AMRAL

Hilbert's Twenty-Three Problems

The Riemann Hypothesis is the first case, not the destination. CASE-0001 hangs under Hilbert's Eighth Problem: PROGRAM-HILBERT-23 → H08 → Riemann Hypothesis → Weil quadratic-form research → Batch 01. The claim is direct: a problem that has already been proven can still be re-researched; the existence of an answer does not end the research path.

Every new case on this program line follows the same rhythm going forward — Batch, W-series engineering packages, sealed volumes, platform import packages — the AMRAL name stays fixed; what grows is only the case count.

The same H08 has also grown an extension line: Critical Line Proportion Ladder (CPL). It doesn't attempt to prove or disprove RH, but extends a real, published Anthropic paper — Claude's "More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line" (2026-08-10), which unconditionally pushes the proportion of critical-line zeros to 67.25%, explicitly without affecting RH itself. Neo.K's follow-up research reconstructs a constant the paper left unaddressed, examining whether it can be pushed past 70%.

The Universal Covering Problem

A different family of classical open problems, running parallel to and independent of Hilbert's Twenty-Three: the Kakeya needle problem, Moser's worm problem, the Lebesgue universal covering problem — different facets of the same axis, how directional change in a one-dimensional object translates into a two-dimensional covering region. PROGRAM-UNIVERSAL-COVERING currently carries three independent cases: direct numerical-optimization attacks on Moser's worm problem; a bridging theory unifying Kakeya, center-generated spirals, and Moser; and a line attacking the Lebesgue universal covering problem itself — a four-stage closure methodology (RCHM/LUC-FC) spanning 37 main-line rounds, alongside four independent verification/search methodologies Neo.K designed specifically for this line (DLMVC, BCODR, LESR, UESFCM). Currently 10 of 77 necessity cells reach a replayable exact certificate; the global lower bound of 0.835 remains unproven. The three are not merged in the record. A neighbouring line, WSFC (Worm Sequence Freedom Closure), treats the broadworm breadth sequence and the seven 2007 conjectures; it is not a Moser-area result. The Lebesgue case also holds a second batch of documents (LESR 09C–09U and MCC-OE) on a local certification at root 6217; the author-defined 'research interval' closure holds only locally, and the global lower bound remains unproven.

P versus NP

PROGRAM-P-VS-NP is another independent problem family, split into two independent cases. p-np-dual/ is the traditional P/NP dual discussion: without pre-committing to a side, it builds both H=: P=NP and H≠: P≠NP to their respective strongest versions, and attacks them round by round under one shared computational model, resource ledger, and correctness standard — 24 rounds covering quantifier compression, cross-representation invariants, clocked diagonalization, WQO, and semantic monotonicity, among other mainstream toolchains, without rushing to declare a proof complete. glc-framework/ is the researcher's own further proposal after round 24 — a "Dynamic Four-Layer Closure Framework" that reprojects the problem onto four layers, GCC／USRT／USEG／GLC, with the core statement "the process is free, the final ledger is not free," accompanied by two handoff plans with differing execution orders, presented side by side. This line currently only lays out the completed research; it has not been pushed further yet.

The BSD Conjecture

PROGRAM-BSD is the fourth independent problem family — the Birch and Swinnerton-Dyer Conjecture, another Clay Millennium Prize problem. The core tool is a curve-level certificate ladder (C0 identity → C9 full strong BSD → C10 family theorem), precisely grading exactly which level each curve, at each prime, has actually been proven to — not just storing a single "BSD true/false" boolean. Phase 0 (the global encircling framework, nine papers) and P5 (strong BSD at a single prime, for the rank-2 curve 389.a1 at p=11, ten papers) are live — one Phase 0 paper is a formal retraction of the researcher's own earlier claim, and P5 has now compressed down to a single open comparison between the leading complex coefficient and the Selmer determinant. Phase 1 (reproducing the Banwait–Huang 2026 algorithmic census) and Phase 2 (an explicit twist family for the non-semistable curve 696.e1) both already have substantial real progress, still awaiting page-by-page publication. Since 2026-09-08 an independent stress-test sub-line (Chinese only) is live: an adversarial independent audit of the existing content plus a finite Kurihara certificate that advances P5's Sha[11^∞]=0 and rank = corank Selmer₁₁ = 2 from CONDITIONAL to VERIFIED_THEOREM_APPLICATION — the full strong-BSD main-term formula remains OPEN, and no claim is made to have proved the whole P5 line.

