The 3D Fourier–Galerkin triad kernel (v3) stores complex Fourier coefficients and applies the −i factor in the trilinear coupling; it conserves energy. The earlier real-coefficient kernel injected up to 15% at N = 7; the defect was found and corrected on 2026-03-24. An FFT pseudo-spectral solver and the triad kernel agree to 2.8×10−17 at N = 4 over T = 0.01. Read the record · Zenodo (v19)

The v3 kernel and its checks

FileLanguageDescriptionResult
solvers/v3/triad_kernel_v3.c C 3D Galerkin truncation: mode enumeration, triad precomputation, Leray projection, complex coefficients with the −i factor Energy conserved by the nonlinear term
experiments/v3_final/experiment_energy_audit.c C Energy conservation test at ν = 0 across several Δt values, real-coefficient kernel vs v3 Real-coefficient kernel injects up to 15% at N = 7; v3 drift is integrator error and halves with Δt
validation/pseudospectral_compare.py Python Cross-architecture check: FFT pseudo-spectral solver against the triad kernel Agreement to 2.8×10−17 at N = 4, T = 0.01
validation/taylor_green_test.c C Taylor–Green vortex with known exact energy decay Matches exp(−6νt) to 10−7
validation/ns_galerkin_3d.py Python Independent NumPy implementation of the 3D Galerkin system E(0) matches C to 10−8; nonlinear energy rate zero
validation/dedalus_ns_test.py Python Cross-validation with scipy RK45 (adaptive, rtol = 10−10) E(T) within 9×10−6
Thermalisation check C Inviscid truncations: Ω/E against the lattice mean ⟨k2⟩ (Lee–Kraichnan absolute equilibrium) Within 1% at every N from 3 to 8; unchanged under time-step halving
ns-proof-experiments/experiment_leray_angle_variance.c C Angular variance Var(cos2αT) of triad interactions on Z3, shell by shell Positive at every shell computed; tends to 4/45

DNS and profile instruments

InstrumentLanguageMeasuresResult
verify_review_claims.py Python Pointwise Kolmogorov closure at t = 0 (check 1), intra-shell dispersion for the shear flow (check 2), amplitude scaling of a per-mode estimate (check 3), Burgers vortex direction and stretching (check 4) Closure fails for a spectral-gap datum (E10 = 0, Π10 = −1.04); dispersion constant nK/2 = 9, 31, 175, 625, 2517 at K = 1, 2, 5, 10, 20; Burgers vortex is unidirectional with sustained stretching
N = 192 closure script Python Pointwise closure for localised vortex blobs of size ℓ and in the exponential dissipation range 13.6× to 55× for ℓ = 0.25 to 0.15; 7.5×, 30×, 98× at K = 40, 48, 52
Six-patch field Python Divergence-free field of six Burgers-type patches: direction second moment and enstrophy-weighted stretching ‖Q‖ = 0, resultant 0, stretching 0.999
E9 · pincer_probe.py Python Mean-strain and strain–direction covariance shares of the vortex stretching on DNS at the enstrophy peak Mean-strain term at most 6% at the flux scale; the covariance carries it (Betchov's identity in explicit form)
A1 · shear_profile.py Python Angular shear G = Pξ⊥Σξ of homogeneous critical profiles ω = |x|−2Ψ(x̂); linearisation and nonlinear minimisation, l ≤ 4 G vanishes to 10−15 on swirl-free axisymmetric profiles only; null space = symmetry count; six minimisations converge to that family; solver validated to 10−14
Spectral3D · zoom_classification.py Python 2/3-dealiased RK4 pseudo-spectral DNS ν = 0 energy drift 10−7

Solver versions

FileStatusDescription
solvers/v3/triad_kernel_v3.c CURRENT Complex Fourier coefficients, −i factor, energy conserved. Use this.
solvers/v2/triad_kernel_v2.c SUPERSEDED Real coefficients, missing −i; injects energy. Kept so the 2026-03-24 audit can be rerun.

Three Fluid Equations: Euler, SQG & MHD Truncations

Fourier–Galerkin truncations of 3D Euler, 3D MHD and 2D SQG integrated with a first-order splitting scheme, with energy, enstrophy and per-shell diagnostics across nested truncation levels. At ν = 0 the truncated systems relax to the Lee–Kraichnan absolute equilibrium. A finite Galerkin system cannot blow up, so bounded enstrophy in these runs is not evidence about any of the three PDEs. Read the record · Zenodo (v2)

EquationConfigurationsRecord
3D Euler (ν = 0)4 IC families, N = 2–8Ω/E reaches ⟨k2⟩ within 1% at every N; the ≈22% enstrophy growth is the equilibration; unchanged under time-step halving
2D SQGCritical (κ = 0.01) and inviscid, N ≤ 15Energy and enstrophy trajectories. Critical SQG is globally regular by theorem (Kiselev–Nazarov–Volberg 2007; Caffarelli–Vasseur 2010); inviscid SQG blow-up is open and the runs bear on it neither way
3D MHD4 configurations, N ≤ 6Kinetic and magnetic energy trajectories of the truncation

