# OMEGA NAUGHT — Mathematical papers and exact reproducibility archive

Matthew Alan Newman. ORCID 0009-0007-5468-3125.

Canonical scoped overview: /archive/OVERVIEW.md

Canonical machine entry (fetch this first): /archive/MACHINE.json
Schema: OMEGA_NAUGHT_MACHINE/1.0
Raw indexes: /archive/claims.json · /archive/artifacts.json · /archive/conformance.json · /archive/papers.json

OMEGA is a marked geometric construction: a circular seed of radius a with a one-arcsecond opening and bilateral golden extensions. The declared construction and selected measurement rule yield G = aφ, C = aπ, Ω₀ = φ + π, and λ = φ/Ω₀. This archive packages the construction, the measurements, the maps they enter, the proofs, and the reconstruction data so the specified consequences can be independently rebuilt. It indexes eight 2026 source works. It does not present those works as one theorem: conformance testing is not proof, historical G₆ packaging is not a joining theorem, none of the works is peer reviewed, and no independent third-party reproduction is currently recorded.

The live site is https://omega-naught.grok.me. The public site exposes no accounts or login interface.

This site accompanies seven Zenodo preprints and one Zenodo report, together with their computational artifacts.
It separates theorem statements and proofs, exact machine-checkable certificates,
and cross-language implementation conformance.

Conformance tests do not establish theorem validity.

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. Not peer reviewed unless a specific work is
later linked to a peer-reviewed publication.

Independent third-party reproduction is currently recorded as none.

The source works are not one theorem. Cubature results do not logically depend on Ω₀
or SRSD. G₆ is a historical reproducibility package of Papers I–VI, not a theorem joining the theories.
SRSD v1.0 is used only as an external arithmetic and resolution-audit instrument.

The primitive SRSD encoder and decoder use coordinatewise quotient-residual decomposition through the floor function, a standard mathematical construction. SRSD does not claim to invent division with remainder or to provide a new foundation for mathematics. "Foundational closure" refers only to the internal sufficiency of the frozen SRSD v1.0 specification for the systems it defines.

The archive makes no blanket claim of novelty, priority, peer review, or independent validation. Any claim of originality must concern a specifically stated formulation or derived result and must be evaluated separately against the relevant prior literature.

## Naming and vocabulary

OMEGA NAUGHT, SRSD, and other project-specific names identify author-defined constructions, specifications, and artifacts in this archive. These names do not claim invention of the underlying standard operations, confer mathematical privilege, establish priority, or imply general adoption.

This site is the maintained index and reproducibility archive for artifacts hosted or linked here. Zenodo deposits remain the source records for deposited works. This site is not an authoritative map of the wider mathematical subject.

Mathematical claims are stated in standard terminology and evaluated individually through their definitions, proofs, exact certificates, computations, independent scrutiny, and relevant prior literature. Proofs, certificates, and computations apply only within their declared scope. Novelty and priority must be assessed claim by claim against prior literature.

Names locate objects. Specifications define them. Valid proofs establish theorems; certificates and computations verify their stated finite or implementation scope.

Standard glosses:
- SRSD: coordinatewise quotient-residual resolution at a declared positive scale, d_i = floor(y_i/s_i), rho = y - Delta_s d, y = Delta_s d + rho.
- Newman's Omega map: Omega(v) = 1 + lambda(v-1), lambda = phi/(phi+pi). The contraction parameter lambda is fixed by the specified construction and selected measurement rule G = a phi, C = a pi, Omega_0 = phi + pi, lambda = G/(G+C). It is not an independently chosen free parameter.
- Omega/JPRV: the rational floor recurrence d_{k+1} = floor(Q d_k / S) with ordered Euclidean residuals rho_k = Q d_k - S d_{k+1}.
- Omega_0 = phi + pi: an author-defined constant used within the specified constructions. Its definition does not make it privileged, universal, or canonical outside those constructions.

## Central purpose

Package mathematical claims so specified computational consequences can be independently reconstructed.

