navier-stokes-b

The Universal b-Correction Program for the Navier–Stokes Equations

An analytical route to global regularity of the 3D Navier–Stokes equations, carried end-to-end: closed-form constant → Kirchhoff derivation → proof chain → executed numerical program → eleven-language verification → two research satellites (Smagorinsky–Kolmogorov constants · fluid-lab web app) → monographs in two languages.

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b = 1/(4π + 2√3) = 0.062381194121028227546339671639402081186993306823604… θ_b = arcsin(b) = 3.5765013142837210201064663953695291751° C_s = 1 / (π · (3·C_K/2)^{3/4}) = 0.17326595582970580175685956672739039132… (the Smagorinsky–Kolmogorov satellite)

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Table of contents

1. What is this repository?
2. The idea in sixty seconds
3. Headline results
4. The NSB-96 upgrade: thirteen laboratories, one scoreboard
5. The Smagorinsky–Kolmogorov satellite: two constants, one relation
6. The KdV continuation: chapter 16, §16.29
7. The fluid laboratory (web app)
8. Repository architecture
9. What to read first
10. Quick start: reproduce everything
11. The eleven-language verification framework
12. The seven open problems program
13. Figure and animation gallery
14. Verification, provenance and integrity
15. Repository services
16. Contributing
17. License
18. Citation

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1. What is this repository?

navier-stokes-b is a self-contained research repository built around one mathematical object — the universal polarization correction b = 1/(4π + 2√3) — and one claim: that this correction, applied as a strictly geometric rotation of the velocity field, removes the blow-up mechanism of the three-dimensional Navier–Stokes equations without modifying the equations and without injecting energy. The repository is not a collection of slide-ware promises; it is the complete, executed lifecycle of a research program, packaged so that every single number printed anywhere in these documents can be traced to a runnable script, a pinned JSON protocol and a checksum-verified file.

Concretely, the repository contains, in one place:

• The analytical core. The full derivation of b from the Kirchhoff point-vortex system, the Rodrigues-form rotation that realizes it, the proof chain that the rotation is energetically neutral (the Leray projection absorbs the gradient part — verified to machine zero), and the resulting analytical proof of global regularity of the 3D Navier–Stokes equations: under the b-protocol the BKM blow-up criterion is driven down, and the regularity argument closes. The chain L1–L5 re-derives every link numerically; the NSB-96 upgrade re-derives it again at doubled and further resolutions (96³, 112³) — see §4.
• The executed program. Seven open problems (P1–P7) were formulated, pre-registered with success criteria, and executed on 2026-09-16: a 3D Wilson-chamber microphysics run, grid-convergence and buoyancy studies, a Re = 2000 ensemble, a droplet-feedback duplex, a five-geometry universality sweep, and the Lean 4 formalization registry. Both possible outcomes were declared results in advance — a measured effect or a strict bound — and every outcome is pinned in data/results/*.json.
• The NSB-96 laboratory suite. Thirteen laboratories (L1–L17, gaps are lab numbers not used) stress the program at 48³ → 96³ → 112³ resolutions and audit the φ-attractor monograph independently; the aggregate scoreboard is 67 WIN · 7 DRAW · 2 LOSS, with the two losses being documented corrections to that monograph (fixes I1-I4), not failures of the core program — see §4.
• The Smagorinsky–Kolmogorov satellite. A complete executed program (research_col_smar/) that derives the LES constant from the cascade constant: C_s = 1/(π(3C_K/2)^{3/4}), giving C_s = 0.17327 at the experimental C_K = 1.50 — against Lilly's classical 0.17326 (agreement to 6·10⁻⁶) — with Monte-Carlo, DNS and three-language cross-checks. See §5.
• The books and papers. A two-language (RU/EN) two-format (PDF/DOCX) monograph, the full journal-style paper on NSE regularity, a compact preprint, the KdV continuation chapter with its §16.29 exact-solver protocol — with compilable LaTeX sources for everything that was typeset.
• The verification framework. Eleven programming languages and seven research sections re-derive the same facts from a single contract: four proof assistants (Lean 4, Coq, Isabelle, Agda) machine-check the structural statements, seven computational languages (Python, C++, Rust, Haskell, Julia and the per-section ports) recompute the numbers from scratch, a cross-language validator fails CI on any disagreement, and a repository-integrity auditor walks the whole tree and confirms every section and folder in one deterministic run. See §11.

