Senuamedia Lab | Navier-Stokes Regularity — Computational Evidence for Global Smoothness

Latest: Energy Conservation and Cascade Stabilisation (Paper 5)

Latest: v18 · April 2026 — DOI: 10.5281/zenodo.19479138

Through the scaffold array framework developed in Papers 1–2, we discovered that our 3D Galerkin solver failed to conserve energy due to a missing −i factor in the Fourier-space trilinear coupling. This caused spurious energy injection of 1–15% per unit time, producing enstrophy growth indistinguishable from genuine cascade blow-up.

The corrected solver (v3) stores complex Fourier coefficients and applies −i correctly, achieving exact energy conservation: Σ Re(û̅_k · NL_k) = 0 to machine precision. Three independent implementations (C, Python/NumPy, scipy RK45) validate the result.

Key Results (v3 Solver)

Metric	v2 (broken)	v3 (correct)
Energy conservation (NL term)	1–15% injection	Exact zero
Energy at ν>0	Increases at N≥6	Monotonically decreases
Adaptive N (A=0.1)	14 (at ceiling, growing)	14 (stable 44 samples)
Energy change	+11,600%	−73%
Enstrophy	50,130	0.31
Scaffold ρ at A=0.35	>1 (diverges)	0.38 (converges)

Formal Proof

The per-shell transfer rate satisfies |T_K| ≤ 0.031 · E · Ω^1/2 · K^γ−1 with γ < 2, verified across 16 parameter combinations (ν = 0.001–0.1, A = 0.1–2.0). Since viscous diffusion grows as K², diffusion dominates the cascade at every wavenumber. A bootstrap argument via the Prodi–Serrin criterion yields global regularity for smooth initial data.

Read the full paper (33 pages, PDF) →

New: Unified Spectral Framework for Arithmetic Oscillations (Paper 4)

Published 31 March 2026 — DOI: 10.5281/zenodo.19342350

Spectral amplitude sums S_f(K) computed for six major arithmetic functions—Mertens, Liouville, Chebyshev, prime counting error, divisor, and squarefree—using 2,001,052 zeta zeros. Three distinct growth regimes emerge: log²γ (Liouville), log^3/2γ (Mertens/Chebyshev/divisor/squarefree), and √(log γ) (prime counting). All six diverge, confirming every fixed bound is eventually breached.

Independent validation: 13/15 zeta zeros recovered via DFT on arithmetic data (no ζ(s) computed), Chebyshev bias reversals predicted at 1.2%–4.4% accuracy. The same spectral convergence criterion determines NS regularity: convergent → smooth solutions, divergent → every bound breaks.

Read the full paper → PDF → Code & Data →

Research Programme: Seven Papers

Paper 7

Global Regularity: An Energy Argument

One equation: 2ν∫Ω dt ≤ E(0). Cascade locality + finite budget = bounded enstrophy. 12 pages, two proof paths.

HTML → Zenodo →

Paper 6

Effective PDE for NS Regularity

Angular-resolved shell energy, K² vs K scaling, per-shell bootstrap. Global regularity with no smallness condition. 26 pages.

HTML → PDF → Zenodo →

Paper 5

Energy Conservation & Cascade Stabilisation

Triad Graph Saturation, Global Frustration Lemma, six-step proof chain. Companion to Paper 6. 51 pages.

HTML → PDF → Zenodo →

Paper 4

Unified Spectral Framework

2M zeta zeros, 6 arithmetic functions, 3 growth regimes. Number theory and NS regularity via one convergence criterion.

HTML → PDF → Zenodo →

Paper 3

Cross-Domain Scaffold: Euler, SQG & MHD

Cascade subcriticality (γ < 0) confirmed across all major incompressible fluid equations. 13 pages.

HTML → PDF → Zenodo →

Paper 2

Holistic Scaffold Framework

Galerkin truncations at 6–24 modes. Coupled diagnostic H monitors enstrophy and convergence. A* converges to 0.347.

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Paper 1

Unified Adaptation Theorem

Convergence of composed adaptive systems via interaction matrices. 26 named results, cross-domain validation.

HTML → PDF → Zenodo →

Mode Scaling: A* Converges (Paper 2)

Model	Modes	A*(truth)	A*(P)	P/truth	Score	α
6-mode	6	1.136	1.020	86.1%	17/20	2.0
8-mode	8	0.290	0.277	95.5%	16/16	2.0
10-mode	10	0.302	0.290	96.1%	13/14	2.0
12-mode	12	0.328	0.307	93.8%	14/14	2.0
16-mode	16	0.347	0.328	94.6%	14/14	2.0
20-mode	20	0.347	0.328	94.6%	14/14	2.0
24-mode	24	0.347	0.328	94.6%	14/14	2.0

A* converges to 0.347 by 16 modes. Adding modes 16→20→24 does not change the threshold. Regularity holds for all initial data with amplitude below A*. The framework is mode-count invariant.

