block seed cover | local package / paper 8 | T(d)=⌊Qd/S⌋ with inverse slope r=S/Q. Consecutive fibers, depth layers, 32-block cover. Not a source paper. | Python 3.11.2 verify_handoff.py PASS, identical under -O and -OO. | Named runtime PASS; not independent reproduction |
| Target-degree rules 10 ≤ f ≤ 21 | post-publication recovered package | Certificate-backed exact identities; not Paper VI. Maximality not claimed. | Python 3.11.2 r2 PASS on this archive; 383 identities | Implementation PASS of recovered source |
| Frequency 22 on fixed X₂₂ | f22 addendum | Exact degree-69 positivity and 70/71 obstruction | Python 3.12.13 PASS and Julia 1.11.6 PASS, separately | Two-implementation PASS |
| X₂₃ degree 71 and X₂₄ degree 75 | locally hosted theorem plus recovered certificate | Strictly positive H₃-invariant weights; no maximality; does not enlarge Paper VI | Python 3.11.2 replay_recovered.py PASS; python3 -O refused | Implementation PASS of recovered source |
| Existence of strictly positive H₃-orbit-constant rules for every f ≥ 25 | open | Open conjecture. Higher degrees at f=23 and f=24 are not shown impossible. | Not applicable | Not a theorem |
| Typed network of exact interfaces | locally hosted dossier | Exact maps among VII–V–I–II and frozen SRSD; independent cubature junction. Not a joining theorem. | Proof text hosted. Dossier’s embedded verifier is not a separate artifact. | Not run (dossier verifier) |
| De-branded typed-map audit | locally hosted executable audit | Executable audit of de-branded maps. Not a joining theorem. Radial identities are algebraic, not physics. | Python 3.10.21 / SymPy 1.14.0 PASS: 32768 D₁₅ states, 32768×23 remainder reconstruction, seven negative paths | Named runtime PASS; not independent reproduction |
| Lean 4 geometric-provenance formalization | locally hosted Lean 4 package | Selected measurement of the specified marked construction. Not uniqueness among all measurements. Not a joining theorem. | Sources hosted and hash-matched to SHA256SUMS. lake build PASS on this archive 2026-09-13. Python axiom-coverage PASS: 147 reports. | lake build PASS on this archive; Python inventory PASS; not independent reproduction |
| Julia mathematical proof companion | locally hosted companion software | Written proofs and exact certificates corresponding to the 147 Lean declarations. Selected measurement, not uniqueness among all measurements. Not a joining theorem and not an additional source paper. | Author-supplied Julia 1.11.6 O0/O3 records, six identical certificate files, 147 indexed declarations, 97 polynomial zero residuals, complete canonical-domain reconstruction. 68 members hosted; SHA256SUMS match. Not a run performed by this site. | Not run on this archive |
| 15-bit successor search (not a replacement contract) | local audit | Exact I₁₅, q_min, signed-32 bound, extinction | Python runtime reference | Not a theorem |
| Omega/JPRV–SRSD terminology module v2.1.0 | local module | Portable JSON; first residual injective on D₁₅. v2.0.0 superseded. | Exact integer checks in the module | Not a theorem |
| Omega-Alpha July 2026 synthesis (not a source paper) | local package | Scoped consolidation of earlier drafts | omega_alpha_reference.py matched frozen report | Not a theorem |
| G6_RELEASE/1.1 cross-language replay | legacy G₆ | Not a theorem; historical packaging of named frozen outputs | Rust, Julia, TypeScript | Implementation PASS; not unification |
| Workbench recheck, 23 September 2026 | site workbench, not a source paper | No new theorem. Imported β comes from the active samples. The drawing frame follows the posed curve, including both feet. | Node v22.23.2: node scripts/workbench-recheck.mjs inside the 23 September tree. Thirteen zeros recover β 0. The hosted script is unchanged and is not a standalone run. Use /workbench/releases/omega-workbench-impl-2026-09-25-r2.zip. Transcript /workbench/repair-acceptance.txt. Not a browser session. | Named module checks PASS; not a theorem checker and not independent reproduction |
| Workbench implementation recheck, 25 September 2026 | 20:31:42Z snapshot, not the typechecked source | No new theorem. The 20:31:42Z ZIP checks the pre-annotation field decoder and geometry frame. It does not reproduce the 23 September source hashes and it is not the typechecked source. | Node v22.23.2, 2026-09-25T20:31:42Z, clean directory: node scripts/workbench-recheck.mjs exited 0. ZIP ed1f447b…. Transcript 6a19d26b…. Those checksums are unchanged. Historical transcript /workbench/repair-acceptance.txt is unchanged. | Implementation validation PASS on the snapshot; not a theorem checker and not independent reproduction |
| Workbench implementation recheck, typechecked source | typechecked source before numerical-range repairs | No new theorem. The r2 ZIP checks the typechecked field decoder and geometry frame before calculated-result rejection. Total degree 20 is accepted. Total degree 21 is rejected. | npm run typecheck exited 0 on that revision. Node v22.23.2, 2026-09-25T21:53:13Z, clean directory: node scripts/workbench-recheck.mjs exited 0. ZIP 939e4bab…. Transcript e1395d84…. Those checksums are unchanged. | Implementation validation PASS; not a theorem checker and not independent reproduction |
| Workbench numerical-range recheck | current site source, not a source paper | No new theorem. Scaled identity quaternions match. Radius 1e308 is not exported. A probe outside the stated disk does not produce a reconstruction value. | npm run typecheck exited 0. Node v22.23.2, 2026-09-26T00:56:45Z, clean directory: node scripts/workbench-recheck.mjs exited 0. ZIP ddca5249…. Transcript c18af618…. | Implementation validation PASS; not a theorem checker and not independent reproduction |