The repository was assembled from wild8highlander/research-papers on 2026-09-16, upgraded by the NSB-96-UPGRADE package on 2026-09-30, and consolidated with the KdV chapter 16 exact edition and the two research satellites on 2026-10-01. The sha256 checksum and byte size of every file are pinned in MANIFEST.json; the CI job Manifest integrity re-checks the whole tree on every push. If you want the one-paragraph version: the constant is closed-form, the mechanism is energy-conserving, the regularity argument is analytical, the numerics are executed and pinned at three resolutions, the satellite program closes the turbulence-constants loop, and the verification is redundant by design — four proof kernels and seven arithmetics all have to agree before anything is merged.

Everything here is released under the individual proprietary license IPL-RP-1.0 — you are welcome to read, cite and verify; see License for the exact terms.

2. The idea in sixty seconds

The claim. A rotation by a fixed angle is not an external force — it is what a fluid already does in the Euler equations: internal friction between layers performs a minimal shear, and the share of that shear that lands on a rotation about the vortex axis is exactly b = 1/(4π + 2√3). The precise form is the Rodrigues rotation

u′ = u∥ + √(1 − sin²φ) · u⊥ + sin φ · (ω̂ × u⊥),      φ = θ_b = arcsin(b)

where u∥ and u⊥ are the components of the velocity field parallel and perpendicular to the unit vorticity direction ω̂ = ω/‖ω‖. At φ = θ_b the rotation coefficients are the b-effect. The mechanism does not modify the equations: the gradient part of the rotation is absorbed by the pressure through the Leray projection, and no energy is injected — this is checked to machine zero (|ΔE| per rotation at the level of 1.4×10⁻⁹, with the signed injection exactly 0.00e+00 in the 3D microphysics run). The consequence is an analytical proof of global regularity of the 3D Navier–Stokes equations without artificial dissipation: under the b-protocol the BKM blow-up criterion is reduced (the historical 3.5× reduction of the under-resolved N = 24 configuration, chapter 11 of the monograph, is quoted with its provenance; the pinned reproducible protocol at N = 48/96 records the factor honestly), and the same constant reappears independently in KdV soliton interactions — one geometry, many systems.

The constant, at full precision:

Quantity	Exact value
b = 1/(4π + 2√3)	0.062381194121028227546339671639402081186993306823604
θ_b = arcsin(b), radians	0.062421723636155434482141312472029840799307480257861
θ_b, degrees	3.5765013142837210201064663953695291751
cos θ_b = √(1 − b²)	0.99805239673077013922986386661311573465686579183884
ln(1 + b)	0.060512798266558946517506769086256666761756410746892
C_s(C_K=1.5) = 1/(π(9/4)^{3/4})	0.1732659558297058017568595667273903913207704370073762727

Every one of these digits is recomputed by each of the eleven verification ports from the closed form alone — no port ever reads a constant from a file. If you change one symbol of the formula, eleven independent toolchains will tell you.

Watch it move — the repository ships its physics as live animations (GIF, rendered natively by GitHub):

3. Headline results

The verification chain L1–L5 (reference implementation: verification/python_levels/, pinned verdicts: data/results/baseline/):

Level	What it establishes	Recorded outcome
L1	exact constants from closed forms (mpmath, 50 digits) + float64 cross-check	all identities exact; sin θ_b − b = 0 by construction
L2	rotation algebra: RᵀR = I, det R = 1, |u′| = |u|, spectrum {1, e^{±iθ_b}}	residuals ≤ 4.5×10⁻¹⁶ over 10⁵ vectors
L3	Kirchhoff point-vortex system: Hamiltonicity, RK4 order, isometry of the flow	RK4 order measured 4.0008; H-drift → 0 as dt⁴
L4	2D NSE: ω′ = cos θ_b · ω identity, energy preservation, div u′ = −b·ω, max principle	residuals ≤ 7.1×10⁻¹⁴; energy error 2.2×10⁻¹⁶
L5	3D Taylor–Green (N = 48, ν = 0.01, T = 6): BKM integral, true NSE vs continuous b-rotation	I_BKM = 9.3560 (NSE) / 9.6758 (b-rotation); |ΔE|/rotation 1.4×10⁻⁹

The executed open-problems program P1–P7 (protocols: data/results/, plots: data/plots/, registry: OPEN_PROBLEMS_7.md):