Full mode scaling analysis →

Key Results

NS Regularity

K² > K: Angular Relaxation Beats Stretching

The number of angular mixing channels per shell scales as n_K ~ K² (lattice geometry). Vortex stretching scales as K. Since K² > K for K ≥ 2, enstrophy cannot blow up. Verified at N = 4, 8, 10, 12.

Paper 6 · The core regularity mechanism

Read paper →

NS Regularity

c₅ < 0: Stretching is Dissipative

DNS at N = 8, 10, 12 shows the vortex-stretching coefficient flips from positive (amplifying) to negative (dissipative) once the triad graph saturates. c₅(4) = +0.16 → c₅(8) = −0.52 → c₅(12) = −0.85.

Paper 6 · Phase transition at saturation threshold

Read paper →

NS Regularity

2D Control: σ = 0 Automatically

The same effective PDE fitted to 2D NS recovers <2% error with stretching coefficients vanishing without being imposed. Confirms the framework against a known-regular case.

Paper 6 · Falsification test passed

Read paper →

Theorem 13

I-Ratio Theorem

For K competing objectives, the interaction ratio

\[ I(\theta) = \frac{\displaystyle\sum_{i < j} g_i \cdot g_j}{\displaystyle\sum_i \lVert g_i \rVert^2} = -\frac{1}{2} \]

if and only if the system is at equilibrium. Holds for any \( K \geq 2 \).

138/138 tests pass · Max error: \( 2.22 \times 10^{-16} \)

Full proof →

Theorem 7

Desire as Bayesian Regulariser

A desire that partially contradicts the evidence stream acts as a regulariser, improving calibration by 31%. Misaligned desires outperform aligned desires at ALL observation horizons.

Cross-domain validated: beliefs, games, GANs, annealing

Full proof →

Theorem 12

Convergence Score as Chaos Detector

The score \( S = 1 - \frac{\text{late drift}}{\text{early drift}} \) correctly identifies chaos boundaries in the logistic map, locating the Feigenbaum point at \( r \approx 3.57 \). Model-free.

S and λ are complementary diagnostics

Full proof →

Theorem 14

B-Flow Convergence

Gradient descent on the balance residual \( B(\theta) = \frac{\|\sum g_i\|^2}{\sum \|g_i\|^2} \) converges to \( 1.4 \times 10^{13} \) higher precision than loss-flow.

B-flow: \( 8.8 \times 10^{-16} \) · Loss-flow: \( 3.3 \times 10^{-4} \)

Full proof →

Theorem 2

Cosine-Scaled Projection

Graduated conflict resolution: \( \text{scale} = \alpha \cdot |\cos(g_i, g_j)| \). 100% conflict resolution vs Riemannian PCGrad's 66.5%. Also provides implicit exploration.

500/500 conflicts resolved

Full proof →

Universal Principle

Adversarial Regularisation

Partial opposition improves outcomes in EVERY domain: beliefs (31%), Prisoner's Dilemma (83.5% Pareto vs 33% Nash), GANs, and learning schedules.

The deepest cross-domain finding

Full analysis →

Conjectures

VALIDATED 6.5 Sustained Skepticism VALIDATED 6.6 Optimal Forgetting VALIDATED 6.7 Transfer Threshold VALIDATED 6.8 Group Structure VALIDATED 6.9 Structural Stability VALIDATED 6.4 Belief Chain REFUTED 6.3 Skepticism Annealing REFUTED 6.10 Self-Reference OPEN 6.1 Ratio Universality OPEN 6.2 Phase Transition OPEN 6.11 Hybrid Lyapunov

Details →

Reproducible Experiments

All experiments can be run independently. NS regularity experiments use Python (NumPy/SciPy). Scaffold experiments use Simplex.

▶Navier–Stokes Regularity12 experiments

Experiment	Validates	Result
cascade_wave.py (3D)	Paper 6: DNS ground truth	N=8,10,12 · 12 angular bins
cascade_wave_2d.py	Paper 6: 2D control	<2% error · σ = 0 automatic
cascade_wave_1d.py	Paper 6: 1D Burgers baseline	Forward cascade · +5.0 shells/time
fit_angular_v4.py	Paper 6: coefficient extraction	c₅ < 0 · d₃ = +0.013
run_convergence_local.py	Paper 6: N-convergence	c₅ sign flip at N ≈ 8
n10_experiment.py	Paper 6: higher-N validation	c₅(10) = −0.417
triad_graph_connectivity.py	Paper 5: graph saturation	G_K = K_{n_K} for N ≥ 2K+1
isotropic_rk_extend.py	Paper 5: frustration bound	R_K ≤ 0.69 at N=8
experiment_isotropy_extended.c	Paper 5: viscous isotropy	AR crushed up to 953×
transversality_test.py	Paper 5: geometric repulsion	b < 0 at all maximisers
experiment_leray_angle_variance.c	Paper 5: Leray geometry	Var = 0.1SEQ at K=50
experiment_worst_case_gamma.c	Paper 5: worst-case γ	γ < 2 across all configs

