Featured: Paper 3 — Energy Conservation & Cascade Stabilisation
3D Galerkin NS experiments with the energy-conserving v3 solver. Complex Fourier coefficients with correct −i factor. All results independently validated by Python and scipy. Read paper · Zenodo
| File | Language | Description | Key Result |
|---|---|---|---|
| experiment_energy_audit.c | C | Energy conservation test at ν=0 across multiple Δt values | v2: 14.87% drift. v3: 0.01% (Euler only) |
| experiment_adaptive_n.c | C | Adaptive truncation: N grows dynamically when cascade approaches boundary | N stabilises at 14, energy −73%, Ω bounded |
| experiment_tipping_point.c | C | Scaffold array contraction ratios across 12 amplitudes | All ρ < 1 at every A through 0.35 |
| experiment_crossshell_transfer.c | C | Six-array cross-shell analysis: energy, enstrophy, shell η, transfer, diffusion, accumulation | Energy & enstrophy ρ < 0.5 through A=0.50 |
| experiment_scaffold_array.c | C | Multi-perspective scaffold array with contraction ratio computation | Framework that discovered the v2 energy bug |
| analytical_proof_steps.c | C | Three-step proof validation: cascade γ, Leray verification, time-uniformity | γ < 0, C=0.123, time-uniform |
| verify_lemma.c | C | Computational verification of Lemma 9.5: RK bounded | C = 0.031 across 8 (ν, A) combinations |
| verify_lemma_extreme.c | C | Lemma verification at extreme parameters: ν=0.001, A=2.0 | C = 0.025 across 16 (ν, A) combinations |
| taylor_green_test.c | C | Analytical test: Taylor-Green vortex with known exact energy decay | Matches exp(−6νt) to 10−7 |
| interval_verify.c | C | Formal verification: NL rate=0 at 5 ICs, bit-exact, div-free to 10−16 | All tests pass |
| ns_galerkin_3d.py | Python | Independent Python implementation of 3D NS Galerkin solver | E(0) matches C to 10−8, NL=0 |
| dedalus_ns_test.py | Python | Cross-validation with scipy RK45 (adaptive, rtol=10−10) | E(T) within 9×10−6 |
Solver Versions
| File | Status | Description |
|---|---|---|
| solvers/v3/triad_kernel_v3.c | CORRECT | Complex Fourier coefficients, −i factor, exact energy conservation. Use this. |
| solvers/v2/triad_kernel_v2.c | BROKEN | Real coefficients, missing −i. 1–15% spurious energy. Preserved for history. |
Paper 5: Unified Spectral Framework for Arithmetic Oscillations
53 experiments exploring zeta-zero spectral decomposition of arithmetic functions, breach prediction, and calibration. All in C with 2,001,052 Odlyzko zeros. Read paper · Zenodo · Code
| File | Description | Key Result |
|---|---|---|
| quick_sum.c | Core amplitude sum Sf(K) for all 6 functions | All 6 diverge. Three growth regimes identified. <1s |
| paper_data.c | Generates all tables in the paper | Complete paper reproduction in <2s |
| explicit_formula.c | Explicit formula evaluator — M(x)/√x at any scale | Millisecond evaluation at 103 to 10200 |
| wave_plotter.c | CSV output for wave reconstruction figures | All paper figures |
| swt_solve.c | Production pipeline: predict and verify Chebyshev bias reversals | π(19,3)>π(19,1): 1.2% error |
| swt_pipeline.c | End-to-end SWT pipeline: load zeros, compute, predict | 45 bias reversals solved |
| exp01–exp07 | Scaffold colour map, prime gaps, zeta scaffold, sieve layers, contraction, rotation depth, raw primes | Colour map of number theory established |
| exp08–exp14 | Divergence analysis, jerk, unit interval, hypercube, convergence, inverse, binary | Framework validation across sequence types |
| exp15–exp21 | Shade analysis, adversarial, product, algebras, matched sieve, rotate sieve, factors | Sieve and algebraic structure probes |