## Primary routes

- /archive/MACHINE.json canonical machine contract (fetch this first; schema OMEGA_NAUGHT_MACHINE/1.0; independent objects, not a joining theorem)
- / OMEGA object, construction, selected measurements, maps, proofs, reconstruction
- /geometry construction, measurements, Construct transfer record
- /proofs navigator for 147 indexed statements, Lean source, Julia companion
- /omega0 normalized scalars from the selected measurement rule
- /dynamics contraction, certified floor, reconstruction
- /cubature cubature claims
- /cubature/f22 frequency-22 audit
- /cubature/f23-f24 X23 degree 71 and X24 degree 75 existence theorem plus recovered exact certificate (e886510e… / 41d83727… / 4d56dc2c… hosted; python3 -O refused)
- /network typed network of exact interfaces (5 September 2026 locally hosted dossier plus a 9 September 2026 de-branded executable audit of the typed maps; not a joining theorem)
- /formal Lean 4 geometric-provenance formalization (omega_provenance v0.1 sources hosted and hash-matched to SHA256SUMS; lake build PASS on this archive 2026-09-13, transcripts at /formal/evidence/current/; Python axiom-coverage PASS, 147 reports, axioms {propext, Classical.choice, Quot.sound}; selected measurement, not uniqueness among all measurements; not an additional source paper and not a joining theorem; cubature 10–21 and f=23,24 packages are hosted separately under /cubature/)
- /julia-proofs Julia mathematical proof companion v0.1 / software 0.1.0 (68 members hosted and hash-matched to SHA256SUMS; written proofs and exact certificates corresponding to the Lean source; author-supplied Julia 1.11.6 O0/O3 evidence is not a site run; not an additional source paper and not a joining theorem; cubature 10–21 and 23–24 packages are hosted separately)
- /branches mathematical branches
- /dynamics Papers I–II
- /dynamics/successors 15-bit successor search (not a replacement contract)
- /dynamics/training canonical terminology module v2.1.0 (current; not a source paper). v2.0.0 is a superseded historical artifact (noindex, noarchive).
- /dynamics/omega-alpha July 2026 Omega-Alpha synthesis (not a source paper; no DOI)
- /geometry Paper VII v1.6 marked reference geometry (construction-to-curve theorem, center-angular fan, exact R³ realization; remaining notebook completion not claimed). The marked kernel is a planar transfer record: unique labeled similarity class in an oriented plane; Construct mode / OMEGA_KERNEL_NODES/1.0 export the kernel, not a unique shell. Orbitable major-arc reconstruction and Construct mode (fifteen-stage presentation; true-ε export; 12° display is non-normative).
- /geometry/drift OMEGA drift experiment v0.1 (falsification control; not a source paper)
- /geometry/boundary-reconstruction aligned 12-node harmonic-quadratic decoder, volume-preserving transport of J, thirteen-sample decoder on F_A; Ψ²=Φ_{τ², 2 log φ}; checker 2026-09-22.r2 exact J² mean plus numerical remainder; not a source paper
- /dynamics/floor-kernel explanatory supplement for the existing 399 rational floor map. Not a new theorem and not a source paper. No ZIP was supplied.
- /geometry/boundary-reconstruction/ten-sample explicit ten-sample decoder for H≤2 ⊕ ℝJ. Written note Ten_Sample_Reconstruction.md, 9973 bytes, SHA-256 967bc8af…. Checker TOTAL 13 PASS. Not an additional source paper and not a DOI.
- /geometry/exact-interface exact field interface. F=Q(√5) inside K=Q(√2,√3,√5), degree 8. Eight rational coefficients, N_{K/Q}=(N_{F/Q})^4, rational reflection of (2,3,0), and the split between injective samples and encoded records. Checker TOTAL 29 PASS. Not an additional source paper. Does not place π in K and does not enlarge Paper VI.
- /srsd Paper III frozen external SRSD audit instrument
- /omega0 Paper V reference identities
- /papers Papers I–VII plus paper 8
- /papers/i /papers/ii /papers/iii /papers/iv /papers/v /papers/vi /papers/vii deposited landings
- /charts exact depth layers, BigInt floor images, twelve-node moments, deformation eigenvalues (display; not a theorem)
- /verify independent source-map audits plus historical G₆ artifact replay
- /reproduce verifiers and run commands. Software-dependency transcription: /docs/OMEGA_NAUGHT_Software_Dependencies.docx (G₆ Rust has no third-party crates; cargo --offline --locked). Not an additional source paper.
- /integration Legacy G₆ reproducibility release (historical packaging, not current architecture)
- /status publication, review, certificate, and reproduction axes
- /report error and reproduction reports
- /artifacts hosted packages, versions, hashes, licenses, and supersession status
- /changelog site and deposit events
- /g6/spec.html legacy G₆ specification (noindex provenance copy; frozen 1.1/1.2 bytes unchanged)

