The Omega–Alpha Framework: Affine Contraction, Exact Quotient–Remainder Witnesses, and Deterministic Multiscale Geometric Probes

Matthew Alan Newman

July 2026 synthesis · not a source paper

Omega-Alpha scoped consolidation

This page hosts a document labeled July 2026, a scoped synthesis of earlier Omega drafts. It is not a source work and does not rewrite Papers I–VII or paper 8. The current Zenodo records I–VII have publication dates in August and September 2026; they do not support a contemporaneous July deposit of those seven works. Any seven-work count attached to this page is a later retrospective annotation on this archive, not a July 2026 deposit history. paper 8 is a later preprint (19 September 2026). Physical, cryptographic, and generic-compression interpretations from older drafts are excluded here.

Status of this package

Archival fixation: locally hosted only. No DOI is assigned. Same-site hashes identify the currently published bytes but do not independently timestamp or externally preserve them.

Inventory of the nine hosted payloads; LaTeX source is not supplied.

Authority inside the package: preprint PDF, then omega_alpha_reference.py, then the frozen JSON report, then the claim-verification matrix, then the version history. Historical Figshare records cited in that history remain historical; they are not a DOI for this package.

The synthesis uses “boundary-witness” for the ordered remainder ledger. Later terminology (internal revision 5 / training module v2.1.0) treats that ledger as a complete execution trace: on the 15-bit domain the first residual already determines the limb, and N = 23 is protocol padding after extinction rather than a minimal encoding.

Current vocabulary: canonical terminology module. Frozen SRSD: Paper III. Finite successor search: 15-bit successor audit. Cubature is independent: cubature audit.

Finite integer kernel

Protocol constants $S=10^{15}$ , $Q=339949771344778$ , $N=23$ . The quotient map

d_{k+1}=\Bigl\lfloor\frac{Q d_k}{S}\Bigr\rfloor,\qquad \rho_k=Q d_k-S d_{k+1},\qquad 0\le\rho_k<S

is many-to-one. Reconstruction uses the terminal value and the ordered remainder ledger. Site-local execution of the frozen Python source reproduced the published JSON report byte-for-byte, including exhaustive checks on $D_{15}=\{0,\ldots,32767\}$ .

Named finite results from omega_alpha_reference.py. Exactness begins after S, Q, and N are fixed.
Check	Result
15-bit reverse round trips	32768 / 32768
Full witness identity	32768 / 32768
Terminal state after 23 steps	0 for every limb
Maximum steps to zero on D₁₅	10
Byte codebook T(0), T(65), T(255)	33386, 1118443, 4290149
Byte increments	{16692, 16693, 16694}
Byte compact-JSON SHA-256	`0fc8922c2d84cfd965c9fe310070e00ffd86ed76117584a9ff2faa85136a0c47`
16-bit codebook last tick	1094021571
16-bit u64be map SHA-256	`c0d90669759bfe4c2efc7692ba74496249f9ff1f63748166afb19232724f8ff8`
Q re-derived at 70 and 90 digits	339949771344778
Overall	PASS

The Python reference checks declared finite identities after the integer protocol constants are fixed. Decimal evaluation of φ, π, and λ is a reproducibility check, not symbolic exactness. Passing tests do not establish theorem validity for Papers I–VI, peer review, or independent third-party reproduction.

Corrected ellipse-coordinate anchors

The table stores Ramanujan’s second-approximation multiplier truncated to 15 decimal places, then a coordinate computed from the declared literal $\lambda_{15}=0.339949771344778$ . It is not a closed form for the exact ellipse perimeter. The geometric line-segment limit $4/\pi$ and Ramanujan’s limit $14/11$ remain distinct.

Selected corrected ellipse rows. The r = 2 multiplier 1.027976282991305 replaces an inconsistent earlier value.
Aspect ratio	Multiplier w	Coordinate v
1	1.000000000000000	1.000000000000000
φ	1.013981238987019	1.041127366939274
2	1.027976282991305	1.082295342869730
10	1.175989279709379	1.517692007890454

Claim matrix

Sixty historical claims were classified. The CSV is the complete artifact. Counts below are a summary, not a substitute.

