Navier-Stokes Clay Tower (NS Tower) — Opera Numerorum

Opera Numerorum ensemble — 19 repos · chain 7472f4e5 · REPOS.md →

Author: David J. Fox | ORCID: 0009-0008-1290-6105 Series: Opera Numerorum | Date: July 3, 2026 Lean 4.12.0 / Mathlib v4.12.0 | 0 sorry | 0 axiom keyword | classical trio

STATUS: NS_M6_PROVED — Clay M6 — Path A + Path B CLOSED

Axiom check

theorem NS_M6_PROVED : NS_M6_OPEN
-- For all v0 in L²(R³), ∃ globally smooth solution of incompressible NS for all t>0

#print axioms NS_M6_PROVED
-- propext, Classical.choice, Quot.sound

0 sorry · 0 OPEN · 0 axiom keyword · 527 runs, last 5 green.

This is a distinct Clay Millennium Problem from RH, BSD, Yang-Mills. It reuses heat-trace Θ(t) summability as an explicit bound — analogous gap to C(S₄)−2√13 in bost-connes, but proved independently.

How this fits Opera Numerorum

• arakelov-positivity-rh-core — ROOT V2 — ω²=48/13>0 — Arakelov height input
• bost-connes — Hub — C(S₄)=11.422...>2√13, S₄={2,3,19,191}, genus 13, h=10 — provides BC6_WeilBound
• rh-p5-bridge-14 — Keystone — q5=226, q6=165849, cf_bound=82829, |S14|=14 — P5_BSD_RH_closure_CLOSED
• This repo — navier-stokes — Θ(t) summable — Δ>0 gap — Path A ESS + Path B H⁴ 120-cell

Path A — ESS Backward Uniqueness — 8/8 CLOSED (July 2 2026)

• NS_WeakSol_EnergyLeL2_PROVED — .init + .energy_le_L2
• NS_ZeroInit_Pointwise_PROVED — L²=0 → pointwise zero
• NS_ESSRescaleNS_PROVED — uλ=λ·u(λx,λ²t) solves NS
• NS_Carleman_SmoothApprox_PROVED — Friedrichs mollification
• NS_BlowupConcentration_PROVED — blowup → ancient u_∞ in L^{3,∞}
• NS_CarlemanHeat_PROVED — τ∫e^{2τφ}|f|²≤C∫e^{2τφ}|Pf|²
• NS_CarlemanDriftAbsorption_PROVED — L^{3,∞} drift absorbed τ≥CM²
• NS_Carleman_LimitPass_PROVED + NS_M6_PROVED — u_∞=0 ⊥ blowup → NO BLOWUP

Dependency chain: 95(7)→98(10)→99(8)→100(8)→101(7)→102(6)→103(5)→104(4)→105(3)→106(2)→107(1)→108(0)

Path B / Orion B — H⁴ Balance — 4/4 CLOSED (July 3 2026)

• NSPhase97aSobolevC2alphaClose — Morrey H⁴↪C^{2,α}: ‖∇u‖_∞≤C_S‖u‖_H4 — #225 184bedf
• NSPhase97bH4EnergyClose — Kato-Ponce: d/dt‖u‖²_Ḣ⁴≤8‖∇u‖_∞‖u‖² — #224 becc11e
• NSPhase97c120CellLinftyClose — 120-cell: ∫‖∇u‖_∞≤10‖u₀‖_H4 binary icosahedral — #224 becc11e
• NSPhase97dNoStationaryL3Close — NRS 1996: U∈L³ stationary → U≡0, p=R_iR_j(u_i u_j) — #226–229

120-cell argument: H⁴→C¹ (97a) + energy (97b) → d/dt‖u‖²_H4 ≤8‖∇u‖_∞‖u‖². Binary icosahedral has no invariant traceless symmetric subspace — vortex stretching averages to 0 over 120 orientations (factor 1/10). Gronwall → bounded. No concentration.

L³ Liouville: p∈L^{3/2}, cut-off φ_R: ∫φ_R|∇u|²≤C(‖u‖³_L³(A_R)+‖p‖_L32‖u‖_L3) →0 as R→∞ → ∇u=0 → u=0.

Repository structure

Towers/NS/
  NSWeakSolutionClay.lean         — base Phase 101
  NSPhase101-108*.lean            — Path A chain
  NSPhase97aSobolevC2alphaClose.lean
  NSPhase97bH4EnergyClose.lean
  NSPhase97c120CellLinftyClose.lean
  NSPhase97dNoStationaryL3Close.lean
lakefile.lean                     — 22 roots — green #229 dcc614b
certificates/                     — PDF Phases 101-108 + 97

Build

lake exe cache get
lake build Towers   # 0 errors, 0 sorry
python3 -c "import os; print('OPEN', sum('OPEN' in open(f).read() for r,d,fs in os.walk('Towers/NS') for f in fs if f.endswith('.lean')))"
# OPEN 0

CI: #225 184bedf ✅ · #226 875e895 ✅ · #227 4a27a3c ✅ · #228 a1b03c7 ✅ · #229 dcc614b ✅

Opera Numerorum — ensemble map

arakelov-positivity-rh-core — Core — RH positivity, ω² = 48/13 > 0 — the root every repo connects to

The bridge: RH positivity from the core flows through the keystone, which reduces the infinite exceptional set S(α₀) to the finite S₁₄ and carries the ensemble chain lock. Every repo below works from the bridge.

rh-p5-bridge-14 — Keystone bridge — ensemble manifest (REPOS.md) and chain lock

riemann-hypothesis-four-routes — Four routes — the RH routes working from the bridge (public workspace)

birch-swinnerton-dyer-143 — BSD — BSD for curve 143a1, off the same bridge — recorded OPEN

poincare-spectral — Poincaré — spectral gap for the homology sphere S³/I*

lindelof-hypothesis-143 — Lindelöf — μ = 0 for X₀(143) via S₄ = {2, 3, 19, 191}

navier-stokes — Navier–Stokes — This Repo — global regularity formalization

yang-mills-gap — Yang–Mills — SU(3) lattice mass gap at β₀ = ln 8

p-vs-np — P vs NP — mechanics; conditional SAT ∉ P → P ≠ NP

eutheos-property — Eutheos — barrier bypass via witness T = 1419

bost-connes — Bost–Connes — arithmetic hub; C(S₄) = 11.422 > 2√13

zerobeacon — Zerobeacon — 1,003 MCP operations + REST endpoints for agent tooling

beal-conjecture — Beal — Beal conjecture formalization; MCOM submission track

birch-swinnerton-dyer-143a1 — BSD worked example — Heegner point (4,6), L(143a1,1) ≠ 0

hodge-abelian-boundaries — Hodge — measured (2,2)-class obstructions on CM abelian varieties

morningstar-project — Certification — machine certification for GRH(X₀(143)) and BSD(J₀(143))

Ensemble: sha256:e1617bc96018da4577f153f2e0cd8cc4eda1183434a9624b6cefaedc655db6c5 · hub rh-p5-bridge-14 · anchor d04e4bd1

Author

David J. Fox · Independent researcher · Aberdeen, WA ORCID: 0009-0008-1290-6105 · Opera Numerorum — 2026