## Source DOIs and records

- I  10.5281/zenodo.21830284 (concept DOI, all versions; v1 record https://zenodo.org/records/21830285; version DOI 10.5281/zenodo.21830285 resolver currently unavailable)
- II 10.5281/zenodo.21843485 (deposit v1; PDF filename revision label v2 is not a separate deposit; record https://zenodo.org/records/21843485)
- III 10.5281/zenodo.21927248 (document 1.0; deposit v1; record https://zenodo.org/records/21927248)
- IV 10.5281/zenodo.22033917 (deposit v1; record https://zenodo.org/records/22033917)
- V  10.5281/zenodo.22048540 (Zenodo report; document 1.0; deposit v1; record https://zenodo.org/records/22048540)
- VI 10.5281/zenodo.22068873 (document 1.3; deposit v1; record https://zenodo.org/records/22068873)
- VII 10.5281/zenodo.22233573 (version 1.6; concept DOI 10.5281/zenodo.22233572; record https://zenodo.org/records/22233573; local mirror /papers/OMEGA_Paper_VII_v1.6.0.pdf; Zenodo publication-date field 2026-09-01, manuscript cover 2026-09-02)
- 8  10.5281/zenodo.22848213 (site label paper 8; manuscript heading Paper VIII; version 1.0; record https://zenodo.org/records/22848213; local PDF /papers/OMEGA_Paper_VIII_Final.pdf)

## Research-branch boundaries

- Dynamics: Papers I–II
- SRSD: Paper III, frozen
- Cubature: Papers IV and VI; independent of Ω₀ and SRSD. 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.
- Ω₀: Paper V report
- Geometry: Paper VII; construction-to-curve theorem, complete center-angular fan, exact R³ realization; remaining notebook completion not claimed. The kernel is a typed transfer record of the proved planar object, not a unique outer shell.
- Integration: G₆ typed assembly of Papers I–VI; not a joining theorem

## Proven versus conjectural cubature ranges

- Paper VI v1.3: 1 ≤ f ≤ 9
- Recovered exact certificate package for 10 ≤ f ≤ 21: hosted, hash-pinned, post-publication; maximality is not claimed; does not enlarge Paper VI
- Frequency 22: positive degree 69; no nonnegative degree 70 on fixed X_22. Is supplement to 10.5281/zenodo.22068873. Does not enlarge Paper VI. Current Python verifier is website revision r2 (refuses python3 -O); original zip/SHA256SUMS member is unchanged.
- f = 23 and f = 24: locally hosted existence theorem plus recovered exact certificate; X_23 exact through degree 71, X_24 through degree 75. Original bytes e886510e… / 41d83727… / 4d56dc2c…. No maximality. Does not enlarge Paper VI.
- f ≥ 25: open conjecture. Higher degrees at f=23 and f=24 are not shown impossible.

