# OMEGA NAUGHT — Canonical archive overview

OMEGA is a marked geometric construction: a circular seed of radius a with a one-arcsecond opening and bilateral golden extensions. Selected measurements of that construction yield G = aφ, C = aπ, Ω₀ = φ + π, and the contraction parameter λ = G / (G + C). OMEGA NAUGHT is Matthew Alan Newman’s public archive of that construction, the maps it enters, Papers I–VII, and paper 8. paper 8 is a Zenodo preprint at 10.5281/zenodo.22848213 (Arabic numeral on purpose; the manuscript heading says Paper VIII). Specified computational consequences should be independently reconstructible rather than merely described. The archive does not present the source 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. Publicly exposed artifacts are indexed with hashes and version information where supplied.

The canonical machine entry is /archive/MACHINE.json (schema OMEGA_NAUGHT_MACHINE/1.0). An agent should fetch that file first. It names objects, theorem names, dependencies, evidence kinds, artifact URLs, hashes, versions, reproduction commands, and claims_not_made. Independent claim axes stay independent. It is not a joining theorem and not a belief instruction.

This overview is a guide to the deployed archive. Theorem statements remain in the source works. Zenodo deposits remain the source records for deposited versions.

## How the site is organized

Primary navigation:

- OMEGA — the marked construction, selected measurements, maps, and reconstruction
- Construction — Paper VII geometry and Construct transfer record
- Mathematics — separately defined theories, SRSD as an external audit instrument, then legacy G₆ packaging
- Proofs — 147 indexed statements, Lean source, Julia companion
- Reproduce — verifier commands and files
- Charts — display of exact integer/rational series; binary64 pixels are not theorems

Additional locally hosted dossier (not a source paper):

- /network — 5 September 2026 typed-network proof dossier. Exact interfaces among Papers VII, V, I, II and frozen SRSD; independent cubature junction. A 9 September 2026 de-branded Python/SymPy script audits those maps. Not a joining theorem.
- /formal — Lean 4 formalization of a selected geometric-provenance measurement. omega_provenance v0.1 sources hosted and hash-matched to SHA256SUMS. lake build PASS on this archive 2026-09-13. Python axiom-coverage PASS (147 reports; axioms {propext, Classical.choice, Quot.sound}). Not an additional source paper, not a joining theorem, not peer review of Papers I–VII.
- /geometry/sampling-extension — Affine dynamics, sampling, and quadrature (20 September 2026). This-archive Python/Java match expected.txt. Author rust stdout hosted; Julia/Rust sources not received. Not a source paper.
- /geometry/boundary-reconstruction — Boundary reconstruction, volume-preserving transport, and an invariant cubic (PDF 21 September 2026; checker 2026-09-22.r2). Written proofs in the PDF and remarks. Exact Fraction J² spherical mean; remaining checker groups numerical. This-archive Python 3.10.21 / NumPy 2.2.6: TOTAL 30 PASS bound to checker ce3e5ea2…. Not a source paper.
- /geometry/boundary-reconstruction/ten-sample — Explicit ten-sample reconstruction. Nine specified icosahedral samples and one extra boundary value determine 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/pair-dynamics — interactive view of the experimental pair R(2πλ) ⊕ R(2πμ). Hypothesis layer; not a Paper VII theorem; not a universal anchor. Recorded drift E4: no universal privilege.
- /geometry/exploration-02 — additional six-mark consequences: perspective cross-ratio inverse, affine/projective decoding bounds, positive degree-three and no nonnegative degree-four baseline integration at the canonical seed. The 36-degree Gauss–Lobatto member is a related family instance. Canonical seed remains one arcsecond. Not a source paper.
- /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 records are not a site run. Not an additional source paper, not a joining theorem. Cubature 10–21 and 23–24 packages are hosted separately under /cubature/.
- /archive/MACHINE.json — canonical machine contract. Fetch first. Independent objects, not a joining theorem.

Footer: CC BY 4.0 for documents and data, MIT for code, ORCID, changelog, artifact index, and a report page.

## The source works

Seven Zenodo preprints and one Zenodo report. Paper V (OMEGA NAUGHT Ω₀) is the report; Papers I–IV, VI, VII, and paper 8 are preprints. They are not one theorem. Zenodo remains the source record.

