← AMRAL · PROGRAM-ACTM
Algebraic Combinatorial Transcendence Methodology · a methodology paper plus 34 rounds of research notes (2026-09-30 to 10-04); no round proves or refutes the Four Exponentials Conjecture
ACTM is a methodology: it studies which algebraic seeds and relation structures acquire transcendence degree under 'escape operators' such as exp, log and powers, with candidate diagnostics, a six-phase plan and a Schanuel-predimension interface. After round 2 the actual content turns to one concrete open problem — the Four Exponentials Conjecture. R01–R02 calibrate classical results (Lindemann–Weierstrass; Gelfond–Schneider and Baker); R03 rewrites the conjecture as an equivalent structural problem (a '2 → 1' dimension gap); R04–R09 study the two types of a hypothetical counterexample (Type P / Type B) and the Binary Anchored Descent Tower BADT; R10–R16 prove that the 'rational-linear Waldschmidt–Masser amplifier' strategy cannot work; R17–R34 follow the route of Dasgupta–Kakde (Adv. Math. 487, 2026), attacking their open implication (w) ⇒ (m) in the real BADT₂ case, and stop at an explicitly open bridge — the 'Cubic One-More-Zero' problem.
Honest status: no round claims to prove the Four Exponentials Conjecture, the Matrix Coefficient Conjecture, Schanuel's conjecture or the impossibility of BADT₂, nor the transcendence, algebraic independence or irrationality of any specific constant (all 35 documents carry a non-claims line; R02's 2^√2 examples are classical Gelfond–Schneider/Baker). The content is of four kinds: (1) correct restatements of classical results (R01–R03); (2) conditional structure theory, 'if a Type-P/BADT state existed, then …' (R04–R09); (3) barrier theorems about proof strategies (R10–R19, R24–R27, R33), where 'NoGo' never means the conjecture fails; (4) exact complexity, asymptotic and degree results (R18, R19, R25, R28–R31, R33, R34), some resting on external theorems (Waldschmidt 1978, Perucca–Sgobba). All of it is unrefereed, AI-executed mathematics. Scope ceiling: BADT₂ is a sub-case of Type P, which is only one of two branches; Type B is never revisited after R05. Complete success on BADT₂ would settle only one corner of the conjecture.
Self-assigned PASS, and the verifier scripts: all 34 round verdicts are self-assigned 'PASS', including rounds whose content is 'this route does not work'; the methodology's own PROVED / KNOWN / DERIVED / CONJECTURAL / UNKNOWN labels are never used in R01–R34. The 24 verifier scripts are finite sanity checks — this site re-ran all of them, all pass, the counts reproduce (5,150 / 14,420 / 40,698 / 31,419,396) and the CSVs of R12–R14 regenerate byte for byte — but R11's bound tables and the 2 + 4 + 12 special cases of R12–R14 are hard-coded, R22 has a vacuous assert, and R28 uses an algebraic stand-in for the transcendental s. No formal-proof artefact exists.
Corrections inside the series, and wording that is easy to misread: R24 is an erratum of R23 (R23's script still asserts the old bound); R30 retracts a heuristic of R29; R16 makes R11–R15 logically redundant (it calls them 'corroboration'); R28's degree bound is superseded by R30/R31. Phrases such as 'completely solved', 'closed in one stroke' and 'solving ASRP' (R16, R20, R22, R23) are each scoped inside their own document and mislead if excerpted. The boxed '2^(π³) must be transcendental' in R04 is conditional on 2^π and 2^(π²) both being algebraic (the classical form: at least one of 2^π, 2^(π²), 2^(π³) is transcendental). Dates and versions also have oddities (the header dates of R07 and R23 precede the file times; R16 is the only v1.0).
Authorship and provenance: the headers of all 35 documents read 'Author: Neo.K; Aletheia (GPT-5.6 Sol); Institution: EveMissLab / 一言諾科技有限公司'; none says who directed and who executed, none records a human review, and none mentions Claude. External checks: Dasgupta–Kakde (arXiv:2408.08178; the n = 2 case is equivalent to the Four Exponentials Conjecture; (m) ⇒ (o) proved; (w) ⇒ (m) open), Floater (arXiv:2603.19900) and Roy–Waldschmidt 1995/1997 exist and match how the series uses them; Waldschmidt's 1978 constants, Perucca–Sgobba and Philippon 1997 were not re-checked.
Reading path: start with the methodology (the non-claims), R03 (the open target), R16 (the strongest theorem with its own scope section) and R34 (the endpoint). To see how a failed route is recorded honestly read R08 and R24; R17 is a map of what is proved and unproved on the Dasgupta–Kakde route; R33's 'what is not proved' is the clearest. R11–R15 are logically superseded by R16 and R23 is partly corrected by R24; the other rounds have no pages of their own and are listed in the table below for download. Some formulas in the source use undefined LaTeX macros (\BKd, \BK, \Log in R17 and R22–R28) and show as red raw source on the pages; the files' bytes are unchanged. The three enumeration CSVs are not offered separately (the scripts regenerate them in about two seconds).
The methodology and 13 rounds have pages of their own; the other 21 rounds are in the table below. All originals are zh-Hant with English terminology. All 14 registered reader bodies now provide complete Chinese-English full texts; 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.
The original bytes of the 14 documents with pages and of the remaining rounds, unmodified; the 20 original zips are attached too. The other 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.