## Reproduction entry points

- /archive/MACHINE.json object-scoped commands, hashes, evidence kinds, and claims_not_made
- /reproduce verifier commands
- /julia-proofs Julia proof companion v0.1 commands (68 members hosted; SHA256SUMS match; author-supplied Julia 1.11.6 evidence is not a site run)
- /verify pinned TypeScript G₆ replay
- /cubature/f22 Python and Julia CERTIFICATE PASS transcripts. Current Python download is Omega_Naught_ClassI_f22_exact_verifier.r2.py (python3 -O guard). Original Omega_Naught_ClassI_f22_exact_verifier.py remains the zip/SHA256SUMS member.
- /cubature/f23-f24 locally hosted X₂₃/X₂₄ existence theorem plus recovered exact certificate (python3 replay_recovered.py; python3 -O must refuse)
- /cubature/classI_exact_certificates_f2_f21.json recovered f=2–21 exact certificate (python3 verify_classI_f2_f21.r2.py … --ledger …)
- /cubature/frequencies/f10.json … f21.json, f23.json, f24.json per-frequency derived records
- /geometry/kernel/omega_reference_geometry.py companion reconstruction generator (defaults write CWD; use --json/--obj for independent generation)
- /geometry/kernel/verify_geometry.py numerical conformance (NumPy; expects generated/JSON and OBJ; not a theorem checker)
- /geometry/kernel/omega_kernel_oriented_r3.jl Julia binary64 materialization of the kernel, 15-angle fan, Gram factorization, and outer scaffold
- /geometry/construct/OMEGA_Kernel_Construct_v1.1.0.pdf technical release paper for Construct mode (not a source work)
- /geometry/construct/omega-kernel-nodes.unit.true.json canonical OMEGA_KERNEL_NODES/1.0 unit log
- /geometry/construct/verify_scalar_interface_exact.py exact scalar Ψ/Φ certificate (scalar layer only)
- /network/typed_connection_proof.py de-branded typed-map audit (Python 3 + SymPy; D_15 floor certificate with rational enclosures of √5 and π; not a joining theorem)
- /network/typed_connection_proof.runtime.txt named runtime of that audit
- /formal/README.md Lean 4 package README
- /formal/Omega/Main.lean Omega.central_provenance_theorem
- /formal/SHA256SUMS package file digests (hosted sources match)
- /formal/scripts/check_axioms.py axiom-report coverage checker (not a proof oracle)
- /formal/check_axioms.runtime.txt named Python 3.10.21 PASS: 147 reports; axioms Classical.choice, Quot.sound, propext
- /formal/evidence/axioms.log author axiom-audit transcript
- /geometry/drift Python/NumPy falsification control

## Artifact manifest locations

- /archive/MACHINE.json
- /archive/claims.json
- /archive/artifacts.json
- /archive/conformance.json
- /archive/papers.json
- /artifacts
- /CITATION.cff
- /citations/omega-naught.bib
- /releases/G6_RELEASE_1.1/RELEASE.json
- /releases/G6_RELEASE_1.2/RELEASE.json
- /cubature/CLASS_I_CUBATURE_1_1.json
- /cubature/RELEASE.json
- /conformance/OMEGA_NAUGHT_1_0.manifest

## Frozen release

G6_RELEASE/1.1 remains frozen at /releases/G6_RELEASE_1.1/RELEASE.json
SHA-256 f63ae1c2d595b4805ba9d36687a828fd87200febe01079b184723f7d5eaf746c
Canonical record SHA-256 92187a21a7ad8ecef384eda3a654aaa49daf46a31422e168eb5871d218c949fa
cubature_frequency_max=9 inside that record.

G6_RELEASE/1.2 is the license-and-documentation correction at
/releases/G6_RELEASE_1.2/RELEASE.json
SHA-256 0235d4615e5a37bff3baf3c98b0f751274fcb584ae4fe6f73174050694b4532f
Docs/data CC BY 4.0; source MIT. Canonical record SHA-256 unchanged.
External archival pending.

Runtime scopes (do not collapse):
- G₆ canonical record reproduced in Rust and Julia, with a pinned TypeScript browser replay
- Separate OMEGA_NAUGHT/1.0 conformance: Python, Julia, Java, Rust

CLASS_I_CUBATURE/1.1 is a proposed additive extension and does not rewrite 1.0.

## Personal notes (not claims)

- /notes — blank notes, to be written
- /notes/notes.md — markdown twin

These pages are not source papers and are not indexed in MACHINE.json.