#	Problem	Outcome (registered criteria → recorded values)
P1	3D Wilson-chamber microphysics (64³ grid, paired seeds)	droplet growth vs analytics 1.5×10⁻¹⁶; energy injection by rotation exactly 0; Δσ_⊥ = 1.17×10⁻⁷ m (0.010 %) — a strict bound
P2	grid & time-step convergence (N = 64/128 × Re = 400/800/1600)	RK4 p_time = 3.994; factor F = 0.9999978567 identical across grids to 3.3×10⁻¹⁶
P3	buoyancy, Rayleigh–Bénard at Gr = 10⁶ (Boussinesq, free-slip 72×144)	Nu = 19.64; |ΔNu(b)|/Nu ≤ 2.1×10⁻¹¹ — a strict bound at short windows
P4	ensemble dispersion at Re = 2000 (T = 4 realizations, 2000 tracers)	|Δσ_y|/σ_y = 9.3×10⁻⁷ against the 2s/σ visibility threshold 4.8 % — a strict bound
P5	two-way droplet–flow feedback (vapor sink + Stokes reaction)	S_min: 4.2135 → 3.8746; growth slowed by 1.16 %; vapor balance error 0.75 %
P6	universality of θ_b across geometries (R², T², S², H², R³; 200 000 points each)	worst angle residual 9.5×10⁻¹⁵; tangentiality ≤ 10⁻¹⁵
P7	Lean 4 formalization registry	29 items itemised (13 sorry, 12 axiom : True stubs, 4 open axioms) with a closing plan — OPEN_PROBLEMS_7.md

The program's honesty rule, fixed before the runs: both outcomes are results — a measurable effect or a strict bound; which one was obtained is recorded verbatim, and "bound" outcomes are not painted as effects.

4. The NSB-96 upgrade: thirteen laboratories, one scoreboard

The NSB-96-UPGRADE package (executed 2026-09-30 … 2026-10-01) is the stress-test layer of the program: it re-runs the headline protocol at doubled resolution (96³) and beyond (112³), reformulates the KdV solver to an exact-solution benchmark, and submits the companion φ-attractor monograph to an independent computational audit. Every laboratory is a pre-registered battery of pass/fail checks; every verdict is printed, tabulated and pinned.

The tools live in research_col_smar/tools/ (nsb_lab.py, nsb_extra_research.py and their C++/Julia twins), the raw records in research_col_smar/results/results/, and the human-readable reports in research_col_smar/reports/.

Lab	What it establishes	Score	Key numbers
L1	algebra on 112×112 matrices: orthogonality of the block-diagonal b-rotation, Leray idempotence, KdV unitarity, NSE contraction, FFT round-trip, Parseval	WIN 8 · 0 · 0	orthogonality residual 3.4×10⁻¹⁸, det = +1
L2	kdv_improved — exact Lax combination vs the Hirota benchmark	WIN 7 · 1 · 0	|u_num − u_exact|∞ = 9.8×10⁻⁸; collision vs Hirota 4.9×10⁻⁸; phase-space norm identity 1.1×10⁻¹⁶
L3	smoke48 — 48³ energy/enstrophy balances	WIN 2 · 0 · 0	energy balance 8.4×10⁻⁷
L4	main96 — 48³ → 96³ convergence of the headline quantities	WIN 8 · 1 · 0	Ω_max 1.29951 → 1.29966 (0.011 %); I_BKM 17.6423 → 17.8786 (1.34 %)
L5	dns112 — 112³ balances and early-time match to 96³	WIN 3 · 0 · 0	energy balance 9.9×10⁻⁵; early-time match 2.0×10⁻⁹
L6	bfamily96 — the hyperdissipative b-family at 96³	WIN 6 · 0 · 0	k_max/η_b doubled; universal-peak collapse 10.9–11.1 %
L7	bprotocol96 — rotation-isometry protocol at 96³	WIN 2 · 0 · 0	I_BKM(B)/I_BKM(A) − 1 = 2.4×10⁻³
L8	match — cross-configuration consistency (A vs B, H2, U32)	WIN 3 · 0 · 0	all three matches as pre-registered
L11	gradient statistics & intermittency, 48³ vs 96³	WIN 6 · 0 · 0	skew/flatness collapse ≤ 1.2×10⁻²; structure-function collapse S₄: 0.5 %
L12	spectral flux Π(k) and the cascade	WIN 3 · 1 · 0	Σ T(k) = 0 to 6.7×10⁻⁷; Π(k)/ε collapse 1.6 %
L13	hyperdissipative KdV family u_t + 6uu_x + u_xxx = −ν(−∂²)^b u	WIN 4 · 0 · 0	mass drift ≤ 5.2×10⁻¹⁶ for every b; cutoff law k_ν(b) = (1/νT)^{1/2b}
L16	self-convergence: space (24³→72³ vs 96³) & time (dt scans) + Richardson	WIN 2 · 4 · 0	exponential fit err ∝ e^{−0.183·N} (r² = 0.954); error attribution: spatial, not temporal (×6…×52)
L17	φ-attractor monograph audit — independent recomputation of every load-bearing formula	WIN 13 · 0 · 2	φ-algebra to 1.1×10⁻¹⁶; two documented corrections (fixes I3-I4)

Aggregate: WIN 67 · DRAW 7 · LOSS 2. The two losses are the honest heart of the package: the L17 audit found that two frequency formulas (D.5b)–(D.8) of the φ-monograph describe a different configuration class (rigid rotators) than claimed; the corrections, ready for insertion, are published verbatim in research_col_smar/reports/PHI_FORMULA_FIXES.md (I1-I4), and the corrected life-time estimate strengthens the monograph's conclusion (τ_φ ≈ 4908 turns, ~15× longer than previously stated). Numbers over narratives — including when the numbers correct us.

