← AMRAL · PROGRAM-LEGENDRE
Legendre Base-Space Prime-Fitting Program · 2026-09-30 to 10-04 · the Program plus Phases 1–32 (33 documents) and 27 code/data packages
Legendre's conjecture: for every n ≥ 1 there is a prime between n2 and (n+1)2. It is open; the known results the package itself measures against are a Baker–Harman–Pintz-type prime-gap exponent of about 0.525 (Legendre needs 1/2) and what the Phase 15 Errata cites, namely that every sufficiently large square interval contains an integer with at most two prime factors. This line of research does not claim to prove it: it rewrites the problem as 'does a global prime-fitting operator FP exist?' (the Program), audits finite ranges, repeatedly retracts or demotes its own more optimistic statements (Phases 3, 5, 10, 12, 15), and then compresses the remaining gap into a very narrow analytic problem. Structure: the Program and Phases 1–4 reconstruction and finite audits (an operator falsified, a historical obstruction theorem, an exact sieve identity); Phases 5–16 the fit-ordered fibre, the obstruction horizon, shell certificates, the semiprime terminal barrier, the P2 boundary and constant ledger, a literature correction, Type-I/II coverage; Phases 17–25 source-class audits, weighted Type-I, the landing matrix, a moment hierarchy, a Schatten-6 double-large-sieve architecture; Phases 26–32 amplitude separation, a base operator, Helson cycles, CRE6, transverse defect, degeneracy closure and a toric three-GAP core. (Unrelated to this site's Lebesgue universal covering problem.)
Honest status: no document claims that Legendre's conjecture is proved, closed or solved, nor any new prime-gap bound or an unconditional 'every (n2, (n+1)2) contains a prime' result. The Program ends with 'Legendre proof status: NOT PROVEN' and Phases 1–32 all say 'Legendre proof status: OPEN' (Phase 11 says 'Prime proof status: OPEN'; the Errata has no such block). The only sentences of the form 'then Legendre holds' are conditional: the candidate theorems A–D of Program §13, Theorem 24.1 of Phase 2, Theorem 20.1 of Phase 3 (a section with gcd(F(n), Hn) = 1 exists — which is the conjecture itself), 'sublinear fit-depth ⇒ Legendre' of Phases 5 and 10 (much stronger than Legendre and demoted in Phase 12), Theorem 20.1 of Phase 20 (assuming a sixth-moment bound M6 ≪ X5/6+ε) and TTG3 of Phase 32. CLOSED / PROVED / PASS in a status block always refers to a sub-lemma or one sector (for example 'Phase 1 verdict: CLOSED' only means the formalization is finished and 'all proper-subsum degenerate CRE6: CLOSED' does not close CRE6).
Self-corrections and retractions inside the package: Phase 2 → 3: the open-loop 'autonomous' operator is falsified as the canonical fitter (position ratio about 0.7904) and the restored original operator uses the true prime history; Phases 5/10: the rank-1 prime section is falsified (hit rate 17.15%) and the 'rank-1 non-lift theorem' is formally retired; Phase 12: sublinear fit-depth is demoted to a 'much stronger' optional route; Phase 15 and the Errata: the statement of Phases 12–14 that 'the unconditional every-square frontier is P3' is judged 'incorrect' — ordinary P2 in every sufficiently large square cell already follows from classical short-interval results, the cube-root-rough set remains a stronger statement, and the genuinely open target is 'prime occupancy / parity closure' (the Errata covers Phases 12–14 only; Phases 6 and 11 carry the same wording unannotated); Phase 16 → 18: the two 'independent gates' WTI and LOC are revised to fuse at the crossing level; Phases 18 → 19 → 20, 27 → 28 and an internal erratum in Phase 30: corrections of wording, spectral targets and demotions.
