← AMRAL · PROGRAM-NS
Navier–Stokes Research Zone · 3D incompressible Navier–Stokes global regularity · a Clay Millennium Prize problem
This is the English entry point for AMRAL's Navier–Stokes research zone. The underlying research — every round, every proof attempt, every source document — was written in Traditional Chinese by the AI research process this site documents; this page is a curated English summary of that work, not a translation of the individual papers themselves. Clicking through to any sub-line below leads to its original Chinese pages. Going forward, newly-authored AI research on this site defaults to English first; this zone's existing ~529 source files predate that policy and are not being retroactively translated.
16 confirmed independent sub-lines, all built. Eleven of them — RFP, CSP, DRC, ANP, CFOP, FCBP, MORP, DCRP, IDRP, TSKR, RKAP — turned out to be a single continuous relay (Cycle I through XI): each round's own closing document names the next program by title, confirmed from primary sources rather than assumed from naming. The other five are independent: NS_O (the original 67-round proof-chain), a novel observer-relativity framework NTLA-O, a large "pure continuous proof route" experiment NS-X72, a process-journal NS-RMRM that turned out to be DCRP's own working log rather than a 12th lineage, and NS-INRS, a DCRP/X72 bridge line that was hiding inside NTLA-O's own source folder until it was correctly identified. Global regularity is fully OPEN across every line — nothing here claims a proof.
Added a 17th sub-line on 2026-09-03: NS-GSM, different in kind from the other 16 — not a research paper, but an executable Reference Runtime software project and audit trail that applies an independent methodology series, Closure-Space Mathematics (CSM), to the existing NS research corpus (a seed dataset plus v0.1 through v0.7, eight versions in total). It adds no new research papers; it only audits existing documents version by version — throughout, the root formal NS, C1, and C2 remain OPEN, and only 13 narrowly-scoped local lemmas/theorems climb the internal authority ladder. CSM itself (ten methodology papers) is a separate, independently-shipped case.
The 17th sub-line, different in kind from the other 16 — not a research paper, but an executable Reference Runtime software project and audit trail applying the Closure-Space Mathematics (CSM) methodology to the existing NS research corpus. Across all eight versions, the root formal NS, C1, and C2 remain fully OPEN; only 13 narrowly-scoped local lemmas/theorems are gradually promoted through the audit tiers.
NS_O · framework + C1-C6 · 67 rounds · EN availableReduces global regularity to two falsifiable claims, Chain Necessity (C1) and Finite Obstruction (C2). All 67 rounds across framework/C1/C2/C3(25)/C4(9)/C5(13)/C6(17) are live. Global regularity remains fully open.
NTLA-O · 9-paper series · EN availableAn original framework by this project's researcher: promotes "are two structures different?" to "different relative to which observer?" Nine formal papers are live; its application to NS problems has its own line, NS-INRS.
NS-INRS · 63 rounds (DCRP43-58/43-QC/60-105) · EN availableA DCRP/X72 bridge line that was sitting inside NTLA-O's own source folder, mislabeled as leftover cross-reference files. Rounds 43-58 run as a parallel branch to the published DCRP 43-58; from round 60 on — where the source text itself declares the compound DCRP/X72 numbering begins — it is genuinely new territory, continuing past both DCRP's and X72's official endpoints to round 105, the current frontier.
NS-DCRP · Cycle VIII · 55 rounds (01-55, 59) · EN availableA single continuous, round-by-round research chain from carrier entropy and concentration recovery through sheet/pancake geometry and viscous lower bounds. Direct continuation of NS-MORP; its raw process journal is NS-RMRM.
NS-RMRM · Cycle VIII process journal · 1 framework round · EN availableNot an independent lineage — the 52-version raw process journal DCRP (Cycle VIII) was drafted in, checkpoint version mapped 1:1 to DCRP round number. Its one genuinely independent contribution, a routing framework for composing mathematician-style research strategies, is built as its own page.
NS-IDRP · Cycle IX · 4 rounds · EN availableFour rounds on the temporal structure of singularity formation; the final audit reduces the surviving obstruction to a tangential singular-impulse phantom and hands off to NS-TSKR.
NS-TSKR · Cycle X · 4 rounds · EN availableAttacks tangential source geometry and the singular residual kernel; the final audit compresses the obstruction to an at-most-one-dimensional hyperbolic mismatch fiber and hands off to NS-RKAP.
NS-RKAP · Cycle XI · 1 round · current frontier · EN availableThe newest round across the entire ten-part relay (CSP through RKAP). Proves covariant-transport transversality and a two-sided PSD lift-tax theorem; a planned RKAP-02 was never written — honestly flagged as the open frontier.
NS-MORP · Cycle VII · 5 rounds · EN availableCompresses the minimal nonzero obstruction to a normal form; the final round proves the survivor is a minimal, zero-tax, kernel-saturated diffuse carrier, and hands off to NS-DCRP.
NS-FCBP · Cycle VI · 6 rounds · EN availableSearches for a forest functional that is simultaneously globally finite, near-critical, and stable under branching; FCBP-03 proves a non-summable slow-scale critical lift window before handing off to NS-MORP.
NS-CFOP · Cycle V · 3 rounds · EN availableProves that standard finite budgets (Leray enstrophy time, H⁻² forcing) are summable per scale and cannot exclude infinite cascades, handing off to NS-FCBP.
NS-ANP · Cycle IV · 10 rounds · EN availableBuilds a causal-lineage theory for pre-singular points; proves a horizon causal-forest necessity theorem, though atomic-level necessity (CN3) stays open, handing off to NS-CFOP.
NS-DRC · Cycle III · 7 rounds · EN availableCloses the reservoir-mechanism classification for the Type-I branch, but the audit finds closure does not imply chain necessity — handing off to NS-ANP.
NS-CSP · Cycle II · 8 rounds · EN availableSynchronizes mid-eigenvalue strain action with moving frequency-window action; closes with an explicit four-mechanism residual core, handing off to NS-DRC.
NS-RFP · Cycle I · 12 rounds · EN availableThe earliest traceable point of the whole relay lineage. Builds a formal architecture, exact parent-lineage machinery, and a unified tax ledger; the final round reduces the danger core to a triple-divergent-action intersection and hands off to NS-CSP. Directly interweaves with NS_O's own C3 series.
NS-X72 · independent methodology · 69 of 71 rounds (missing 11, 14) + 2 foundational papers · EN availableThe largest single sub-line on the site, and fully independent of the eleven-part relay. A "Generalized Structural Continuum Hypothesis" treats discrete-looking structure as innocent until proven essential, then 71 rounds each attack regularity using only continuous tools until hitting a locatable STOP/TRANSITION/ILLEGAL node. Round 71 is the live frontier, not a dead end — it lists nine concrete next steps in its own text.
This landing page is the first English-language surface on AMRAL, built 2026-08-31, starting with NS-RKAP
(Cycle XI) as the first sub-line fully translated — hub and detail pages, sourced from a Gemini 3.1 Pro
translation of the original Chinese .md files, verified for structural fidelity (headings,
LaTeX math blocks, and boxed statements all match the Chinese originals exactly). As of 2026-09-02, all 16
of the site's NS sub-lines are translated (fifteen of them this way; NS-INRS's own source papers were
already written directly in English, so only this site's Chinese commentary needed translating there). The
pattern (a mirrored /en/<path>/ tree, reciprocal hreflang
tags, and a language-switch control in the nav) is the same one RKAP first proved out.