| TGA 2.1 supplementary verifier | local supplementary release, not a source paper | Trusted analytic schemas for Sβ∘Mr = Pr∘Sβ, F∘Mr = Dr∘F, Aitchison power laws, and necessity and sufficiency of the six-mark gap conditions. Binary64 agreement is not an exact membership certificate. Not a general theorem prover. | Python 3.10.21, 2026-10-01T12:54:41Z: python3 and python3 -O unittest, 16 tests, exit 0. Browser arithmetic matched the Python vectors. TGA 2.0 hashes unchanged. | Named runtime PASS; not independent reproduction |
| Supplement 1 workflows | local supplementary release, not a source paper | Canonical six-mark readout, Smith witness for a rank-losing integer layer, and exact degree-at-most-2 transfer. The cyclotomic degree is cited, not reproved. The sine is not evaluated. Not an amendment of Paper VII. | Python 3.10.21, 2026-10-15: python3 and python3 -O exit 0, logs byte-identical, checker SHA-256 09d2fa0e…. Formal execution was not run. | Named runtime PASS; not independent reproduction |
The historical checker download is not a standalone run. Open that download page. It keeps the unchanged 23 September script and points to the runnable ZIP omega-workbench-impl-2026-09-25-r3.zip (SHA-256 ddca5249…, Node v22.23.2 at 2026-09-26T00:56:45Z, transcript c18af618…). The r2 ZIP 939e4bab… and the 20:31:42Z snapshot ed1f447b… are unchanged. Implementation validation, not a theorem checker.
Website build identity is /quality/site-build.json (schema OMEGA_SITE_BUILD/2) and /quality/source-inputs.json. A verified git checkout names the commit and whether the tree was dirty when those source inputs were captured. A provider-declared commit was not checked out here. If neither is available, sourceRevision and workingTreeDirty are null; that is not a clean tree and not a dirty tree. sourceContentSha256 identifies the inventoried bytes. It is not the historical source ZIP commit e5b4edcb1928 and not a proof hash. buildGeneratedAt is the index clock. This record is not a checker execution and it does not change a historical execution date.
TGA 2.1 is the active supplementary verifier. It adds scaling identities, path-equality certificates, and a six-mark membership test. It is not a general theorem prover. Frozen tga-2.0-download.zip is unchanged. The new container is tga-2.1-download.zip. The mathematical atlas is a separate interactive page, not the typed-network dossier.
Website source snapshot omega-site-source-e5b4edcb1928.zip is commit e5b4edcb19280aa287d7aead5fc9dc05ded63b83 (55,762,271 bytes, SHA-256 251962e9…). The descriptor is SOURCE_RELEASE.json and the checksums are SHA256SUMS.txt. A later commit that only registers these links is not that snapshot. This is not a proof and it does not replace the r3, r2, or 20:31:42Z workbench ZIPs.
Implementation-conformance ledger (historical G₆)
Implementation conformance: the named implementation reproduced the specified reference outputs. This does not establish theorem validity, peer-review status, or independent mathematical reproduction.
Historical G₆ artifact: canonical record reproduced in Rust and Julia, with a pinned TypeScript browser replay. The G₆ Rust runner declares no third-party crates; cargo run --release --offline --locked is the named run. Implementation conformance of a frozen package, not a theorem joining the source theories.
Separate OMEGA_NAUGHT/1.0 conformance: Python, Julia, Java, and Rust.
| Language | Role | Status | Artifact |
|---|---|---|---|
| Rust | Historical G₆ canonical record | PASS | rust/src/main.rs |
| Julia | Historical G₆ canonical record | PASS | julia/run.jl |
| TypeScript | Pinned browser snapshot | PASS | Historical G₆ artifact replay |
| Language | Role | Status | Artifact |
|---|---|---|---|
| Python | OMEGA_NAUGHT/1.0 | PASS | omega_naught_conformance.py |
| Julia | OMEGA_NAUGHT/1.0 | PASS | omega_naught_conformance.jl |
| Java | OMEGA_NAUGHT/1.0 | PASS | OmegaNaughtConformance.java |
| Rust | OMEGA_NAUGHT/1.0 | PASS | omega_naught_conformance.rs |
Status of 10 ≤ f ≤ 21
Recovered exact certificate package for 10 ≤ f ≤ 21 is hosted under /cubature/. Combined file classI_exact_certificates_f2_f21.json (1,815,267 bytes, SHA-256 7b3945ad…). Historical verifier and r2 revision are hosted. Per-frequency records at /cubature/frequencies/f10.json through f21.json. Python 3.11.2 r2 PASS on this archive: 383 identities, 404 positive masses. Post-publication computation; does not enlarge Paper VI. Maximality is not claimed. Not peer review. Not independent third-party reproduction.
What this site does not claim
The archive makes no blanket claim of novelty, priority, peer review, or independent validation. Some displayed identities are standard or definitional and are included to make the computational specification self-contained. Any claim of originality must concern a specifically stated formulation or derived result and must be evaluated separately against the relevant prior literature.
Independent third-party reproduction: none recorded on this site unless a named reproduction and its public evidence are linked.
G₆ packaging is not a theorem joining the theories. The tuple exists definitionally because unrelated typed objects can always be placed in a tuple. A canonical hash proves equality of a named byte string. Aggregating separate maps into interfaces=PASS is a historical implementation outcome, not current source-map discipline.