Arithmetic Oscillations

Truncated Odlyzko–te Riele sums over the non-trivial zeros of ζ, with computed ζ′(ρn) over Odlyzko's 2,001,052 zeros, and recovery of the first zeta zeros by Fourier analysis of M(x)/√x. All in C. Read the record · Zenodo (v2) · Code

FileDescriptionResult
quick_sum.cTruncated amplitude sum Σn≤K 2/|ρnζ′(ρn)| with computed ζ′(ρn)Mertens sum crosses 1 at K = 58 and 2 near K = 1,040; growth consistent with the conjectural (log T)5/4 law
paper_data.cGenerates the tables in the recordReproduction in <2 s
explicit_formula.cExplicit-formula evaluator for M(x)/√xMillisecond evaluation at 103 to 10200
wave_plotter.cCSV output for the reconstruction figuresFigure data
exp33_wave_decomposition.cDFT of M(x)/√x over arithmetic dataFirst zeta zeros recovered from M(x) without computing ζ(s)
exp01–exp53Exploratory sequence, sieve, wavelet and calibration probes on prime and arithmetic dataExploratory; retained as a record of what was run

Shell-Model Experiments (Simplex)

A hand-constructed cubic shell model with state variables (ui, ωi) on 3 to 12 scalar wavenumbers coupled by a relaxation term, a cubic term and a signed cascade term, and a multi-level diagnostic H built from its trajectories. The shell model is not a Galerkin truncation of Navier–Stokes and does not conserve energy in the inviscid limit; H is an in-sample classifier whose weights are learned from the outcomes it reports. The pages record what the model did under the stated protocol; no conclusion about the 3D equations follows from them. Read the record · Zenodo (v3)

Shell models by wavenumber count

FileModelRecord
exp_ns_6mode_solved.sx Shell model, 3 wavenumbers (3+3 variables) Finite-window enstrophy trajectories across initial amplitude
exp_ns_8mode_solve.sx Shell model, 4 wavenumbers Finite-window enstrophy trajectories across initial amplitude
exp_ns_10mode_solve.sx Shell model, 5 wavenumbers Finite-window enstrophy trajectories across initial amplitude
exp_ns_12mode_solve.sx Shell model, 6 wavenumbers Finite-window enstrophy trajectories across initial amplitude
exp_ns_16mode_solve.sx Shell model, 8 wavenumbers Finite-window enstrophy trajectories across initial amplitude
exp_ns_20mode_solve.sx Shell model, 10 wavenumbers Finite-window enstrophy trajectories across initial amplitude
exp_ns_24mode_solve.sx Shell model, 12 wavenumbers Finite-window enstrophy trajectories across initial amplitude
Mode scaling summary All shell models compared The finite-window excursion amplitude depends on window length, threshold value and solver; it is a property of the protocol, not of the equations

Diagnostic H and related pages

PageExaminesRecord
The Scaffold Multi-level diagnostic H on shell-model trajectories In-sample classifier; weights learned from the reported outcomes
The Holistic View H, H′ and H″ as one coupled system Diagnostic construction
Viscosity Spectrum Shell model across viscosity Trajectory record across ν
3D Kernel with Vortex Stretching First real-coefficient 3D Fourier–Galerkin kernel Mode enumeration, triad precomputation and Leray projection later carried into the v3 kernel; this version omits the −i factor
2D Four-Mode System Four-mode 2D shell system Trajectory record

Unified Adaptation Theorem

Experiments on sequentially composed contractive updates on a shared parameter. The convergence statement is block contraction with a non-negative coupling matrix (Bertsekas–Tsitsiklis) plus LaSalle; the diagnostics are the identity B = 1 + 2I for the balance residual, the convergence-score lemma S ≥ 1 − 2σT/2/(1−σ)2, the balanced-matrix eigenvalues and the local Hessian of B. Read the record · Zenodo (v3)

Convergence conditions and diagnostics

FileExaminesResultStatement
exp_contraction.sxContraction of five individual update types (C1)5/5 contract, β < 1Condition C1
exp_gradient_interference.sxPCGrad-type projection on conflicting gradient pairsProjection identity on 500 pairs; first-order descent of each objective preservedProjection lemma
exp_lyapunov.sxLyapunov decrease along the composed update0 increases observedLaSalle step
exp_invariants.sxInvariant set under strong gradients0 violations / 15,000 stepsInvariant set
exp_timescale.sxFast/slow timescale separationSeparation maintained in two- and three-timescale settingsCondition C1
exp_composition.sxFull composed systemConverges within tolerance to a single fixed pointBlock contraction
exp_interaction_matrix.sxBlock-coupling matrix estimated from trajectoriesConverges in 5 cyclesCondition C2
exp_convergence_order.sxConvergence score SS → 0.9997Convergence-score lemma
exp_iratio_proof.sxThe identity B = 1 + 2I for K = 2…20Holds to 2.22×10−16Identity
exp_iratio_proof_statistical.sxThe identity B = 1 + 2I on 70 random problemsHolds on all 70Identity
exp_balance_residual.sxGradient flow on the balance residual BReaches machine epsilon; local Hessian ∇2B = 2JTJ/DLocal Hessian
exp_pcgrad_refinement.sxPCGrad vs cosine-scaled projectionCosine-scaled projection is the graduated variant; limit points differ between the twoProjection lemma