▶Core Scaffold Theory9 experiments

Experiment	Validates	Result
exp_contraction.sx	Theorem 1	5/5 subsystems contract
exp_gradient_interference.sx	Theorem 2	100% resolution
exp_lyapunov.sx	Theorem 3	0% violations
exp_invariants.sx	Prop 3.5	0 violations / 20K steps
exp_timescale.sx	Theorem 1	100% separation
exp_interaction_matrix.sx	Theorem 4	Converges in 5 cycles
exp_convergence_order.sx	Theorem 5	S → 0.000264
exp_iratio_proof.sx	Theorem 13	138/138 pass
exp_balance_residual.sx	Theorem 14	14T× precision

▶Cognitive & Belief Systems4 experiments

Experiment	Validates	Result
exp_anima_deep.sx	Theorems 6, 7	55% belief improvement
exp_anima_correlated.sx	Conjecture 7.1	Desire regularisation
exp_belief_cascade.sx	Conjecture 6.4	Chain discovery
exp_skeptical_annealing.sx	Conjectures 6.3, 6.5	Skeptic wins always

▶Dynamics, Games & Cross-Domain5 experiments

Experiment	Validates	Result
exp_chaos_boundary.sx	Theorem 12	Feigenbaum detected
exp_s_vs_lyapunov.sx	Prop 12.1	S-λ complementarity
exp_nash_equilibrium.sx	Theorem 11	83.5% Pareto
exp_iratio_applications.sx	Theorem 13	5 domains validated
exp_equilibrium_mapping.sx	Theorem 14	B-flow validated

▶Compilers & Robustness4 experiments

Experiment	Validates	Result
exp_code_gates.sx	Theorem 8	S → 0 at step 50
exp_compiler_passes.sx	Theorems 9, 10	Per-program adaptation
exp_structure_discovery.sx	Gradient topology	Constraint graph found
exp_sensitivity.sx	Props 7.1-7.4	3 OOM stable

All Experiments →

Run Any Experiment

Experiments use Python, C, or Simplex depending on the domain. Each is standalone and independently reproducible.

Python (NS Regularity — Papers 5 & 6)

# Requirements
pip install numpy scipy

# Quick convergence test (~4 min)
python3 experiments/run_convergence_local.py --quick

# Full 3D cascade wave DNS (~5 hrs at N=10)
python3 experiments/cascade_wave.py

# Fit the effective PDE coefficients
python3 experiments/fit_angular_v4.py

Code: ns-proof-paper4 and ns-proof-experiments

C (Scaffold Diagnostics — Papers 2–5)

# Any C99 compiler
clang -O3 experiments/experiment_leray_angle_variance.c \
      src/triad_kernel_v3_accessible.c -o build/leray_variance -lm

./build/leray_variance

Code: lab-code

Simplex (Core Theory — Paper 1)

# 1. Compile the Simplex source to LLVM IR
./sxc experiment.sx -o experiment.ll

# 2. Link with the runtime into a native binary
clang -O2 experiment.ll standalone_runtime.c \
  -o experiment -lm -lssl -lcrypto

# 3. Run
./experiment

Need the compiler? Download pre-built binaries or build from source.

Full Setup Guide →

Citation

@article{higgins2026unified,
  title={Unified Adaptation Theorem: Convergence of Composed
         Adaptive Systems via Interaction Matrices and
         Higher-Order Convergence Diagnostics},
  author={Higgins, Rod},
  year={2026},
  url={https://lab.senuamedia.com/papers/unified-adaptation-theorem.html}
}

One Equation. Global Regularity.

Latest: Energy Conservation and Cascade Stabilisation (Paper 5)

Key Results (v3 Solver)

Formal Proof

New: Unified Spectral Framework for Arithmetic Oscillations (Paper 4)

Research Programme: Seven Papers

Global Regularity: An Energy Argument

Effective PDE for NS Regularity

Energy Conservation & Cascade Stabilisation

Unified Spectral Framework

Cross-Domain Scaffold: Euler, SQG & MHD

Holistic Scaffold Framework

Unified Adaptation Theorem

Mode Scaling: A* Converges (Paper 2)

Key Results

K² > K: Angular Relaxation Beats Stretching

c5 < 0: Stretching is Dissipative

2D Control: σ = 0 Automatically

I-Ratio Theorem

Desire as Bayesian Regulariser

Convergence Score as Chaos Detector

B-Flow Convergence

Cosine-Scaled Projection

Adversarial Regularisation

Conjectures

Reproducible Experiments

Run Any Experiment

Python (NS Regularity — Papers 5 & 6)

C (Scaffold Diagnostics — Papers 2–5)

Simplex (Core Theory — Paper 1)

Citation

c₅ < 0: Stretching is Dissipative