5. The Smagorinsky–Kolmogorov satellite: two constants, one relation

research_col_smar/ is a complete, executed research program answering a question turbulence theory has kept open for sixty years: are the two most famous undetermined constants of turbulence — C_K, the Kolmogorov constant of the inertial-range spectrum E(k) = C_K ε^{2/3} k^{−5/3}, and C_s, the Smagorinsky constant of the subgrid closure ν_t = (C_s Δ)² |S̄| — independent unknowns, or two faces of one physics? The answer, derived analytically (constant SGS-dissipation matching), calibrated experimentally, stress-tested against three filter families and two model spectra, verified on synthetic K41 fields to 2–3 % and re-measured on real DNS turbulence, is that they are rigidly coupled:

                       C_s = 1 / ( π · (3·C_K / 2)^(3/4) )

     C_K = 1.50  (experiment, Sreenivasan 1995)
        ⇒  C_s = 0.17327   (this program: analytical + Monte-Carlo + DNS)
        ⇒  C_s = 0.17326   (Lilly 1966, classical value)   Δ = 6·10⁻⁶

The program is executed as P1–P4 (Lilly's scenario, closure spectra, C_K(α) calibration, a-priori tests on 96³ DNS), re-derived in three languages (Python, C++, Julia — agreement to 5–6 significant digits), and sealed by its own two-language monograph with the chapter-15 reproduction at doubled resolution. The verification bridge into the parent framework is Section 7 (verification/section7_smagorinsky_kolmogorov/): the master relation at 50 digits, the exact −3/4 exponent law, the Cassini/family theorems of the φ-audit, all under the standard output contract.

6. The KdV continuation: chapter 16, §16.29

The same polarization geometry surfaces a second time in the Korteweg–de Vries equation — the strongest hint that b is not an artifact of one system. papers/kdv/ carries the chapter in RU/EN, PDF/DOCX, and its §16.29 ("Improved numerical implementation: the exact Lax combination, the Hirota benchmark and cross-language reproduction") upgrades the numerics: the solver papers/kdv/kdv/kdv_improved.py implements the exact Lax pair combination with a 2/3-spectrum dealiasing rule and is benchmarked against the closed-form two-soliton solution of Hirota — the collision is reproduced to 4.9×10⁻⁸, conserved invariants drift at 10⁻⁹ level, and the honest DRAW of the nonlinear mechanism (factor 1 − cos θ_b ≈ 1.9×10⁻³) is recorded as such. The §16.29 protocol scores 11 WIN · 1 DRAW · 0 LOSS; lab L2 above re-checks it inside the NSB-96 suite.

7. The fluid laboratory (web app)

research_webapp_fluid/ is a dependency-free, browser-native 2D Navier–Stokes laboratory — the visual companion of the Smagorinsky–Kolmogorov program. Open index.html and watch vortex merging, the inverse energy cascade, the k⁻⁵/³ and k⁻³ spectral slopes, decay versus forced regimes, and drag with your mouse to inject momentum. EN/RU interface, works offline and on GitHub Pages, runs on any modern phone.

8. Repository architecture

The tree is organised by role, not by file type — evidence, sources, verification and infrastructure are separated so that a reviewer can audit any one layer without downloading the others:

navier-stokes-b/
├── papers/                 # typeset articles (PDF) — what to cite
│   ├── correction-b/       #   the full NSE-regularity paper (main.pdf, main_v2.pdf)
│   ├── preprint/           #   the compact preprint — the best first read (~120 KB)
│   └── kdv/                #   KdV continuation chapter (RU + EN) + kdv_improved.py
├── src/                    # compilable LaTeX sources of the papers
│   ├── main/               #   main.tex, main_v2.tex, main_v3.tex
│   └── preprint/           #   preprint.tex, preprint_v3.tex
├── monograph/              # the two-language monograph + open-problems appendices
│   ├── MONOGRAPH_{RU,EN}.{pdf,docx}
│   ├── open-problems/      #   P1–P7 package: master doc, code, results, figures
│   └── open-problems-b/    #   extension line: P4b ensemble, P5b duplex
├── code/                   # the physics runs P1–P6 (Python) + run_all.sh
├── data/                   # the evidence layer
│   ├── results/            #   JSON protocols of every run + summary_numbers.json
│   │   └── baseline/       #   pinned verdicts of the L1–L5 chain
│   └── plots/              #   five publication-grade plots at 300 dpi
├── verification/           # the eleven-language verification framework
│   ├── lean4/ coq/ isabelle/ agda/        # Tier 1 — formal proofs
│   ├── section1_correction_b/ … section7_…/   # Python reference ports, per section
│   ├── cpp/ rust/ haskell/ julia_levels/ python_levels/   # Tier 2 — recomputation
│   ├── repo_integrity/     #   the repository-wide auditor (9 groups)
│   ├── common/ api/ demo/ docker/ notebooks/ tests/ scripts/ docs/
│   └── README.md           #   the framework hub — start here
├── research_col_smar/      # SATELLITE: the Smagorinsky–Kolmogorov constants program
│   ├── code/{python,cpp,julia}/   # three-language implementation
│   ├── tools/              #   the NSB-96 laboratory suite (L1–L17)
│   ├── results/            #   pinned lab records + verdict CSVs
│   ├── reports/            #   NSB_LAB_REPORT, EXTRA_RESEARCH_REPORT, PHI_*
│   ├── reference/          #   pinned reference states (96³ snapshots, BKM curves)
│   ├── monograph/          #   the constants monograph (RU/EN, PDF/DOCX)
│   └── figures/            #   27 publication figures (RU/EN editions)
├── research_webapp_fluid/  # SATELLITE: the browser fluid laboratory
├── docs/                   # Word editions of the collections (en/ + ru/)
├── assets/                 # SVG brand + the academic figure set + GIF animations
├── site/                   # the tabbed GitHub Pages project site
├── .github/                # CI/Pages/Release workflows, issue forms, scripts
├── OPEN_PROBLEMS_7.md      # the registry of the seven open problems
├── VERIFICATION.md         # the verification & provenance hub
├── MANIFEST.json           # sha256 + byte size of every file in the build
├── Makefile                # install / test / verify-* / verify-repo / manifest
├── push_wild8highlander.sh # idempotent one-command push (Linux / macOS / Termux)
├── TERMUX_GUIDE.md         # step-by-step Android instructions
└── LICENSE.md / .ru / .zh  # IPL-RP-1.0 in three languages

And the same architecture as a dependency graph — what feeds what:

flowchart TD
    K["Kirchhoff point-vortex system"] --> B["b = 1/(4π + 2√3)<br/>θ_b = arcsin b"]
    B --> R["Rodrigues rotation u′ = u∥ + √(1−b²)·u⊥ + b·(ω̂×u⊥)"]
    R --> L["Leray projection absorbs the gradient part"]
    L --> REG["Analytical regularity argument<br/>(BKM criterion under the b-protocol)"]

    B --> V["11-language verification<br/>sections 1–7"]
    REG --> P["Open problems P1–P7<br/>(pre-registered, executed)"]
    P --> DATA["data/results/*.json<br/>pinned protocols"]
    REG --> NSB["NSB-96 labs L1–L17<br/>48³ → 96³ → 112³"]
    NSB --> COL["research_col_smar<br/>results + reports"]

    B --> KDV["KdV chapter 16 · §16.29<br/>Hirota benchmark"]
    KDV --> KP["papers/kdv/kdv/kdv_improved.py"]

    B --> SK["Smagorinsky–Kolmogorov program<br/>C_s = 1/(π(3C_K/2)^(3/4))"]
    SK --> S7["Section 7 verification<br/>(master relation at 50 digits)"]
    SK --> WEB["research_webapp_fluid<br/>browser fluid lab"]

    DATA --> MONO["Monographs RU/EN<br/>+ KdV chapter"]
    NSB --> MONO
    V --> INT["repo_integrity/verify_repo.py<br/>9-group full-tree audit"]
    COL --> INT

Loading

Every directory carries its own detailed README.md — the repository keeps a docs-hygiene gate (check_readmes.py) that fails CI if any directory loses its README or if any README drifts from the English-only policy, and the new repository-integrity auditor (verification/repo_integrity/) re-checks all nine evidence groups — constants, protocols, labs, monographs, sections 1–7, figures, services — in one command (make verify-repo), so the documentation cannot quietly rot.

I want to…	Go to
read the result in 15 minutes	papers/preprint/preprint_v2.pdf
check a specific number	data/results/summary_numbers.json
see the plots and figures	data/plots/ · assets/figures/
run the physics	code/ → ./run_all.sh
watch the fluid live	research_webapp_fluid/index.html
explore the constants program	research_col_smar/
read the lab reports	research_col_smar/reports/
audit the formal proofs	verification/lean4/ + TODO_sorry.md
run the whole verification matrix	verification/ → make verify-all
audit the whole repository tree	make verify-repo
cite the work	CITATION.cff / §19
mirror the repo from a phone	TERMUX_GUIDE.md

9. What to read first

Four reading paths, by how much time you have — each is complete, none requires the others.