Navier–Stokes Global Regularity

PROGRAM-NS is the sixth independent problem family — the global regularity problem for the 3D incompressible Navier–Stokes equations, another Clay Millennium Prize problem. Its founding document opens by listing eight explicit non-claims: it does not claim global regularity has been proven, does not claim a finite-time blow-up has been constructed, and does not claim True ETN (an infinite-dimensional tension field) or the X-integral alone can imply PDE regularity. The core tool compresses the problem into two complementary, falsifiable propositions: C1 (Chain Necessity) — a blow-up must generate a source-traceable, scale-by-scale legal X-legal UV concentration chain; C2 (Finite Obstruction) — any such chain must be genuinely blocked by N–S structure at some finite scale. If both are proven, then ¬Blowup.

Status: C1a/C1b (UV-escape necessity, the causal source of nonlinear replenishment) are CLOSED; C2 has proven that a purely scalar additive budget is not enough to rule out blow-up (an abstract cascade-ledger counterexample), and the frontier has shifted to C3, cross-scale coupling rigidity, unfolding into 25 independent sub-investigations. C4 (9 rounds, synchronization and shared-event coupling), C5 (13 rounds, recording windows and compensated-drift state space), and C6 (17 rounds, minimal recursive-defect cycles and sink-SCC extraction) followed in sequence — framework + C1/C2/C3/C4/C5/C6, all 67 papers, are now live. Global regularity remains fully OPEN; no round claims it has been resolved. There is also a machine-readable symbol cross-reference spanning the whole series, covering every symbol defined across all 67 papers.

The NS zone is the first part of this site to be fully translated — as of 2026-09-02, all 16 of its independent research sub-lines are live in English. See the NS zone's own English hub for the full breakdown.

The Collatz Conjecture

PROGRAM-COLLATZ is the fifth independent problem family — the Collatz conjecture (the 3x+1 problem). Neo.K and Aletheia (collaborative write-up) co-authored nine core papers that start from the exact finite affine structure of parity words, build a Local Affine Atlas, prove that every finite word corresponds to an exact affine operator of the Collatz map, and then generalize the structure from Collatz's specific coefficients (3, 1) to the general Residue-Class Operation Translation (RCOT) class. The series is capped at Paper 09, which states plainly that the gap of the global quantifier has not been crossed; the Hard-Zeta research program developed afterwards is a proof-research outline aimed at that gap, not a tenth paper of the series. It does not claim to prove or disprove the Collatz conjecture, and every paper has its own section on what it does not prove.

The Hodge Conjecture (Chinese only)

PROGRAM-HODGE is the seventh independent problem family — the Hodge conjecture, another Clay Millennium Prize problem. The core is a three-phase reformulation: Phase I Measure (a measure classification of the space of candidate Hodge classes), Phase II Legitimacy (which relation between algebraic cycles and Hodge classes carries a verifiable witness), Phase III Scale–Category Coupling (whether the conclusions of the first two phases remain compatible when scale and category are enlarged; the core quantity κ_s,t is targeted at 0). The 57 execution rounds run in four lines: a 28-round core framework (the M/L/C phases), a dedicated 10-round counterexample hunt, a dedicated 10-round proof attempt and a 9-round deep dive on Weil-type constructions (one round's original file is missing, noted on its page). No round anywhere claims to have proven or disproven the Hodge conjecture itself — the counterexample hunt twice locked on a candidate and both times ended 'not yet'; the proof-attempt line compressed the problem to a real, unsolved subquestion in the 2025 literature (Question 11.2.2 of a paper of Markman) and likewise did not solve it. The framework itself shows two solid self-corrections. The framework was written chiefly by Neo.K with ChatGPT/Aletheia assisting; every one of the 57 rounds was carried out autonomously by Aletheia (GPT-5.6 Sol). The Hodge pages are in Chinese only.