1. Paper I. Newman’s Quantized Omega Contraction Theorem. Exact contraction map, integer/rational representation, and parameter intervals. Concept DOI 10.5281/zenodo.21830284.
2. Paper II. Ordered Euclidean Remainders for Exact Orbit Reconstruction and Parameter Intervals in Rational Floor Maps. Quotient and remainder data, forward and reverse reconstruction, finite-domain agreement and first-disagreement boundaries. 10.5281/zenodo.21843485.
3. Paper III. Scale-Resolved State Dynamics: Foundational Closure Specification. Frozen SRSD v1.0, used as an external arithmetic and resolution-audit instrument. “Foundational closure” means internal sufficiency of that specification, not a new foundation for mathematics. 10.5281/zenodo.21927248.
4. Paper IV. Radial Icosahedral Sampling and Its First Harmonic Obstruction: Exact degree-six activation and Euler–Maclaurin coefficients. 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. 10.5281/zenodo.22033917.
5. Paper V. OMEGA NAUGHT Ω₀, a Zenodo report. Definitions and identities associated with Ω₀ = φ + π. 10.5281/zenodo.22048540.
6. Paper VI. Exact Positive Icosahedral Cubature on Fixed Radial Class-I Grids: Quadratic-Field Interoperability, Exact Replay, and Typed Compatibility with Papers I–V. Positive H₃-orbit-constant cubature for 1 ≤ f ≤ 9. Typed compatibility with I–V is interoperability, not logical dependence. 10.5281/zenodo.22068873.
7. Paper VII. OMEGA Marked Reference Geometry: Construction-to-Curve Theorem for a One-Arcsecond Circular Seed with Bilateral Golden Extensions, version 1.6. 10.5281/zenodo.22233573. Concept DOI 10.5281/zenodo.22233572.
8. paper 8. Exact Arithmetic Certification of Geometric Reconstruction and Transport. Zenodo preprint 10.5281/zenodo.22848213. Site label is paper 8; the manuscript heading says Paper VIII. Companion ZIP cbdf7a29… is the corrected release; review hash 5e47b7d5… is the original submitted archive, a version difference, not a corrupted upload. A localized-flow audit of Theorems 7.3–7.6 and 8.6 is hosted at /papers/8/audit-01; it is additional work, not an additional source paper, and not peer review.

Each deposited Roman work has a landing at /papers/i … /papers/vii. paper 8: /papers/8.

## Architecture: four theories, one instrument, one historical package

Mathematical Branches draws the live architecture as follows.

Four separately defined theories:

- Dynamics (Papers I–II). Newman’s Omega map Ω(v) = 1 + λ(v − 1) with the declared parameter λ = φ/(φ + π), and the rational floor recurrence d_{k+1} = ⌊Q d_k / S⌋ with ordered Euclidean residuals ρ_k = Q d_k − S d_{k+1}. Related but not source papers: a 15-bit successor search, terminology module v2.1.0, and a July 2026 Omega-Alpha synthesis with no DOI.
- Cubature (Papers IV and VI). Exact linear algebra over ℚ(√5) on frozen Class-I grids. Independent of Ω₀ and of SRSD.
- Ω₀ (Paper V). Reference identities. Definition of a constant does not make it privileged outside the constructions that use it.
- Geometry (Paper VII). Euclidean marked kernel. Does not join Papers I–VI.

Then:

- SRSD (Paper III) is an external audit instrument applied to other theories. It is not a fifth source theory in the same sense, and it is not the foundation of a combined object. Applying SRSD to a cubature vector only proves generic exact representability D_s E_s(C) = C, which holds for every admissible numerical object. That is not a mathematical connection between cubature and SRSD.
- Legacy G₆ is historical packaging of Papers I–VI. 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.

The live browser replay targets G6_RELEASE/1.1. The artifact index also records G6_RELEASE/1.2 as historical/superseded. The frozen bytes of both envelopes are unchanged.

## Cubature ranges, kept separate

Proved, certified, and open ranges are listed as separate rows. They are not one theorem for f = 1, …, 24.

- Harmonic obstruction (Paper 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. Finite-frequency computations and a degree-six Euler–Maclaurin expansion are also given.
- 1 ≤ f ≤ 9 (Paper VI). Theorem: strictly positive H₃-orbit-constant rules; frozen CSV replayed inside G6_RELEASE/1.1.
- 10 ≤ f ≤ 21. Recovered exact certificate package hosted (classI_exact_certificates_f2_f21.json, SHA-256 7b3945ad…). Per-frequency records at /cubature/frequencies/f10.json through f21.json. Python 3.11.2 r2 PASS on this archive: 383 identities. Post-publication computation; does not enlarge Paper VI; maximality is not claimed.
- Frequency 22. Certificate-backed: positive degree 69; obstruction at 70/71. Python and Julia pass separately.
- f = 23 and f = 24. Locally hosted existence theorem: X₂₃ exact through degree 71, X₂₄ through degree 75. No maximality. Original certificate (1,407,835 bytes, SHA-256 e886510e…), verifier (6,818 bytes, SHA-256 41d83727…), ledger (10,905 bytes, SHA-256 4d56dc2c…), and launcher replay_recovered.py are hosted under /cubature/. Python 3.11.2 replay PASS on this archive; python3 -O refused. Printed finite-audit extract and LaTeX-emitted JSON 3ecbf665… remain distinct. Does not enlarge Paper VI.
- f ≥ 25. Open conjecture. Higher degrees at f = 23 and f = 24 are not shown impossible.

