Prime Atlas
An atlas of recurring prime-power coordinates.
PrimeMind is a visual and computational atlas of recurring prime-power coordinates, integer factor fingerprints, and prime-to-prime structure.
Every prime power pa is a permanent coordinate that repeats with exact period pa. An integer's fingerprint is the set of coordinates it switches on. Take 72 = 2³ · 3²: it lights five cells — 2¹, 2², 2³, 3¹, 3² — so Ω(72) = 5, it occupies two prime columns, so ω(72) = 2, and the lit cells reconstruct 72 exactly.
The laboratories on this site let you build that grid through large N, walk from one prime boundary to the next, audit the correspondence row by row, and compare the structure against controls and against finite sums over zeta zeros.
pa divides n. Step through 71 → 72 → 73 to see a prime light exactly one cell.How to read a claim on this site
Every statement, chart, and export is tagged with one of three marks.
Exact
Follows from proof or from unique factorization.
- Each coordinate
parepeats with periodpa. - Active cells reconstruct the complete factorization; their count is
Ω(n). - Factor occurrences and active nested coordinates correspond one-to-one.
- The global layer maximum through
Nis⌊log₂ N⌋. - LPF branch ownership and the branch-layer envelope.
Empirical
Verified on named finite computations, with the run recorded.
- Reduced-coordinate filtering experiments.
- Finite zeta-zero reconstruction and frozen threshold tests.
- Large prime-gap balance and half-balance scans.
Open
Research questions. Not theorems, and not claimed as such.
- Compression of the complete coordinate state.
- Next-prime prediction from a reduced fingerprint.
- Predictive use of completed gap skeletons.
- Scale-independent zeta classification.
- Any implication for the Riemann hypothesis.
Laboratories
Self-contained pages. Nothing runs until you press Build, and every export is computed in your browser.
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Prime-Factor Power Matrix v2.5 · flagship
The fixed coordinate grid, built through any
Nyou choose. Inspect one integer's fingerprint, step through consecutive integers, read the completed gap skeleton between two primes, measure how a gap's structural mass is distributed against shuffle controls, and test the simplest boundary rollout with a reduced coordinate set.fingerprintgap sequencegap skeletongap balanceboundary rolloutJSON · CSV · PNG exportsexactempiricalopenStart with the defaults, then inspect
n = 72and walk71 → 72 → 73. The boundary test from3643with horizon 40 shows why cutoff 40 leaves the false survivor3649 = 41 · 89and cutoff 41 removes it. -
Factor–Power Certificate and Gap Half-Balance v1.0
A row-level audit of the exact fact that each factor occurrence has one partner among the active nested coordinates — with tamper demonstrations that fail on purpose — kept deliberately separate from the open, measurable question of whether a prime gap's mass falls in its earlier or later half. Includes a bulk scan of every completed odd-prime gap through ten million.
integer certificategap certificatehalf balancecoordinate decompositionbulk scanexactempiricalopenVerify
72, run one tamper demonstration, then open one earlier-heavy and one later-heavy gap from the bulk scan. -
Wavefield and Finite Zeta Reconstruction v3 · advanced
From a frozen seed prime, continue the prime-power strike field forward and view it as waves, factor skeletons, gap clumps, and spectra; then compare the exact prime-power signal with a finite reconstruction from tabulated zeta zeros. The research-export tabs keep a visible boundary between what is known before a target is revealed and what is only known after.
strike fieldfactor skeletongap clumpsspectrumfinite zetaresearch exportsexactempiricalopenFinite-zeta results are empirical. The million-zero view reads a local copy of a zero table in your browser and never uploads it; current Chrome or Edge is recommended for
.gzfiles, or use the uncompressed text file.
Branch geometry
Four smaller labs on largest-prime-factor branches, layers, and the exact envelope law. About the branch labs
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Branch Envelope Law
The terminal-branch envelope law, stated and drawn: in layer
k, a branch owned by primepruns fromp · 2k−1topk.exact -
3D Branch Space V3
Complete branch layers, long segments compared on one shared scale, and a lane-by-lane prime-power wavefield, all rotatable in three dimensions. Large page; best on desktop.
exactempirical -
Branch Rays
The complete terminal branch of a seed prime and the rays into it, with a choice of horizontal coordinate for comparing branches.
exactempirical -
Branch Clumps
How branch-layer clumps grow: every integer appears once,
Ωgives its layer,log LPFfixes its column, andlog nfixes its height inside that layer.exactempirical
Paper and methods
The preliminary draft, Prime-Power Recurrence Atlas v0.2, states the definitions and exact results, records the finite experiments, and lists the open questions in the same three marks used here. The Overleaf source and the research journal are downloadable alongside it.
The methods page fixes the vocabulary — recurrence coordinate, prime boundary, factor fingerprint, gap skeleton, largest prime factor — and states the information boundary that every prediction experiment must respect: what is allowed before a target is revealed, and what only exists after.