A separate case, RCHM × Hodge Research Round, applies Neo.K's Relational Constraint–Handoff Methodology (RCHM) to the Hodge conjecture: a methodology draft plus 27 rounds of internal research notes (2026-09-17 to 09-18) — exact computations on constructed examples, replays of cases proved by others, and lattice and period bookkeeping for the Küchle c5 Fano fourfold. No round proves or disproves the Hodge conjecture; it is not part of the 57 rounds above and the two are not merged.

Number Theory: Goldbach, Legendre and the Perfect Cuboid

PROGRAM-GOLDBACH, PROGRAM-LEGENDRE and PROGRAM-PERFECT-CUBOID are the eighth to tenth independent problem families — three lines of research about the integers, routed together under Number-Theoretic Conjectures. The Goldbach Reverse-Structure Closure Program (three sub-series RG-SCP, SRBRD and TCDC, 77 documents) writes Goldbach as structural constraints on a counterexample, makes an exact finite re-encoding and records at each step where the argument stalls at the sieve and parity barriers; every series ends with 'GOLDBACH STILL OPEN'. The Base-Space Prime-Fitting Program for Legendre's conjecture (the Program plus Phases 1–32) rewrites 'there is always a prime between consecutive squares' as a prime-fitting operator; every document says 'Legendre proof status: OPEN', and the package contains its own retractions (the Errata in Phase 15). The perfect cuboid is rewritten as a rational-point problem: elliptic curves and Chabauty machinery on the fixed slice k = 7/32, bounded exact searches, the Radical-FDRS line and the FORC / GERES / SIO-X4 / X4-DMSD method lines. None of the three lines claims to prove its conjecture or to have found or excluded a counterexample; their own pages say item by item what they did and did not do, which figures were reproduced and which were corrected by later papers.

CCM Computational Composite Methodology

PROGRAM-CCM (Computational Composite Methodology) is about how to systematically combine computational exploration, formal verification and human review to produce research state that is traceable, falsifiable and handoff-able; its core firewall theorem: heuristic search can be as unrigorous as it likes, but the final verdict must be gated by a sound verifier. The foundational theory paper is live; a set of 13 calibration benchmarks and an 18-round series using Hilbert's Third Problem as an application case are both scoped, but not yet built out page by page.

CSM Closure-Space Mathematics

PROGRAM-CSM (Closure-Space Mathematics) is about how the research state itself should be organized: once a research effort has accumulated many propositions, hypotheses, proof attempts, counterexamples, obstacles and bridges, can they form a verifiable, replayable, updatable relative-global space, with typed verdicts on which regions are closed, which remain open and which are only locally blocked? Three principles run through the series: Observed Proof Space ≠ Admissible Proof Space ≠ Mathematical Reality; Route Closure ≠ Theorem Proof; and the Globality Typing Principle — every 'global' must say along which axis it is global. Ten papers run from formal foundations to instantiating the whole theory on the Navier–Stokes research corpus (NS_GSM v0.1), each drawing its boundary with a full non-claims list or a series of No-Go theorems.

AMRR Autonomous Mathematical Research Runtime (Chinese only)

PROGRAM-AMRR (Autonomous Mathematical Research Runtime) asks a more downstream question: what mechanisms must a system have so that, when a problem is incomplete, a definition has a gap or the available methods are insufficient, it can diagnose what is missing, supply the missing mathematics under constraints and show that it has not quietly swapped the problem? Three mechanisms are proposed — a Mathematical Domain Gap Map (a 13-domain diagnosis), Constrained Mathematical Domain Completion and a Problem Identity Protocol (no mathematical laundering) — layered on a general cognitive runtime, ACR. This site's independent check: the code, schemas and tests of stages M0–M2 (schema freeze, state transitions, versioned storage) are genuinely built, but the nine stages M3–M11 that diagnose, generate, trace, route and run the autonomous research loop have not a single line of code — for now the 'autonomous research runtime' is only a filing cabinet.