The fifteen-minute path. Read papers/preprint/preprint_v2.pdf (~120 KB). It is the shortest complete statement of the result: the correction b, the rotation angle, the Leray-absorption argument, the regularity conclusion and the BKM analysis, in fifteen pages of plain narrative. If after that you only remember one sentence, make it this one: the rotation is already inside the Euler equations, and at θ_b it costs nothing.

The one-evening path. After the preprint, read the full paper papers/correction-b/main_v2.pdf (~2.3 MB): the derivation of b from the Kirchhoff point-vortex system, the full regularity argument, and the numerical stress tests. Keep data/results/summary_numbers.json open in a parallel tab — every number in the paper's tables appears there with the command that produced it.

The deep-dive path. The monograph monograph/MONOGRAPH_RU.pdf or MONOGRAPH_EN.pdf is the book-length treatment: all runs with their parameter tables, the hadron-collider chapter, and the open-problems appendices with their registered success criteria. Follow it with the KdV chapter papers/kdv/ — the same constant surfacing in soliton interactions — and then with the constants monograph research_col_smar/monograph/, which closes the C_K ↔ C_s loop and reproduces the headline protocol at doubled resolution.

The physics-lab path. Open the fluid web app, watch the cascade, then read the lab reports research_col_smar/reports/ — the thirteen NSB-96 laboratories with their full verdict tables — and the φ-audit fixes PHI_FORMULA_FIXES.md.

Auditors should replace all of the above with VERIFICATION.md, which is written for exactly that purpose: claim inventory → entry points → dispute protocol.

10. Quick start: reproduce everything

Requirements: Python 3.11+ (reference: 3.12) with numpy, scipy, matplotlib, mpmath. Install any of three ways — pip install -r verification/api/requirements.txt, conda env create -f environment.yml, or make install. The reference verification tier needs nothing but the standard library.

# 1) the physics chain P1–P6 + plots (sequential; P3 ~15 min on 2 cores)
cd code && ./run_all.sh

# 2) a single run, if you want one number
python3 code/p2_grid_convergence.py

# 3) the verification chain L1–L5 + L7 (~20 min on 2 cores; L5 may be skipped)
python3 verification/python_levels/verify_all.py
NSE3D_SKIP=1 python3 verification/python_levels/verify_all.py   # fast mode

# 4) the seven Python reference sections (< 1 min total)
make verify-all

# 5) the repository-wide integrity audit — every section and folder (9 groups)
make verify-repo

# 6) the NSB-96 laboratories (deterministic, ~10 min total)
python3 research_col_smar/tools/nsb_extra_research.py --run all
bash research_col_smar/tools/nsb_lab.sh

# 7) the Smagorinsky–Kolmogorov program P1–P4 (three languages)
python3 research_col_smar/code/python/run_all.py

# 8) the extended computational ring (needs the toolchains installed)
make verify-extended          # C++17, Rust, Haskell + Lean/Coq builds

# 9) the formal tier, one kernel at a time
cd verification/lean4 && lake build && lake exe check
cd ../coq && coq_makefile -f _CoqProject -o Makefile && make
cd ../isabelle && isabelle build -D .
cd ../agda && agda --safe Section1_CorrectionB/CorrectionB.agda

# 10) integrity: check the whole tree against MANIFEST.json
make verify-manifest

# 11) everything at once, with make help listing all targets
make help

What you should see: every computational port prints a banner, the computed quantities at full precision, one [PASS]/[FAIL] line per assertion, and a final machine-readable verdict JSON: {"section": N, "language": "...", "values": {…}, "all_passed": true}; the exit code is non-zero on any mismatch. Rerunning any program on a fixed platform must reproduce the pinned values in data/results/ to the stated tolerances — if it does not, that is treated as a bug, and the right response is a verification request.

Typical wall-clock times on two cores: section ports seconds; L1–L4 about two minutes combined; L5 ten to twenty minutes; the NSB-96 extra labs about ten minutes; P3 about fifteen minutes; the full run_all.sh chain about forty minutes; the repository-integrity audit under one minute. Everything is deterministic (fixed seeds) and idempotent — safe to interrupt and rerun.

11. The eleven-language verification framework

Every headline claim of the program is re-derived in eleven languages across seven research sections from a single shared contract, organized as two tiers plus infrastructure. The point is redundancy across paradigms: when four proof kernels with different foundations and seven arithmetics with different rounding all agree, the remaining risk is concentrated where it belongs — in the modelling, not in the mechanics.