## Synthesis 1.0 review (not a source paper)

- /synthesis — focused second-review page
- /synthesis/omega_synthesis_claim_check.py — stdlib companion checker (SHA-256 9397c530…)
- /synthesis/omega_synthesis_claim_check.stdout.txt — this-archive Python 3.11.2 PASS in normal/-O/-OO
- /synthesis/ENVIRONMENT.json — this-archive run record
- /synthesis/REVIEW.md — written corrections and Farey certificate

OMEGA_Synthesis(3).pdf was not supplied. Claimed SHA-256 01f0d19c… is unverified here.
Not Lean, not cubature, not an additional source paper.

## paper 8 Proof Audit 01 (not a source paper)

- /papers/8/audit-01 — localized forced incompressible flow, Theorems 7.3–7.6 and 8.6
- /papers/8/audit-01/OMEGA_Localized_Flow_Complete_Proof.pdf — 306508 bytes, SHA-256 e3985cb5…
- /papers/8/audit-01/OMEGA_Paper_VIII_Proof_Audit_01_Forced_Flow.md — SHA-256 4617fd3f…
- /papers/8/audit-01/independent_checks.py — SHA-256 380c32bd…
- /papers/8/audit-01/independent_results.json — supplied Python 3.12.14 results
- /papers/8/audit-01/ENVIRONMENT.json — this-archive Python 3.10.21 / SymPy 1.14.0, 29 groups PASS

TEX source and original independent_checks.log were not supplied.
Additional work, not a relabeling of the adversarial review, not peer review, not a blowup theorem.

## Affine dynamics, sampling, and quadrature (not a source paper)

- /geometry/sampling-extension
- PROOFS.md — 28146 bytes, SHA-256 99d6b58e…
- verify.py / Verify.java this-archive match expected.txt 7ef550cb…, TOTAL 21 PASS
- Author rust stdout hosted (same SHA-256 as expected.txt). verify.jl and the Rust crate not received

## OMEGA pair dynamics (not a source paper)

- /geometry/pair-dynamics — interactive R(2πλ) ⊕ R(2πμ)
- λ = φ/Ω₀, μ = π/Ω₀, λ+μ = 1, torus slope π/φ
- Hypothesis layer used by /geometry/drift. Not a Paper VII theorem. Not a universal anchor.

## OMEGA object exploration 02 (not a source paper)

- /geometry/exploration-02 — projective recognition, landmark decoding, baseline integration
- /geometry/exploration-02/OMEGA_Object_Exploration_02.md — 20104 bytes, SHA-256 12f3106a…
- /geometry/exploration-02/OMEGA_Object_Exploration_02.py — 15581 bytes, SHA-256 4c1b3cfa…
- /geometry/exploration-02/this_archive_exploration_results.json — this-archive Python 3.10.21 / SymPy 1.14.0, 21 groups PASS

Canonical seed remains one arcsecond. The 36-degree Gauss–Lobatto member is a related family instance.
Named author hashes 6c35365d… / e50f3443… were not in the upload.
Not a camera-noise model. Does not enlarge Paper VII or paper 8.

## Navier–Stokes geometric extension (not a source paper)

- /extension — OMEGA–Navier–Stokes geometric transport 3.0
- /extension/OMEGA_NS_Geometric_Extension_v3.pdf — received SHA-256 98a3f747… (does not match notes claim 42802e00…)
- /extension/verify_omega_ns.py — SHA-256 f054d343…
- /extension/audit_omega_sources.py — SHA-256 7bb07240…
- /extension/test_omega_auditor.py — SHA-256 9cae8025…
- /extension/verify_omega_ns.stdout.txt — this-archive 62 exact checks PASS
- /extension/v1/ — historical Extension 1.0 (34 checks)

Not Paper VII rewritten. Not a blowup result. SRSD remains frozen. No fresh Lean or Julia run.

## Floor-map fibers (not a source paper)

- /fibers — consecutive fibers, depth layers, 32-block seed cover
- /fibers/OMEGA_Fiber_Proofs_and_Grok_Handoff.docx — written theorems
- /fibers/verify_handoff.py — SHA-256 9f7179c5… (matches Appendix C)
- /fibers/verify_handoff.stdout.txt — this-archive RESULT PASS

Reported 0.2 cover/certificate files were not supplied. Seed blocks, not cosets.