RCIG Recursive Constraint Infinity Game (Chinese only)

PROGRAM-RCIG (Recursive Constraint Infinity Game) was produced entirely by Aletheia (GPT): a 176-round autonomous research log (2026-09-09 to 09-13) that at each round adds one traceable constraint to 'infinity', tests whether it still survives, extracts the residual variable that lets it survive, and then constrains that variable. Run 176 marks Phase I CLOSED, but that is a voluntary process freeze chosen by the author, not a 'real closure' defined by the game's own rules and obtained by argument — Run 176 itself has a whole section listing the conclusions that were not established. From about Run 118–130 the game turns from pure argument to a real machine-checked verification pipeline, yielding 34 K3 formal kernels with runnable code and adversarial attack reports. The 176 rounds are a continuous log, not an independent collection; 9 curated runs plus 4 core deep-dives are live, and the complete 749 files are sealed as originals for download.

NIOT Numerical Information Object Theory

PROGRAM-NIOT (Numerical Information Object Theory) proposes that a usable numerical value is only a finite projection of a richer 'numerical information object', which can be observed ever more finely by a family of 'information probes' placed in matrices that are finite at each point in time but have no fixed upper bound in dimension. Ten v0.1 DRAFT documents (2026-09-22, signed Neo.K, with no AI collaborator stated in the originals) stay at the level of definitions and specification: this site's inventory finds that about 37 statements marked 'theorem' are elementary consequences of definitions, only 3 carry a 1–5 line proof, there is no runnable code, machine formalization or external citation, and the series' own core test (whether it is really more than a 'scalar with annotations') cannot yet be answered by any implementation. All ten documents are live, each on its own page.

PPS Projection Possibility Space

PROGRAM-PPS (Projection Possibility Space) is five methodology papers (Papers 02–06; Paper 01 was not received) that treat the dynamic, projective and perturbative structure of a geometric problem as something to be tested, running from a 3D carrier, dynamic projection, the principal–guest–edge triple and worm-style high-freedom perturbations to projection-induced hidden geometry and adversarial closure, and requiring a counter-certificate. It has no results and no experiments, and every paper has a non-claims list; Lebesgue is only a one-to-three-section case sketch in each paper, and what it develops is Parts VI–IX of the Lebesgue case's open-problem collection.

ACTM Algebraic Combinatorial Transcendence Methodology

PROGRAM-ACTM (Algebraic Combinatorial Transcendence Methodology) is a methodology plus 34 rounds of research notes (2026-09-30 to 10-04): which algebraic seeds and relation structures acquire transcendence degree under 'escape operators' such as the exponential, the logarithm and powers. After round 2 it turns to a concrete open problem, the Four Exponentials Conjecture: R03 rewrites it as an equivalent structural problem (a '2 → 1' dimension gap), R04–R09 study the two types of a hypothetical counterexample and a binary anchored descent tower (BADT), R10–R16 prove that the strategy of a 'rational-linear Waldschmidt–Masser amplifier' cannot work, and R17–R34 follow the route of Dasgupta–Kakde to attack the BADT₂ case, stopping at one explicitly open bridge. No round proves the Four Exponentials Conjecture, Schanuel's conjecture or the transcendence of any specific constant; a 'NoGo' never means the conjecture is false. Unrefereed, AI-executed mathematics.

DMSD Decimal Multiscale Serializable Domain

PROGRAM-DMSD (Decimal Multiscale Serializable Domain) is a software and representation architecture: a theoretically unbounded integer domain is layered by decimal scale levels L_k = [10^k, 10^k+1) and expanded lazily, with an exact summary algebra and a 'No False Closure' epistemic status on each block. It comes with finite-range experiments that use the primes as a stress test (n < 10⁷); the package states and proves no theorem, the prime phases end in null results, and no claim is made about infinitely many primes or about any conjecture. This site recomputed the main counts from scratch and all of them match.