Frequency 22 has its own audit. An earlier summary named (22, 0, 0) as the minimum positive degree-69 orbit after binary64 conversion. Exact comparison in ℚ(√5) gives (11, 11, 0). The theorem and the two canonical solution digests did not change. The unique forced negative degree-71 mass remains at (21, 1, 0).

The same binary64 cancellation affected candidate f=23,24 minima. Format CLASS_I_CUBATURE_F23_F24_EXACT/1.1 replaces floating minimum selection with exact comparison. Exact weights, identities, positivity, and rank witnesses are stated to be unchanged.

A 5 September 2026 locally hosted dossier types exact interfaces among Papers VII, V, I, II and frozen SRSD, with an independent Q(√5) cubature junction. It is a typed diagram of maps between distinct theorem packages, not a joining theorem. A 9 September 2026 Python/SymPy script, typed_connection_proof.py, audits those maps in de-branded form: symbolic identities, exact integer/rational claims, and exhaustive floor equality on D={0,...,32767} using rational enclosures of √5 and π. Homothety of the six kernel marks is Euclidean scaling, not a unique shell. Radial template identities in the script are algebraic and are not a physical theorem. Page: /network.

A Lean 4 formalization of a selected geometric-provenance measurement is hosted at /formal. The package names Omega.central_provenance_theorem as the main result and takes the construction and measurement rule as explicit inputs: it does not assert uniqueness of the measurement among all possible measurements of an unmarked outline. omega_provenance v0.1 sources, lakefile, lean-toolchain, and evidence logs are hosted and match SHA256SUMS. lake build PASS on this archive 2026-09-13 (build.log SHA-256 ddc10580…; axioms.log SHA-256 82dae064…, byte-identical to the author log). scripts/check_axioms.py PASS: 147 theorem reports; axioms {propext, Classical.choice, Quot.sound}. That checker is not a proof oracle. This is not an additional source paper, not a joining theorem, not peer review, and not independent reproduction of Papers I–VII. Cubature 10–21 and f=23,24 exact packages are hosted separately under /cubature/; they are not produced by this Lean run.

A Julia mathematical proof companion v0.1 (software 0.1.0) is hosted at /julia-proofs. Sixty-eight recovered members match SHA256SUMS (74,508 source lines). It preserves the 147 declarations and the 29 Lean reference files, with written proofs and exact certificates. Author-supplied Julia 1.11.6 O0/O3 records are not a run performed by this archive. Cubature 10–21 and f=23,24 packages are hosted separately under /cubature/. Companion software, not a source paper and not a joining theorem.


The 10 ≤ f ≤ 21 results are no longer the exception to “indexed with hashes and versions”: the recovered combined certificate, historical verifier, r2 revision, and per-frequency records are hosted and hash-pinned. CLASS_I_CUBATURE/1.1 records that package as publicly_hosted_exact_certificate_package. They remain a post-publication computation, not Paper VI.

## Geometry (Paper VII)

Paper VII is separately defined Euclidean geometry. Input: a center α, a scale a > 0, an oriented plane, and an ordered pair d, b on the radius-a circle with central separation ε = 1″. The chord direction is then extended outward by a and by aφ on both sides. Those extensions are declared rules, not consequences of the arcsecond. The six marks lie on the line y = −a cos(ε/2), not on a diameter through α. The complete curve is the existing arc-and-feet object Γ: the retained major arc joined to two outward feet. The seed chord of length 2as is not a segment of Γ. This is the Paper VII construction, not a different object.

The Construction-to-OMEGA theorem proves that these data, together with the selected major arc, determine exactly one marked kernel: six baseline marks, every distance among those marks and from them to α, all fifteen pairwise center angles, rank-two Gram data, and an oriented frame in ℝ³.