Exploratory: beliefs, dynamics, games

FileExaminesResult
exp_anima_deep.sxBelief interaction, consolidation, target misalignmentSingle-run observation
exp_anima_correlated.sxCorrelated beliefs with misaligned targetsExploratory
exp_belief_cascade.sxChain, circular, delayed topologiesChain topology recovered from data
exp_skeptical_annealing.sxSkeptic vs believer across horizonsConjecture 6.3 refuted; 6.5 consistent
exp_memory_dynamics.sxForgetting rate, transfer, self-referenceMeta-gradient recovers near-optimal λ*
exp_liquid_hive.sxLiquid neural network in a hive architectureDynamic reconfiguration converges
exp_chaos_boundary.sxConvergence score S on the logistic mapExploratory; S does not separate periodic from chaotic orbits
exp_s_vs_lyapunov.sxS against the Lyapunov exponentComplementary diagnostics
exp_nash_equilibrium.sxIterated Prisoner's Dilemma with an added termExperiment outcome; not a theorem
exp_gan_convergence.sxGAN training via projectionGenerator–discriminator balance
exp_ode_solvers.sxLearned blending of ODE solversAdaptive blend outperforms fixed methods
exp_prime_gaps.sxPrime gap derivative seriesExploratory
exp_iratio_applications.sxBalance residual across 5 domainsIdentity holds in each
exp_collatz_analysis.sxCollatz convergence diagnosticsExploratory

Code / compiler

FileExaminesResult
exp_code_gates.sxCode-structure convergence via gatingS → 0 at step 50
exp_compiler_passes.sxPer-program compiler pass adaptationPer-program adaptation
exp_structure_discovery.sxGradient topology as a structural probeConstraint graph recovered
exp_equilibrium_mapping.sxB-flow equilibrium in an optimisation landscapeMatches the local Hessian

Stress / robustness

FileExaminesResult
exp_sensitivity.sxStability across 3 orders of magnitude in learning rateStable
exp_stress_test.sxRosenbrock + Rastrigin combinedConverges
exp_stress_rosenbrock.sxBanana valley at 4–10 dimensionsConverges in high dimension
exp_stress_adversarial.sxAnti-parallel gradient objectivesProjection resolves the pair; limit point not unique
exp_symmetry_breaking.sxGroup discovery, perturbation recoveryRecovery within O(10) cycles
exp_stochastic_projection.sxDeterministic vs stochastic projectionDeterministic matches stochastic
exp_stochastic_rastrigin.sxStochastic projection on a multimodal landscapeFinds the global region
exp_lyapunov_refinement.sxTighter Lyapunov boundsTighter bounds achieved
exp_learnable_projection.sxLearnable projection scaleLearned scale converges
exp_learnable_projection2.sxMomentum-augmented learnable projectionFaster convergence
exp_convergence_ratios.sxRatio series and entropy analysisRatios converge (not universal)

Compiler Math Validation

188 tests confirming the Simplex compiler produces correct numerical results for every mathematical operation used by the Simplex experiments above.

FileTestsCoverageOperations
test_math_arithmetic.sx75f64/i64 arithmetic, casts, edge cases+, -, *, /, %, int↔float
test_math_comparisons.sx23All 6 comparison operators, both types<, ≤, >, ≥, ==, !=
test_math_transcendental.sx66Transcendental functions and identitiessqrt, sin, cos, exp, ln, pow, tanh
test_math_loops.sx10Loop-based numerical computationAccumulation, Newton, series
test_math_functions.sx14Function composition and recursionComposition, recursion, dot product

Total: 188/188 pass.

How to Run

Navier–Stokes kernel and instruments (C & Python)

git clone https://github.com/senuamedia/lab-code.git
cd lab-code

# C: v3 kernel + Taylor-Green check
gcc -O3 -c solvers/v3/triad_kernel_v3.c -o kernel.o
gcc -O3 -c validation/taylor_green_test.c -o tg.o
gcc -O2 tg.o kernel.o -o run_tg -lm
./run_tg

# Python: independent implementation and FFT cross-check
pip install numpy scipy
python3 validation/ns_galerkin_3d.py
python3 validation/pseudospectral_compare.py

# Python: closure, dispersion, scaling and Burgers-vortex checks
python3 numerical/verify_review_claims.py

Arithmetic oscillations (C)

cd domains/spectral-wave-theory
make              # builds all experiments
./quick_sum       # truncated amplitude sums (<1s)
./paper_data      # tables (<2s)

Shell-model and adaptation experiments (Simplex)

cd simplex && ./build.sh
cd ../theorem-proof
./run_all.sh            # adaptation-theorem experiments
./run_math_tests.sh     # 188 compiler math tests