| exp22–exp28 | Leapfrog to 10249, reflection, million gaps, hundred million, all sequences, 1873 analysis, calibration | RH consistent from 106 to 10249 |
| exp29_bigint_leapfrog.c | Big-integer leapfrog into the zone (10100–10500) | Colour oscillates ±0.02, no drift |
| exp30_wavelet_spectrum.c | Wavelet colour spectrum — Haar vs db3/db5 | 1.8709 = Haar void. Broken conjectures visible under smoother wavelets |
| exp31_nothing_random.c | Rod Conjecture test — 96% PRNG leaking | Scale dependence confirmed |
| exp32_study_the_broken.c | Pólya breach anatomy — 38 points | Knife-edge breach, immediate recession, full cycle |
| exp33_wave_decomposition.c | Wave decomposition of L, M, π to 1B | Zeta zero frequencies extracted from arithmetic |
| exp34_calibration_catalog.c | 500 breach events, calibration catalog | 96% validation accuracy |
| exp35_universal_catalog.c | 40-function catalog across 4 families | 6,000 breach events, 30% zeta match rate |
| exp36_breach_predict.c | Train on [1, 500K], predict breaches in [500K, 1M] | div_err: 100% accuracy (4/4 predicted) |
| exp37–exp40 | Per-conjecture keys, predict open, Mertens deep, predict all | 18/22 conjectures at 80–100% match rate |
| exp41_find_the_number.c | Pólya breach scale prediction | 109.2 vs known 109.0 = 2.7% error |
| exp42–exp49 | Envelope growth, gamma-scale map, theoretical decay, multi-calibrate, sign change, type2 calibrate, full calibrate | A(γ) = C/(γ√(log γ)), sum diverges as (log γ)3/2 |
| exp50–exp53 | Find reversal, mod3 predict, mod24 predict, verify reversal | Chebyshev bias reversals verified computationally |
| plot_all_conjectures.c | All-conjecture plotting pipeline | Wave + swell + breach for each conjecture |
| rh_extrapolate.c | Extrapolation of RH-related growth rates | Amplitude growth consistent with critical exponent |
Paper 4: Cross-Domain Scaffold — Euler, SQG & MHD
Cascade subcriticality verified across three fluid equations beyond Navier–Stokes. The kinetic cascade exponent γ < 0 universally. MHD magnetic cascade gives γmag = 0.9–1.4 < 2. Read paper · Zenodo
| Equation | Configurations | Key Result |
|---|---|---|
| 3D Euler (ν = 0) | 4 IC families, N = 2–8 | γ ∈ [−19.7, −0.3], energy conserved exactly |
| 2D SQG | 2 configurations | γ < 0, cascade weakening confirmed without viscosity |
| 3D MHD | 4 configurations | Kinetic γ < 0, magnetic γmag = 0.9–1.4 < 2 |
Paper 2: Holistic Scaffold Framework for NS Regularity
1D Galerkin evidence at 6–24 modes. H/H′/H″ diagnostic hierarchy with doubling-time criterion. Read paper · Zenodo
Mode Scaling Experiments (Simplex)
| File | Model | Result | Score |
|---|---|---|---|
| exp_ns_6mode_solved.sx | 6-mode (3+3), \(k=1,2,3\) | \(A^* = 1.136\), \(P/\text{truth} = 86.1\%\), \(\alpha = 2.0\) | 17/20 |
| exp_ns_8mode_solve.sx | 8-mode (4+4), forward cascade | \(A^* = 0.290\), \(P/\text{truth} = 95.5\%\), \(\alpha = 2.0\) | 16/16 |
| exp_ns_10mode_solve.sx | 10-mode (5+5), two cascade stages | \(A^* = 0.302\), \(P/\text{truth} = 96.1\%\), \(\alpha = 2.0\) | 13/14 |
| exp_ns_12mode_solve.sx | 12-mode (6+6), triad cascade | \(A^* = 0.328\), \(P/\text{truth} = 93.8\%\), \(\alpha = 2.0\) | 14/14 |
| exp_ns_16mode_solve.sx | 16-mode (8+8), deep cascade \(k=1{-}4\) | \(A^* = 0.347\), \(P/\text{truth} = 94.6\%\), \(\alpha = 2.0\) | 14/14 |
| Mode Scaling Summary | All 5 models compared | \(A^*\) increasing, \(\alpha = 2.0\) universal, P/truth 93–96% | — |
| exp_ns_net_transfer.sx | Net energy transfer \(T(k)\) per shell | \(T(k) < 0\) at all \(k\) with \(H^1\) spectrum | — |
