Methods and terminology

The words are fixed so that a picture, an export, and a sentence in the paper always mean the same thing.

Vocabulary

Prime-Power Atlas: The full system of permanent prime-power coordinates.
Prime-Factor Power Matrix: The fixed grid: prime bases increase to the right, exponent levels increase upward. A cell's address never changes; only its state does.
Recurrence coordinate, prime-power coordinate: One cell p^a. It is active at n exactly when p^a | n, and it repeats with period p^a.
Prime-base coordinate: The level-1 cell p¹ of a prime column.
Factor fingerprint: The set of active cells for one integer. It reconstructs the integer exactly.
Factor-power level: The exponent row a. Higher cells in a column are nested inside lower ones.
Inherited support: Coordinates an integer receives from its divisors: the fingerprint of m is contained in the fingerprint of n exactly when m | n.

Prime boundary: A prime, seen as the wall between two gaps. A prime is a new-base event: no earlier prime-base coordinate is active there.
Gap sequence, gap skeleton: The ordered factor structure of the integers between two consecutive prime boundaries.
Global layer frontier: The largest Ω reached so far. Through N it equals ⌊log₂ N⌋.
Factor–power correspondence, certificate: The exact pairing of copy k of prime factor p with coordinate p^k, audited row by row.
Gap balance, half-balance: Descriptive measurements of how a chosen structural mass is distributed between the earlier and later halves of a completed gap.
Largest prime factor (LPF): Owns the branch. Always the largest here; some software uses a similar abbreviation for the least prime factor.

Notation

Symbol	Meaning
`n`	An integer being displayed or tested; normally `n ≥ 2`.
`LPF(n)`, `SPF(n)`	Largest and smallest prime factor.
`Ω(n)`	Prime factors counted with multiplicity; the factor layer. `Ω(12) = 3`.
`ω(n)`	Distinct prime factors. `ω(12) = 2`.
`1`	The empty multiplicative state: no active coordinates, `Ω(1) = ω(1) = 0`, no LPF.
`u = n / LPF(n)`	Canonical cofactor.
`q`, `a`, `v_q(n)`	A prime acting as a coordinate base (not necessarily the LPF); an exponent; the exponent of `q` in `n`.
`S`, `H`, `N = S + H`	Freeze boundary, horizon, and endpoint of a run.
`P_s < P`, `g = P − P_s`	Consecutive primes bounding one gap, and the gap length; the interior holds `g − 1` integers.
`Λ(n)`	The prime-power source weight: `log p` at `n = p^a`, zero otherwise.

Claim boundary

Every page marks its content with one of three tags. The same tags appear in the lab headers and in the paper.

Exact

Follows from proof or from unique factorization.

Each coordinate p^a repeats with exact period p^a.
Active cells reconstruct the complete prime factorization; their number is Ω(n).
Factor occurrences and active nested coordinates correspond one-to-one.
The global layer maximum through N is ⌊log₂ N⌋.
The complete required base-coordinate set identifies every prime in a declared bounded interval.
LPF branch ownership and the branch-layer envelope.

Empirical

Verified on named finite computations; the run and its settings are 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.

Compression of the complete coordinate state.
General next-prime prediction from a reduced fingerprint.
Predictive use of completed gap skeletons.
Scale-independent zeta classification.
Any implication for the Riemann hypothesis.

PrimeMind is not presented as a proof of the Riemann hypothesis or as a proven faster-than-standard prime algorithm. Where a lab compares against a conventional method, the word sieve is used for that method and nothing else.

Information boundary for prediction experiments

Allowed before reveal

The starting boundary and the declared horizon.
Previously established coordinates.
Residue and phase state of those coordinates.
Fixed external zero data.

Post-reveal only

Target prime labels and target factorization.
SPF, LPF, Ω, ω of the target.
The future completed-gap boundary.
Target factor geometry.

The visual labs compute the truth internally in order to draw descriptive views. Their research-export tabs keep the pre-reveal and post-reveal tables apart as a disciplined export boundary. That is not the same as a physically isolated two-process predictor; a page is only labeled predictive when a separate freeze → predict → reveal engine is used, with its metric, normalization, controls, and decision rule declared before the target gap is opened.

Using the labs

Each lab is a single self-contained page. Nothing heavy starts on load; you press Build. In the heavier labs, long jobs run in a background worker with a progress indicator and a Cancel button. Exports (CSV, JSON, PNG) are generated in your browser and keep raw numeric precision; rounding happens only on screen.

The labs work within JavaScript's safe integer range, 2⁵³ − 1. Inputs are not extended past that silently.

The Wavefield & Zeta lab can read a local copy of Andrew Odlyzko's zero tables. The file is opened by your browser and never sent to this server. Current Chrome or Edge is recommended for .gz files; if decompression is unavailable, use the uncompressed text file instead. A one-million-zero comparison is expensive; keep the small defaults for a first run and expect a large job to take a while.

Current desktop Chrome, Edge, and Firefox are supported. Every control is reachable on a narrow screen, but the heavier labs are better used on a desktop.