The Geometry page reconstructs that object and explains it as a typed transfer gate:

- Shape: a marked planar kernel (major arc, six collinear marks, distinguished center). Not a volume object and not a unique outer shell.
- Structure: six baseline marks, every distance among them and from them to α, all fifteen pairwise center angles, rank-two Gram data, and an oriented frame in R³.
- Transfer gate: Construct mode and OMEGA_KERNEL_NODES/1.0. A receiving system can rebuild the unique labeled similarity class in R³ from the node log. Kernel data may pass; the adopted scaffold only if separately jumped; exclusions (outer circle, clam, shell, extrusion, signed depth) and 12° display coordinates do not travel. The scalar map (φ, ε) ↔ (τ_Ω, λ) is a side channel of numbers, not a reconstruction of the geometric realization.
- Construct mode: fifteen sourced presentation stages, Step/Back/Replay, kernel jump, optional scaffold jump. Replay cannot enter the scaffold. Node-log export is OMEGA_KERNEL_NODES/1.0 at true ε = 1″; a 12° picture is display-only.
- a six-ray fan (the true 1″ seed is invisible; a schematic opening can be toggled)
- an orbitable Oriented ℝ³ view: major arc C(t), golden feet, six center rays, Definition 11.1 scaffold, Space / Echo / Loco
- a notebook outer-scaffold diagram
- fourteen browser identities that hold as Float64 evaluations of the closed formulas

The companion Julia script materializes the same unit-scale object: marks, arc, feet, 15-angle fan, Gram factorization Ḡ = c² 11ᵀ + zzᵀ, and constructed scaffold widths. rank(Ḡ) = 2 is a numerical check of the symbolic identity UᵀU = diag(6c², ∑ zᵢ²). The proof is in the paper.

The metric outer scaffold is an adopted transcription of notebook a, aφ annotations. Remaining outer curves and signed-depth completion are not claimed.

A related page, Drift, hosts a falsification control: an experimental 4D OMEGA rotation is not a privileged memory anchor. That experiment is not a theorem of Paper VII.

## How indexed claims are stored

Each indexed claim record uses independent axes that are never collapsed into one “verified” badge:

- mathematical status (theorem, certificate-backed, adopted definition, conjecture, experimental control)
- evidence (paper proof, exact weights, named run, …)
- machine-checkable scope and its limits
- publication kind; peer-reviewed is false for these works
- independent third-party reproduction, currently recorded as none
- implementation conformance (pass / fail / not run / not applicable), with release id and expected output when pass

A Python PASS does not fill the peer-review cell. A hash match does not enlarge a theorem’s range. Not every sentence or unpackaged result has a complete database record.

## Audits, replay, and frozen bytes

The Audits page runs independent source-map checks in the browser, plus a pinned TypeScript replay of the historical G₆ canonical record. In a browser execution on 3 September 2026, the four source-map audits passed separately, the floor-map audit reported zero mismatches on D₁₅, and the historical G₆ replay completed with PASS, including 8/8 historical interfaces and the named canonical hash. The page initially displays NOT RUN; those PASS results belong to that execution, not to a permanent site state. The Geometry page’s fourteen reconstruction identities are likewise browser evaluations of closed formulas. These remain implementation and reconstruction results — not proof, and not independent third-party reproduction.

Reproduce lists the actual commands and files: cubature verifiers, SRSD cross-language listing, the Paper VII generator / verify_geometry.py / Julia script, Construct-mode node log and scalar certificate, the drift experiment, Omega-Alpha, successor search.

Artifacts is the inventory: filename, path, language, byte count, SHA-256 where supplied, which claims an item supports, and whether it sits inside frozen G₆.

## Other rooms

Older G₆ lab pages remain as provenance: Release, Contract, Proof-system audit, Global object, Exact field K, Q/S 15-bit bridge, Ledgers. They document frozen packages. They do not redefine the current architecture.

Supporting pages:

- Status — full matrix across papers, cubature ranges, G₆, and successors
- Changelog — site and deposit events
- Report — how to file an error or reproduction note
- Training module v2.1.0 — terminology, not a source paper; v2.0.0 is superseded and noindexed
- Omega-Alpha — July 2026 synthesis, no DOI, does not rewrite I–VII

Licensing is split: documents and data CC BY 4.0; site code MIT.

## How to read it

Start on Overview. The homepage figure is Frozen SRSD in ℝ³; the status snapshot and live G₆ replay sit below.

Then pick a theory or instrument:

- For cubature, open Cubature Audit, then the frequency-22 page for the certificate package, or X₂₃ and X₂₄ for the hosted existence theorem. Keep 1 ≤ f ≤ 9, 10 ≤ f ≤ 21, frequency 22, f = 23 and 24, and f ≥ 25 as separate claims.
- For the marked kernel, open Geometry, switch to Oriented ℝ³, drag to orbit, and read the fourteen identities. The companion list hosts the generator, JSON/OBJ, verifier, and Julia script.
- For the papers themselves, open Papers and follow the DOI. Local PDFs are mirrors; Zenodo is canonical for deposited versions.

If a page says it is not a theorem checker, it means it. Rank, floating-point comparison, and browser self-checks evaluate a binary64 image of published formulas. The proofs remain in the preprints.