← AMRAL · PROGRAM-PERFECT-CUBOID
Perfect Cuboid · 2026-09-27 to 10-06 · the core (the Global Closure series, the G-line, Phase 1 searches) plus Radical-FDRS and four method lines
The perfect cuboid: is there a box whose three edges a, b, c are integers and whose three face diagonals and space diagonal are integers as well? It is open (the package: 'the classical integer version has neither an existence nor a nonexistence proof'). The best-known near-miss is the Euler brick (44, 117, 240): its three face diagonals are integers and its space diagonal is not. This package rewrites the problem as a rational-point problem (homogeneity makes integer and rational existence equivalent) and works around one seed (448, 495, 840; 1073): six of its seven lengths are integers and only the face diagonal √445729 = √(233·1913) is not — it is not a perfect cuboid. The core (this page): the Global Closure series (01–07 and the center-axis theorem), six-channel dual squares, the G-line (G2A–G2E.6: a two-parameter reduction, ranks of elliptic curves, Mordell–Weil sieves and a bielliptic genus-2 reduction on the fixed slice k = 7/32, ending with a list handed to local exact verification) and the bounded exact searches and classification of the space diagonal of Phase 1 (pe and pq are excluded, p2q is the frontier). Two branches: Radical-FDRS (R0–R23, elliptic–Chabauty machinery on the same fixed slice, ending with a specification whose quadratic-Chabauty computation was not executed) and the method lines (FORC, GERES, SIO-X4, X4-DMSD, IGPC, TIFV: exact-rational runtimes, bounded certificates and frameworks).
Honest status: no sentence anywhere in the package (about 200 documents) says that a perfect cuboid, or a 'candidate perfect cuboid', was found, and none says it does not exist or that the problem is solved; disclaimers such as these are everywhere: 'so the classical Perfect Cuboid is still not solved by this paper', 'the global problem remains open because k itself is free to vary over Q ∩ (0, 1)', 'the fixed slice k = 7/32 has not yet been globally eliminated', 'this is not a nonexistence result'. Every computation is bounded: G2A exhausts 121,590,395 reduced h (denominator ≤ 20000) on the fixed slice k = 7/32 with no double hit; Phase 1D exhausts primitive cuboids with space diagonal g ≤ 5,000,000 with no solution ('a finite-range exclusion', far below published search ranges); Phase 1 scans 478,765 shapes without a perfect cuboid. Even if the fixed slice k = 7/32 were eliminated completely, k can still vary and the perfect cuboid problem would not be settled.
What 'closure', 'PASS', 'certified' and 'EXACT' mean: Global Closure in the titles means closing the conceptual framework (Phase 0 — Conceptual Closure); '6/6 closure' and 'one defect away' mean that six of the seven lengths are integers; 'rational closure' means existence of a rational solution, a target rather than an achieved result; 'certified', 'theorem' and 'PASS' are outputs of the authors' own exact-integer code (the G2D checkpoint says their runtime had no Sage / PARI / mwrank / Magma, and G2E.6 itself asks for local re-verification with those tools). Nowhere in the package is a Chabauty or quadratic-Chabauty computation actually carried out; only the rank inequality r < g + ρ − 1 is checked. 'Candidate' (GERES-P5) means an integer triple with exactly two integer face diagonals and an integer space diagonal, and every record is tagged claim_status NOT_PERFECT_CUBOID. 'Height' means different things on different lines (g, HE, hypotenuses of face triples, …).
What the pre-publication read-through reproduced (with independent code; PARI 2.15.4): the seed arithmetic and t = (25/39, 17/28, √(233/1913)); the G2A gate scan (v ≤ 1500) and the points and torsion of E_f; the 200 orbit states, largest denominator of 5,206 digits and 0 second-gate hits; ellrank / torsion / root numbers of E_f, E_0, E_0′, E_− (rank 1 with root number −1; Sel₂(E_−) of dimension 5) and the I, J invariants; the clean-prime table and the power-of-2 primes; the point counts of C_{7/32} at p = 11, 19, 31; the boundary lifts of G2E.3, the τ involution and E_+ = E_0; w(m), A4, B4 and m = 25, 39, 31200/1073 of G2E.6; the genus-3 scan for v ≤ 1200; the Chebyshev rational roots of Phase 1 (e = 3…10) and the 24,813,268 pair checks and 7,817 near-hits of Phase 1D — all matched. Not redone: the full 121,590,395-state G2A scan, the sieves themselves (only their data were checked), the 8 classes of the provisional descent in G2E.6, the p²q orbit counts 57/42/22, the attribution to Korec 1992 and the 'ProofAtlas' record.
Superseded and duplicate files: G2A→G2E.5 is a chain of cumulative bundles that are strict supersets (the last bundle of 45 members supersedes the previous seven; members are byte-identical); G2D.2 is superseded by G2D.3; the early G2D checkpoint records the E_0 torsion wrongly; Phases 1F-B / 1F-C / 1G / 1H are covered by 1F-D; the Phase 1 packages form a chain of subsets with 9 / 43 / 52 members (v3 supersedes the first two). Missing: paper 08 of the Global Closure series ('Upper and lower bounds of the solution space and a second expansion') is announced in every header but is not in the package; verify_g2d1.py and g2e3_power2_scan.cpp are stubs, so those two certificates cannot be regenerated from the package alone. Not on this site: the code, CSV, JSON and SQLite inside each zip (about 1.3 MB of de-duplicated data, a 53.9 MB SQLite in FORC-P6 and an 11.4 MB SQLite in X4-DMSD-P13) are kept in AMRAL's research archive; this site hosts only the md documents.
Authorship and provenance: no core document carries an author, provenance or AI-role line; TIFV says 'Neo.K / EveMissLab'; no file says who directed and who executed, and this site does not guess. The Global Closure series and the G-line are dated 2026-09-28 to 10-02 (zip member times of the G-line 09-30), the Phase 1 packages about 09-27. Reading path: 01 (reframing) → the center-axis theorem → six-channel dual squares → G2A → G2B/G2C → G2D (ranks) → G2E.1/E.2 (the tower) → G2E.3/E.4 (the sieves) → G2E.5/E.6 (the hand-off) → the Phase 1 searches → the diagonal restriction → the p²q frontier; then the two branches. The other 19 documents have no pages of their own and are listed in the table below for download with a one-line claim and a flag. The originals are Traditional Chinese with English terminology (the G-line and Phase 1 are almost entirely English).
Beyond the core: the Radical-FDRS line on the same fixed slice, and the four method lines.
The Global Closure series, the center-axis theorem, six channels, the G-line (G2A–G2E.6) and the searches and classification of Phase 1. All 14 reader pages and 26 complete bodies of the core are now fully available in Chinese and English. The originals remain authoritative for their research claims; translation is not independent mathematical verification or an upgrade of evidence. Another 19 download-only original documents are listed in the table below and are outside the full-text bilingual scope stated above.
The original bytes of the documents with pages and of the other 19 (md files taken from the zips and loose md files), unmodified; the code, CSV, JSON and SQLite inside the zips are not on this site. The others carry a one-line claim and a flag ('none' = no overclaim found; 'mild' = wording a reader could over-read, or superseded by a later paper). The complete archive is kept in AMRAL's research archive and verified per file by SHA-256.