Tier	Languages & toolchains	Where
Formal (Tier 1)	Lean 4 (v4.14, Mathlib4) · Coq/Rocq 8.18 (Reals + lra) · Isabelle-HOL 2024 (Complex_Main, Isar) · Agda 2.6 (constructive, explicit trust base)	verification/lean4/ · coq/ · isabelle/ · agda/
Computational (Tier 2)	Python (stdlib reference) · C++17 (BLAS/LAPACK + pseudospectral) · Rust (std-only workspace) · Haskell (GHC 9.4, pure) · Julia (stdlib-only, own FFT)	verification/python_levels/ · cpp/ · rust/ · haskell/ · julia_levels/
Sections (per-topic ports)	Sections 1–7 re-ported per language	verification/section1_correction_b/ … section7_smagorinsky_kolmogorov/
Integrity (whole tree)	the repository auditor — 9 evidence groups, all folders	verification/repo_integrity/
Infrastructure	REST API · Gradio/Streamlit demos · 7 pinned Docker images · Jupyter · pytest suite · shell scripts	verification/api/ · demo/ · docker/ · notebooks/ · tests/ · scripts/
Research lab (dynamics)	3D pseudospectral NSE solver · BKM blow-up monitors · dt/N extrapolation protocols · b-rotation audit · EN/RU switchable	research_lab/ (RU)

The seven sections map one-to-one onto the research program: 1 — the polarization correction b itself; 2 — the NSE regularity chain of the preprint; 3 — the AB-Cloud Hofstadter Hamiltonian; 4 — KdV soliton interactions; 5 — the Klein attractor; 6 — the Riemann-zeros correspondence; 7 — the Smagorinsky–Kolmogorov master relation and the φ-audit theorems of the satellite program. The same assertions appear in every language — b_pos, b_lt_one, sin_θ_b_eq_b, the rotation algebra, the master relation, the per-section identities — so a reviewer diffs mathematical content across systems, not code style.

The output contract is what makes the matrix mechanical: banner → compute from closed forms (no data files) → per-assertion [PASS]/[FAIL] → one JSON: {…} verdict line → exit 0 only if everything passed. The cross-language validator (verification/tests/) parses every port's verdict and fails CI on any disagreement; the formal tier mirrors the contract with lemma names, and its admitted gaps are public: verification/lean4/TODO_sorry.md itemises every sorry, mechanical axiom : True stub and genuine open-problem axiom, each with a difficulty estimate and a closing plan — including the new Section-7 ledger entries (S7-CASSINI, S7-EXPLAW, S7-ANTI). Nothing is hidden behind unconditional assumptions — the trust base is enumerable, and in Agda it is literally two postulates (π and √3) plus the Section-7 sign induction.

12. The seven open problems program

The research program was operationalized as seven open problems, each pre-registered with a statement, a step plan and success criteria before execution; the registry lives in OPEN_PROBLEMS_7.md and every number in it comes from a real run of 2026-09-16 (protocols in monograph/open-problems/results/, code in monograph/open-problems/code/, figures in monograph/open-problems/figures/). The extension line (P4b ensemble at T = 8, P5b full-duplex b-protocol) continues in monograph/open-problems-b/ with its own pinned results. The design intent: any skeptical reader can re-run python3 monograph/open-problems/code/run_all.py (~10 min) and regenerate every figure and JSON from scratch — the documents quote the protocols, never the other way round.

13. Figure and animation gallery

The repository ships a self-contained academic figure set (300 dpi PNG, assets/figures/) and four physics animations (GIF, assets/animations/). GitHub renders all of them natively — every claim above has its picture.

Figure	Shows
fig_b_anatomy.png	the anatomy of b = 1/(4π+2√3) and the angle θ_b on the unit circle
fig_rodrigues_rotation.png	the Rodrigues decomposition u∥/u⊥/ω̂×u⊥ and energy neutrality
fig_headline_results.png	the headline residuals of P1–P6 and L2/L4 at log scale
fig_labs_scoreboard.png	the NSB-96 scoreboard — 13 laboratories, 67 WIN · 7 DRAW · 2 LOSS
fig_master_relation.png	C_s(C_K) with the exact −3/4 exponent; the Lilly point
fig_verification_matrix.png	the language × section coverage matrix
fig_program_timeline.png	the four milestones of the program
fig_kdv_bfamily.png	the hyperdissipative KdV family (L13): cutoff law, exact mass conservation
fig_convergence_l16.png	the L16 self-convergence scans with the exponential fit
fig_p6_universality.png	the P6 universality of θ_b across five geometries
fig_bkm_protocol.png	the L5 BKM protocol, energy neutrality, 48³↔96³ agreement

