← AMRAL · PROGRAM-RCHM-HODGE
RCHM × Hodge Research Round · a methodology draft plus 27 rounds of internal research notes (2026-09-17 to 09-18); no round proves or disproves the Hodge conjecture
RCHM (Relational Constraint–Handoff Methodology) is Neo.K's method for writing 'intersection', 'infinite', 'finite' and 'exist' as typed relations and handoffs. This case carries the first batch of documents applying it to the Hodge conjecture: a zh-TW methodology draft plus 27 rounds of English internal research notes (R01–R27). The content falls into four phases: (A) R01–R06, a literature-based rewriting of Hodge as a witness / handoff / carrier problem (Kloosterman, Duque Franco–Villaflor Loyola, Movasati–Sertoz, Cifani–Pirola–Schlesinger); (B) R07–R13, exact SymPy experiments that reconstruct curves on explicit quartic surfaces from Hodge-annihilator signatures; (C) R14–R19, a 'bridge search compiler' calibrated on cases already proved by others (Weil fourfolds, cubics, Gushel–Mukai, Verra, Küchle c7/d3); (D) R20–R27, lattice and period bookkeeping for the Küchle c5 Fano fourfold, ending on one unresolved integer ρ_per. No round states a Hodge-conjecture result for c5 or for any open case.
Honest status: no round claims to prove or disprove the Hodge conjecture or any open case; the word 'counterexample' appears nowhere in the package. The content is of three kinds: (i) exact computations on constructed examples, which reproduce — this site re-ran all 22 verifier scripts, the stdout is byte-identical to the shipped output, the 29 JSON files are semantically identical and all MANIFEST sha256 values match; (ii) replays of results proved by others (R14–R19, calibration on known cases); (iii) an unreviewed, unresolved lattice and period analysis of Küchle c5 (R27: 'Round 27 does not claim: rho_per=0', which it calls 'the strongest safe closure currently available'). This site also recomputed c5's foundation numbers with its own code (Gram [[20,16],[16,14]], b₄ = 26, χ(Ω³(H)) = −5, χ(Ω^p) = −1, 4, −4) and they agree; the blow-up pushforwards, the Wu-class step and R27's BWB data were not independently checked.
Errors and corrections stay in the text: R20's rank-24 block is corrected to rank 22 in R21; R22's '3A₂' and R23's 'det 648' are retracted in R24 §26; R24's closing negative conclusion (direct integral K3^[2] isometry = FAIL) is reversed by R25 (R24 used q_T = −q_A, valid only in an even unimodular ambient lattice); R11 collapses R10's cross-type atlas. A verifier's PASS means arithmetic on typed-in inputs: R22 and R23 print 'ALL … EXACT CHECKS PASSED' for the values later retracted, R15–R19 encode literature facts as fixed PASS, and R27 re-adds a stored certificate and ships no BWB/HLYY generator. This site keeps these texts as written and puts an erratum banner on the R24 page.
Authorship and provenance: the documents carry no author line at all — none of the 28 documents and scripts contains a word identifying an author or an AI role (no author, Neo, Aletheia, GPT, Claude and so on). The only trace of origin is an identical sandbox traceback at the end of the 21 verification_output.txt files, showing that the computations ran in a hosted execution environment, not on Neo's machine. Who executed them and with which model is not stated; this site does not guess and will add the line once it is confirmed. Version numbers are round counters (R10 = v1.0, R27 = v2.7), not separate releases; the rounds have no claim/status field, and their status vocabulary evolves from informal hedges to evidence typing, PASS/UNKNOWN/FAIL, T0–T2 and REUSED/REDUCED/CONDITIONAL/NOVEL.
Relation to this site's Hodge case: this is a parallel new line, neither a duplicate nor a continuation. The Hodge Conjecture case (Chinese only) is MLRSC (Neo.K's framework, 57 rounds executed by Aletheia, Weil abelian six- and eight-folds, Markman and so on); the name 'RCHM' never appears on its pages. This site's Lebesgue case once said RCHM is the same underlying framework as the Hodge case; these documents are the first material that explicitly says 'RCHM × Hodge', and they post-date the Hodge case's launch (2026-09-16). The contact points are only: the same two Weil-fourfold papers, Kuga–Satake, cubic fourfolds, and a few shared words with different meanings (carrier, witness, gate, bridge, Pareto). The site has no other content on Küchle, Gushel–Mukai, EPW or Verra fourfolds.
Reading path: start with R27 (the endpoint) and R24 → R25 (a complete error, correction and reversal); to see how the method starts read the methodology draft, R01, R02, R06; for reproductions on known cases read R07, R08, R12, R14, R16; for the progress on Küchle c5 read R19 → R21 → R24 → R25 → R27. The other rounds have no pages of their own: they are listed in the table below and available for download. The methodology draft calls itself 'internal'; this site publishes it at Neo's instruction.
The methodology draft and 14 rounds have standalone pages, all now offering complete Chinese and English bodies. Research claims remain authoritative in the Chinese methodology original and the English R01–R27 originals; the other 13 rounds are original downloads only, with no new full-text translations.
The original bytes of the 15 documents with pages and of the remaining rounds, unmodified; the 21 original zips of R07–R27 are attached too. The other 13 rounds carry a one-line claim and a flag ('none' = no overclaim found; 'mild' = wording a reader could over-read, or corrected later in the series). The complete archive is kept in AMRAL's research archive and verified per file by SHA-256.