AMRAL

Research Framework: Five Independent Axes

Every research case is described by five mutually independent axes — Case × Methodology × Protocol × Autonomy × Validation — rather than a single mandatory pipeline.

Looking Ahead: 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%.

Program Line Two: 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 two independent cases: direct numerical-optimization attacks on Moser's worm problem, and a bridging theory unifying Kakeya, center-generated spirals, and Moser. The two are not merged in the record.

Program Line Three: 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.

Program Line Four: 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.

Program Line Five: CCM Computational Composite Methodology

PROGRAM-CCM (Computational Composite Methodology) differs in nature from the four lines above — it doesn't attack a single conjecture, but is a methodological theory for mathematical research itself: how to systematically combine computational exploration, formal verification, and human review to produce research state that is traceable, falsifiable, and handoff-able. 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.

Program Line Six: 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.