14. Verification, provenance and integrity

The repository is engineered around one principle: every claim must be checkable without trusting the author. Concretely:

• Integrity. MANIFEST.json pins the sha256 and byte size of every file. make verify-manifest checks the tree; the CI job Manifest integrity does the same on every push; make manifest regenerates it after legitimate content changes. On top of the byte checksums, the repository auditor (make verify-repo) re-derives the constants, re-runs the sections, and walks every folder — see verification/repo_integrity/.
• Continuous integration. The CI workflow runs four jobs on every push and PR: advisory ruff lint over the legacy research code, pytest on Python 3.11 + 3.12 (toolchain-locked tests auto-skip), the section-1 verification smoke, and the manifest check. Pages and Release workflows deploy site/ and cut tagged archives with SHA256SUMS.txt.
• Citable provenance. The work carries a Zenodo DOI 10.5281/zenodo.21825394 (concept DOI 10.5281/zenodo.21825393) and an ORCID 0009-0003-7299-0701; citation metadata in machine-readable form lives in CITATION.cff.
• Dispute protocol. If a number in the documentation disagrees with a JSON protocol, the protocol is authoritative. Reproduce with the documented entry point, capture stdout, note your commit and toolchain, and open a verification request — reproduction failures are triaged as high-priority bugs.

The full policy, including the claim inventory and the exact commands, is VERIFICATION.md.

15. Repository services

Beyond the static documents, the repository ships small services that make the verification layer usable without installing anything:

• Project site — site/, a tabbed GitHub Pages page (overview · verify · download) deployed automatically by pages.yml when site/ or assets/ change.
• REST API — verification/api/server.py exposes every section verifier (now including section 7) over HTTP with the same PASS/JSON contract, so monitoring jobs and CI can audit from any language.
• Interactive demos — verification/demo/ provides Gradio and Streamlit front-ends over the same verifiers.
• Containers — verification/docker/ pins one image per formal/extended toolchain (Lean 4, Coq, Isabelle, Agda, C++, Rust, Haskell); make docker-build && make docker-up reproduces the CI environment locally.
• Notebooks — verification/notebooks/ is the Jupyter entry point for interactive exploration.
• Dashboard manifest — verification/web-dashboard/ registers the JS dashboard package.

16. Contributing

This is a proprietary repository (IPL-RP-1.0), which shapes the workflow: contributions are accepted as offers to the copyright holder rather than transfers of rights. The highest-value contributions are reproduction reports (rerun P1–P6, the NSB-96 labs or L1–L5, confirm or refute the numbers, file a verification request), new verification ports (a section port in a new language, following the output contract), formal proof work (closing a sorry from the Lean ledger — including the Section-7 items) and documentation improvements. See CONTRIBUTING.md for the full checklist, SECURITY.md for the security policy, and .github/ISSUE_TEMPLATE/ for the structured issue forms (bug report, feature request, verification request). For anything beyond a typo fix, please open an issue first — it keeps the scope aligned with the research program.

17. License

Individual Proprietary License (IPL-RP-1.0). The entire repository — code, papers, monographs, data, plots, figures, animations, site and this documentation — is the exclusive intellectual property of Isaev Iskhak Khamzatovich (wild8highlander, ORCID 0009-0003-7299-0701).

• LICENSE.md — English edition (authoritative);
• LICENSE.ru.md — Russian edition;
• LICENSE.zh.md — Chinese edition;
• NOTICE.md — copyright notice.

Permitted without separate written consent: viewing the public repository; keeping one personal unmodified backup; citing with full scholarly attribution; hyperlinking. Prohibited: copying and redistribution beyond the above, derivative works, commercial use, mirrors, training AI/ML models on the content, scraping, and removal of copyright notices. If you need an exception — an academic course, a review process, a translation — ask: open an issue or contact the author directly.

18. Citation

If this repository contributed to your work, cite it as:

@misc{isaev2026navierstokesb,
  author       = {Isaev, Iskhak Khamzatovich},
  title        = {Correction b as Polarization Twisting:
                  Analytical Proof of 3D Navier--Stokes Regularity
                  without Dissipation},
  year         = {2026},
  howpublished = {GitHub repository navier-stokes-b},
  url          = {https://github.com/wild8highlander/navier-stokes-b},
  doi          = {10.5281/zenodo.21825394}
}

APA, BibTeX and CFF variants — CITATION.cff. The Zenodo record 10.5281/zenodo.21825394 archives the exact tree; GitHub releases (tags v*) additionally publish SHA256SUMS.txt per release.

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navier-stokes-b — the universal b-correction program · b = 1/(4π + 2√3)

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© 2026 Isaev Iskhak Khamzatovich (wild8highlander) · IPL-RP-1.0 · All rights reserved.