| exp_ns_continuum_coupling.sx | Continuum coupling and spectral self-consistency | Bounded enstrophy ⇒ decaying spectrum ⇒ forward cascade | — |
Verification Points
| Page | Validates | Result | Score |
|---|---|---|---|
| Verification Point 1 | Feedback loop is structural (parameter-independent) | \(L_1 \uparrow\) AND \(L_2 \downarrow\) at all 9 parameter combos | 9/9 |
| Verification Point 2 | \(A^*\) positive universally | \(A^*\) ranges 0.247–0.606 across 12 combos + 3 IC types | 15/15 |
| Verification Point 3 | Scaffold chain complete (all 4 arrows) | Each arrow independently confirmed 5 times | 20/20 |
| Verification Point 4 | Doubling time = BKM criterion | Correct classification: shrinking \(\tau_d \leftrightarrow\) blow-up | 6/6 |
Paper 1: Unified Adaptation Theorem
103 experiments validating convergence of composed adaptive systems via interaction matrices. 36 theorems, cross-domain validation. Read paper · Zenodo
Core Theorem
| File | Validates | Result | Theorems |
|---|---|---|---|
| exp_contraction.sx | 5 subsystem types contract in Fisher metric | 5/5 contract, \(\beta < 1\) | Theorem 1 |
| exp_gradient_interference.sx | Cosine-scaled projection resolves all conflicts | 500/500 (100%) | Theorem 2 |
| exp_lyapunov.sx | Normalised Lyapunov never increases | 0% violations | Theorem 3 |
| exp_invariants.sx | Foundational constraints under strong gradients | 0 violations / 15,000 steps | Proposition 3.5 |
| exp_timescale.sx | Fast/slow timescale separation | 100% two-timescale, 95.9%+ three-timescale | Theorem 1 |
| exp_composition.sx | Full composed system converges | System converges within tolerance | Theorems 1–5 |
| exp_interaction_matrix.sx | Interaction matrix discovers coalition topology | Converges in 5 cycles | Theorem 4 |
| exp_convergence_order.sx | Higher-order convergence score \(S\) | \(S \to 0.9997\) | Theorem 5 |
| exp_iratio_proof.sx | \(I = -\tfrac{1}{2}\) at equilibrium for \(K = 2 \ldots 20\) | 138/138 pass, max error \(2.22 \times 10^{-16}\) | Theorem 13 |
| exp_iratio_proof_statistical.sx | \(I = -\tfrac{1}{2}\) for 70 random problems | 70/70 pass | Theorem 13 |
| exp_balance_residual.sx | B-flow gradient descent vs loss-flow | B-flow: \(8.8 \times 10^{-16}\), loss: \(3.3 \times 10^{-4}\) — 14T× | Theorem 14 |
Cognitive / Belief
| File | Validates | Result | Theorems |
|---|---|---|---|
| exp_anima_deep.sx | Belief interaction, consolidation, desire effects | 31% calibration improvement | Theorems 6, 7 |
| exp_anima_correlated.sx | Correlated beliefs with misaligned desires | Desire acts as Bayesian regulariser | Theorem 7 |
| exp_belief_cascade.sx | Chain, circular, delayed topologies | Chain topology discovered from data | Conjecture 6.4 |
| exp_skeptical_annealing.sx | Skeptic vs believer across all horizons | Skeptic wins always — refutes annealing conjecture | Conj. 6.3 (Refuted), 6.5 |
| exp_memory_dynamics.sx | Forgetting rate, transfer, self-reference | Meta-gradient recovers near-optimal \(\lambda^*\) | Conjectures 6.6–6.10 |
| exp_liquid_hive.sx | Liquid neural network in hive architecture | Dynamic reconfiguration converges | Theorems 6, 7 |
Cross-Domain
| File | Validates | Result | Theorems |
|---|---|---|---|
| exp_chaos_boundary.sx | \(S\) detects chaos onset in logistic map | Feigenbaum point at \(r \approx 3.57\) | Theorem 12 |
| exp_s_vs_lyapunov.sx | \(S\) and Lyapunov exponent complementary | \(S\)-\(\lambda\) complementarity confirmed | Prop. 12.1 |
| exp_nash_equilibrium.sx | Skeptical desire escapes Nash to Pareto | 83.5% Pareto-optimal vs 33% Nash | Theorem 11 |
