Prime-Power Recurrence Atlas

Integer factor fingerprints, composite propagation, prime boundaries, and preliminary spectral evidence. Discussion draft v0.2, September 2026.

The draft defines the fixed matrix of prime-power coordinates — column i belongs permanently to the ith prime, row a to exponent a, and cell (i, a) to the condition p_i^a | n — and works out what follows exactly: each cell is periodic with period p_i^a, higher cells in a column are nested inside lower ones, the active set at n is a complete factorization fingerprint, and a prime is a new-base event. It then adds two secondary organizations, largest-prime-factor branches with their sharp envelope and prime-gap skeletons, reports a high-scale reduced-coordinate experiment and a finite analytic bridge through prime-power source events, and closes with the experimental protocol, a list of what is established and what remains open, and questions for mathematical review.

Open the PDF Overleaf project (zip)

Where the work stands

Paraphrased from the research journal's own checkpoint of 8 September 2026.

The exact identities are solid: every coordinate has an exact period, the fingerprint is a complete factorization, the branch envelope is a theorem, and the complete required base-coordinate set identifies every prime in a declared bounded interval — which is a segmented sieve written in coordinate language.

Two finite experiments are recorded in full. A partial bank of the 25 prime bases through 97, frozen at seed 459,824,117 with horizon 5,000, kept all 256 primes while removing 92.75% of composites; every next prime fell within the first ten survivors of its gap. A separately frozen finite-zeta threshold rule, declared before the target interval was opened, then classified all 50 primes and 450 composites of one untouched 500-integer window correctly at seed 70,001.

That last result is a real out-of-sample validation at one scale. It is not a general theorem, not an efficiency result, and it has to be replicated across disjoint windows and increasing scales before anything more is said about it.

The project does not claim a universal next-prime shortcut, a proof of the Riemann hypothesis, or historical novelty for the branch-envelope theorem. Negative controls and failed reduced predictions are kept in the record on purpose.

Downloads

Everything below is a plain file; nothing requires an account.

Preliminary draft v0.2 (PDF)docs/The_Prime_Recurrence_Principle%20_D.M.Dorsey.pdf544 KB
Draft source (LaTeX)docs/Prime_Power_Recurrence_Atlas_Preliminary_Draft_v0.2.tex44 KB
Overleaf projectdocs/PrimeMind_Prime-Power_Recurrence_Atlas_v0.2_Overleaf.zip2.1 MB
Research journal and technical handoffdocs/PrimeMind_RECORDER_current_2026-09-10.md — notation, the branch theorem and its proof, the frozen-frontier coverage result, the explicit-formula connection, the prediction protocol, and every recorded run. Markdown with LaTeX math.132 KB
Factor Matrix v2.5 notesdocs/lab-notes/PrimeMind_Gap_Balance_Lab_v2_5_README.md — the Gap balance tab: quantities, controls, charts, validation.6 KB
Certificate v1 notesdocs/lab-notes/PrimeMind_Factor-Power_Certificate_Half-Balance_Lab_v1_README.md — exact certificates, why a gap need not be centered, the scan through ten million.6 KB
Wavefield research-export notesdocs/lab-notes/PrimeMind_Wavefield_Research_Exports_v2_README.md — the pre-reveal / post-reveal export boundary and recommended first runs.6 KB
ChecksumsCHECKSUMS_SHA256.txt — SHA-256 of every public lab file and document.2 KB

Version history

Released files are never overwritten; a change gets a new version. Full detail in CHANGELOG.md.

Component	Version	Source file	Notes
Site shell	0.1	`index.html`, `assets/`	First static shell: shared header, home, branches, paper, methods, archive. Lab snapshot of 2026-09-10.
Factor Matrix	2.5	`prime-power-factor-matrix-gap-boundary-lab_v2_5.html`	Adds the Gap balance tab to the matrix, gap-sequence growth, gap skeleton, and boundary-rollout lab. Versions 2 through 2.4 are historical backups.
Factor–Power Certificate	1.0	`prime-factor-power-certificate-half-balance-lab_v1.html`	First release. Bulk counts matched an independent C++ implementation through ten million.
Wavefield & Zeta	3	`prime-composite-wavefield-lab_RESEARCH-EXPORTS_v3_1M-ZERO-LOADER.html`	Research exports v2 with a local million-zero table loader. Earlier wavefield versions are archived, not current.
3D Branch Space	V3	`prime-branch-space-3d-labV3.html`	Secondary lab.
Branch Rays	—	`prime-branch-ray-lab.html`	Secondary lab.
Branch Clumps	—	`prime-branch-clump-wave-lab.html`	Secondary lab.
Branch Envelope Law	—	`prime-branch-envelope-geometric-law.html`	Exact law page.
Preliminary draft	0.2	`The_Prime_Recurrence_Principle_D.M.Dorsey.pdf`	Discussion draft, September 2026.