Disposition counts in claim_verification_matrix.csv (60 rows).
Status	Count
PROVED	24
RETAIN	9
REJECT	10
PROVED_AND_VERIFIED	4
EXHAUSTIVELY_VERIFIED	3
VERIFIED	3
NOT_ESTABLISHED	2
PROVED_AND_TESTED	1
PARTIAL	1
NOT_PROVED	1
NOT_REQUIRED	1
UNSUPPORTED	1

Claims rejected, unsupported, or not established. These are not theorems of this archive.
ID	Claim excluded or rejected
C009	Bit-perfect floating-point invertibility
C026	N = 23 universally necessary for reversibility
C033	A hash proves reversibility or authorship
C040	Byte decoder C = 16693 on the full 16-bit domain
C042	Repeated floor expansion is the exact inverse
C044	Unsigned 64-bit multiplication suffices
C046	New elementary closed form for ellipse perimeter
C052	Complete physically accurate collision predictor
C054	Scalar tick replaces vector clocks
C057	Physical ontology (entropy, QM, cosmology, consciousness)
C058	Transformed value plus residual is compression
C059	Deterministic regeneration plus SHA-256 is a cryptographic primitive
C060	15 decimal digits are a universal portability horizon

Hosted files

Omega_Alpha_Master_Preprint.pdf464000 bytes · PDF
Locally hosted July 2026 synthesis. Not a source paper. No DOI.
SHA-256: f773a1fc24cfb70e1c036e65b55313dbbb9024caeb720b218a6af0ca0366b717
omega_alpha_reference.py20144 bytes · Python 3
Standard library only. Exhaustive 15-bit kernel, byte and 16-bit codebooks, corrected ellipse table. Site-local run matched the frozen JSON.
SHA-256: bc795467cf108f0a0efefc74de73a363c449056b2f6ef7cc2eed544e27359e7e
reference_test_report.json3315 bytes · JSON
Machine-readable PASS report.
SHA-256: 37b3b87e3bcc3e63139f74af191da7478f3a62a50262f1db100629f638f90b8c
reference_test_stdout.txt3315 bytes · text
Captured stdout; same SHA-256 as reference_test_report.json.
SHA-256: 37b3b87e3bcc3e63139f74af191da7478f3a62a50262f1db100629f638f90b8c
claim_verification_matrix.csv16296 bytes · CSV
Sixty historical claims with RETAIN/PROVED/REJECT and related dispositions. Not a source-paper theorem list.
SHA-256: ed19074129e88ad0fbaaf218ad735c73aff1829887499e809d7bf15f65a0c38c
version_and_correction_history.md13068 bytes · Markdown
Provenance map for the synthesis. Historical documents are not controlling specifications.
SHA-256: e5f1c984969850429120566794904ab90ca99f6e279735503d2ca57e8671c6ce
authoritative_reference_and_test_report.md7208 bytes · Markdown
Narrative of the finite tests. JSON remains the machine report.
SHA-256: d199afaea8a9b0a47c1a32108bb90dcf99a189222f4acda516662a9fe189ed9e
original_jprv_self_test.txt45 bytes · text
Result from the uploaded JPRV v0.2.4 source. Historical capture, not a current verifier.
SHA-256: a811a14a1ae35b1b202d87512de0f8a3fce8df54f0c006c80a29d4b6b25eb4d5
README.md1200 bytes · Markdown
This file lists the nine hosted payloads.
SHA-256: d735a27a42c8e6e3969134547d896ce0e618d054e0652e0a6628e4835e5f3fd0
SHA256SUMS.txt849 bytes · text
Checksums of the nine payloads; this file is not one of the nine.
SHA-256: 2f0a7029d3652767debdeb9134351f4a0cfc38e8e5525888df1a4114ceba29e1

Reproduce

Standard library only. A successful run reports "overall": "PASS" and must match the frozen JSON.

python3 omega_alpha_reference.py --pretty --report reference_test_report.json

Compare: reference_test_report.json · SHA256SUMS.txt · all verifiers