allen-proxmire/Primes: Primes Repo V2.2

Abstract

v1.2 — September 2026 The collection audited itself, and gained a way to be read. Seventeen commits. Two claims retracted, two verified, one small new proposition, and — probably the most useful part — a layer of plain language so the work can be understood without the notation. New: you can now read this without a maths degree Three documents, no notation in any of them: The Movie — what is actually going on. Every prime is a clock; clocks join the orchestra at their own square; each new clock multiplies the pattern's length rather than adding a tooth. From 7 onward the pattern is longer than the stretch of road where it applies — so it can never be caught repeating. That, and not rule complexity, is why the primes look random. The rule itself fits on a napkin. What We Found — the results, each labelled proven / verified-not-proven / measured / a reading / retracted. How We Know — why any of it should be believed, and the two times it shouldn't have been. Each of the five series folders also gained a short README. Results A hard rule, promoted to the front of its paper. The Switchback Law was rewritten around what survives: two consecutive prime gaps can never both be ≡2, nor both ≡4 (mod 6). A prohibition, not a tendency, falling out of the fact that a prime cannot make the same residue-class switch twice running. Those transitions measure exactly 0.0% against 28.4% under a matched null; gap-changes of 6 and 12 are cut to half their expected rate while every other value is elevated; and no gap outside the multiples of 6 ever repeats immediately. This was §5 of that paper, written in July. It is now the front of it. The windowed gap memory is fully the wheel. The +1.97 percentage points that survived the differencing null had been left at "plausibly the wheel." Now measured: 99% reproduced from a wheel of depth 100, while a deliberately shallow wheel reaches only 76% — so the test can see a shortfall, and doesn't find one. The angle-record theorem verified, and it is clean. By PG II's Theorem 6 this is logically equivalent to the Twin-Prime Bertrand Postulate, making it the load-bearing claim of series 3. Checked with exact integer arithmetic over every consecutive prime pair below 10⁸: 440,312 angle-records, 440,312 twin pairs, zero non-twin records — and the two lists coincide exactly rather than one merely containing the other. One new proposition, which was free. PG II proved records ⊆ twins at the full strength of TPB and left the reverse inclusion unstated. It costs nothing — Proposition 6A, now added, shows every twin is a record unconditionally. The asymmetry reads better than the original claim: the record sequence always contains every twin, and TPB is exactly the assertion that it contains nothing else. Equivalently, and usably: TPB fails ⟺ some angle-record is not a twin pair. The balance ratio is a redundancy, not a signal. Koide's quantity applied to prime triples reduces by exact algebra to the consecutive-gap covariance. No residue class carries an excess. And its hexagonal form's divisibility by 3 turns out to be exactly the mod-6 traffic rule above, reached from the Eisenstein integers — two routes, one rule. Two new synthesis papers. The Wheel Is the Whole Story states the collection's central claim and audits it: ten probes, six genuinely independent and four provably the same quantity in other notation. The Null-Model Discipline catalogues five ways to fool yourself, each caught in the act here. Corrections The switchback run-length law is retracted as a statement about primes. It compared how often the wobble's direction flips against a fair coin. Wrong baseline: differencing forces extra alternation onto any sequence, so a coin was never the opponent. Against a matched null the effect vanishes — real 64.00/27.25/7.18/1.34%, null 64.66/27.31/6.74/ 1.13%, with the null's reversal odds slightly higher. It describes any increasing sequence with bounded steps. A table row corrected. The jitter recoil published +2.7 points of genuine effect. Thirty candidate definitions were tested; several reproduce the published real figure, several the published null figure, none reproduces both. They were computed at different thresholds. A theorem's threshold is off by one prime. PG I states that (p₍ₙ₊₂₎² − pₙ²)/12 has last digit 0, 4 or 6 for pₙ ≥ 5. The triple (5, 7, 11) gives 8 — the sole counterexample below 5×10⁶. The proof's mod-5 step has nothing to stand on at 5 itself. True for pₙ ≥ 7. A percentage that needs a range attached. "The small primes give you 68% of everything knowable" is right at N = 10⁴ and drifts to 59% by 10⁶ — not because the wheel weakens but because the denominator grows. The underlying 1.70 bits is stable across three decades. Reproducibility 2_One_Wheel_Many_Shadows/repro/ regenerates every measured table in the three gap papers from a sieve. It is what found the first two corrections above, on its first run. Series 1 gained a script for the entropy budget — worth noting that the upstream reproducibility/tables/ tree turned out to be nine stubs returning placeholder values, so those numbers had never actually been re-derived. Series 3 gained two verification scripts and its original analysis code. Series 5 has a note explaining what reproduction would require and why a token script would misrepresent it. Housekeeping Markdown-native. The 13 PDFs that merely mirrored a markdown file are gone; git retains them. Remaining PDFs are sole copies. The Prime Geometry trilogy now has markdown, converted from the original LaTeX recovered from the upstream repo — not retyped from the PDFs. RESULTS.md rebuilt — 54 tagged results, up from 26, every link repaired. Eighteen of twenty-three had pointed at a folder layout that no longer exists. Seven referenced figures restored; every internal link now resolves. Honest note Two retractions and a corrected theorem is an uncomfortable changelog. But all four corrections were found here, by this collection, using methods this collection had already written down and finally got around to running — and the claims that survived went through the same process that killed the others. A body of work that can find its own errors is worth more than one that has not been asked to. Still unverified: the twin-gap exponent and GBP envelope in series 3, and all of series 5. Given the hit rate so far, that is a real gap rather than a formality.