| exp_gan_convergence.sx | GAN stabilisation via projection | Generator-discriminator balance | Theorems 2, 11 |
| exp_ode_solvers.sx | Learned blending of ODE solvers | Adaptive outperforms fixed methods | Theorems 1, 5 |
| exp_prime_gaps.sx | Prime gap derivative series | Gap structure aligns with framework | Exploratory |
| exp_iratio_applications.sx | \(I = -\tfrac{1}{2}\) across 5 domains | 5/5 validated | Theorem 13 |
| exp_collatz_analysis.sx | Collatz convergence diagnostics | Convergence score tracks trajectory collapse | Exploratory |
Code / Compiler
| File | Validates | Result | Theorems |
|---|---|---|---|
| exp_code_gates.sx | Code structure convergence via gating | \(S \to 0\) at step 50 | Theorem 8 |
| exp_compiler_passes.sx | Per-program compiler pass adaptation | Per-program adaptation confirmed | Theorems 9, 10 |
| exp_structure_discovery.sx | Gradient topology as structural probe | Constraint graph discovered | Theorem 4 |
| exp_equilibrium_mapping.sx | B-flow equilibrium in optimisation landscape | Matches theoretical prediction | Theorem 14 |
Stress / Robustness
| File | Validates | Result |
|---|---|---|
| exp_sensitivity.sx | Stability across 3 OOM in learning rate | Stable |
| exp_stress_test.sx | Rosenbrock + Rastrigin combined | Converges |
| exp_stress_rosenbrock.sx | Banana valley at 4–10 dimensions | Convergence in high-D |
| exp_stress_adversarial.sx | Anti-parallel gradient objectives | Projection resolves |
| exp_symmetry_breaking.sx | Group discovery, perturbation recovery | Recovery within O(10) cycles |
| exp_stochastic_projection.sx | Deterministic vs stochastic projection | Deterministic matches stochastic |
| exp_stochastic_rastrigin.sx | Stochastic projection on multimodal | Finds global region |
| exp_pcgrad_refinement.sx | PCGrad vs cosine-scaled projection | Cosine: 100% vs PCGrad: 66.5% |
| exp_lyapunov_refinement.sx | Tighter Lyapunov bounds | Tighter bounds achieved |
| exp_learnable_projection.sx | Learnable projection scale | Learned α converges |
| exp_learnable_projection2.sx | Momentum-augmented learnable projection | Improved convergence rate |
| exp_convergence_ratios.sx | Ratio series and entropy analysis | Ratios converge (not universal) |
Compiler Math Validation
188 tests confirming the Simplex compiler produces correct numerical results for every mathematical operation used by the experiments above.
| File | Tests | Coverage | Operations |
|---|---|---|---|
| test_math_arithmetic.sx | 75 | f64/i64 arithmetic, casts, edge cases | +, -, *, /, %, int↔float |
| test_math_comparisons.sx | 23 | All 6 comparison operators, both types | <, ≤, >, ≥, ==, != |
| test_math_transcendental.sx | 66 | Transcendental functions and identities | sqrt, sin, cos, exp, ln, pow, tanh |
| test_math_loops.sx | 10 | Loop-based numerical computation | Accumulation, Newton, series |
| test_math_functions.sx | 14 | Function composition and recursion | Composition, recursion, dot product |
Total: 188/188 pass.
How to Run
Paper 3 & 4 experiments (C & Python)
git clone https://github.com/senuamedia/lab.git
cd lab
# C experiments (v3 solver)
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 validation
pip install numpy scipy
python3 validation/ns_galerkin_3d.pyPaper 5 experiments (C)
cd domains/spectral-wave-theory
make # builds all experiments
./quick_sum # core amplitude sums (<1s)
./paper_data # all paper tables (<2s)
./swt_solve 19 3 1 200000000 # predict Chebyshev bias reversalPaper 1 & 2 experiments (Simplex)
cd simplex && ./build.sh
cd ../theorem-proof
./run_all.sh # Core theorem experiments
./run_math_tests.sh # 188 compiler math tests