What the pre-publication read-through reproduced: Phase 4: every n = 1…3000 has a prime in its cell (minimum 2, at n = 1, 2, 3, 5); the horizon theorems of Phases 6/10: zero violations over 2,398,173 composite 6k±1 candidates for n = 2…3000; the shipped audit of Phase 12 re-run is identical to its summary.json; Phase 5 reproduced exactly with a patched copy — the shipped experiment script never terminates at its default n = 10; the explicit example of Phase 28 (R = 24, K = 5) has a zero sum of exactly 0 and the invariants of Phases 30/32 (Δ = 336, A1A3A5 = A2A4A6) are as stated. Not checked: the closed-loop fit of Phase 3 (the ε-table construction is missing), every analytic bound in Phases 8 and 15–32, and all external 2025–2026 citations (Campbell arXiv:2603.10356, Yang arXiv:2608.13299, Croot–Yip arXiv:2607.15311, Pascadi arXiv:2505.00653, Bloom–Kuperberg 2025, a θ = 0.52 preprint, Baier 2026, Merikoski, Wu 101/232). Nothing that ships is a proof artefact: there is no Lean / Coq / interval-arithmetic certificate and no list of primality certificates, only small Python audit scripts, CSV diagnostics and JSON summaries, and all script headers say 'experiment, not a proof'.
Where the series ends: Phase 32 (2026-10-04) says 'full CRE6: OPEN; BA6 exact-zero provider: OPEN; TTG3 provider: DEFINED / OPEN; Legendre proof status: OPEN' and requests a Phase 33 that does not exist. There is no final verdict or synthesis, the Program itself (v0.1, 2026-09-30) was never revised, and its own Phase 1–9 roadmap was not followed after Phase 4. In this site's reconstruction from the phases' own ledgers (not a claim of the package): Legendre ⇐ Provider A (a square-anchored cube-root rough lower bound) and Provider B (a short-hyperbola semiprime upper bound) with crossing constants, or the parity-signed route (Phases 13, 16); Phases 19–32 work only on the weighted Type-I / Type-III remainder inside Provider A (landing matrix → M6 → BA6 → the CRE6 exact-zero sector → TTG3), so even a complete TTG3 would leave open at least the off-zero oscillatory BA6 provider, the |h| > K tail, the model-operator Schatten-6 estimate, the second gate (LOC1/2) and Provider B / parity.
Authorship and provenance: all 33 documents carry 'Author: Neo.K × AI collaboration; institution: EveMissLab / 一言諾科技有限公司'; no AI is named anywhere in the package and there is no AI-role line (the Errata has no author line). Named by the package but not in it: the epsilon_full paper v2.0 (Appendix A, the construction of the epsilon table), the 'V3 / prime geometry' source, the three-in-one predictor, the TCGQT Prime Triplex Lab, RG-SCP-07 and SRBRD-08, 11–22 (which are on this site: RG-SCP, SRBRD) and Phase 33. There is no README or manifest; Phases 1–4, 32 and the Program are loose md files while Phases 5–31 are zips, so the packaging is inconsistent. The selected 12 reader pages and 13 bodies (including the Phase 15 Errata) now provide complete Chinese-English full texts; the authoritative originals and full-text translations can be downloaded separately. Translation is not independent mathematical verification and does not upgrade the original evidence level. The download-only originals and packages below are outside this full-text bilingual scope and retain their original bytes.
Reading path: the Program → Phase 1 (canonicalisation) → Phase 3 (self-correction) → Phase 4 (the exact sieve identity) → Phase 5 (an honest negative) → Phase 6 (the horizon) → Phases 12, 13 (the P2 boundary and constant ledger) → Phase 15 with the Errata (the literature correction) → Phase 16 (the coverage polytope) → Phase 20 (a conditional theorem) → Phase 32 (the end). The other phases are offered as zips (with their code and data) for download, each with a one-line claim and a flag.
The Program, canonicalisation, the operator audit and self-correction, the exact sieve identity.
An honest negative, the P₂ boundary and constant ledger, the Errata in Phase 15, the coverage polytope.
A conditional theorem and the end of the series; the other papers of Phases 17–31 are in the table below.
The documents with pages (the Program and Phases 1/3/4/32 are loose md files; the others come from the zips of their Phases), the md of Phase 2 and the 27 Phase package zips, original bytes, unmodified. The others carry a one-line claim and a flag ('none' = no overclaim found; 'mild' = wording a reader could over-read, or corrected by a later paper). The complete archive is kept in AMRAL's research archive and verified per file by SHA-256.