Experiment Index

103 reproducible experiments validating the Unified Adaptation Theorem and the Holistic Scaffold Framework for Navier-Stokes regularity. All implemented in Simplex. 188 compiler math tests confirm arithmetic correctness.

How to Run

All experiments live in the theorem-proof/ directory alongside the Simplex compiler source.

Run all experiments at once

git clone https://github.com/senuamedia/lab.git
cd simplex && ./build.sh
cd ../theorem-proof
./run_all.sh            # Core theorem experiments
./run_math_tests.sh     # 188 compiler math tests

Run a single experiment

# Compile
../simplex/build/sxc exp_iratio_proof.sx -o build/exp_iratio_proof.ll

# Link with runtime
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_iratio_proof.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_iratio_proof \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

# Run
./build/exp_iratio_proof

Replace exp_iratio_proof with any experiment filename below. On Linux, omit the OPENSSL_PREFIX line and use -L/usr/lib instead.

Experiments by Category

Navier-Stokes Mode Scaling

Galerkin models of the 3D Navier-Stokes equations at increasing mode counts. The H/H'/H'' framework with doubling time criterion \(P\) achieves 93–96% accuracy and \(\alpha = 2.0\) at every scale. See the complete scaling summary.

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%	—

Core Theorem

Direct validations of the main theorem conditions: contraction, Lyapunov stability, interaction matrices, convergence diagnostics, and the I-ratio / B-flow results.

File	Validates	Result	Theorems / Conjectures
exp_contraction.sx	5 subsystem types contract in Fisher metric	5/5 subsystems contract, \(\beta < 1\)	Theorem 1 (Contraction)
exp_gradient_interference.sx	Cosine-scaled projection resolves all conflicts	500/500 conflicts resolved (100%)	Theorem 2 (Cosine Projection)
exp_lyapunov.sx	Normalised Lyapunov function never increases	0% violations over all trials	Theorem 3 (Lyapunov Stability)
exp_invariants.sx	Foundational constraints hold under strong gradients	0 violations / 15,000 steps	Proposition 3.5
exp_timescale.sx	Fast/slow timescale separation maintained	100% for two-timescale, 95.9%+ for three-timescale	Theorem 1 (Timescale)
exp_composition.sx	Full composed system converges	System converges within tolerance	Theorems 1-5 (Composition)
exp_interaction_matrix.sx	Interaction matrix discovers coalition topology	Converges in 5 cycles	Theorem 4, Conjecture 6.8
exp_convergence_order.sx	Higher-order convergence score \(S\) decays	\(S \to 0.9997\)	Theorem 5 (Convergence Order)
exp_iratio_proof.sx	\(I = -\tfrac{1}{2}\) at equilibrium for \(K = 2 \ldots 20\)	138/138 tests pass, max error \(2.22 \times 10^{-16}\)	Theorem 13 (I-Ratio)
exp_iratio_proof_statistical.sx	\(I = -\tfrac{1}{2}\) for 70 random multi-objective problems	70/70 pass	Theorem 13 (I-Ratio)
exp_balance_residual.sx	B-flow gradient descent on \(B(\theta)\) vs loss-flow	B-flow: \(8.8 \times 10^{-16}\), loss-flow: \(3.3 \times 10^{-4}\) — 375B× precision	Theorem 14 (B-Flow)

Cognitive / Belief

Experiments on Bayesian belief systems, desire regularisation, skeptical annealing, memory dynamics, and liquid hive architectures.

File	Validates	Result	Theorems / Conjectures
exp_anima_deep.sx	Belief interaction, consolidation, and desire effects	31% calibration improvement with desire regularisation	Theorems 6, 7 (Belief, Desire)
exp_anima_correlated.sx	Correlated beliefs with misaligned desires	Desire acts as Bayesian regulariser	Theorem 7, Conjecture 7.1
exp_belief_cascade.sx	Chain, circular, and delayed belief topologies	Chain topology partially discovered from data	Conjecture 6.4 (Belief Chain)
exp_skeptical_annealing.sx	Skeptic vs believer across all observation horizons	Skeptic wins always (not just early) — refutes annealing conjecture	Conjectures 6.3 (Refuted), 6.5 (Validated)
exp_memory_dynamics.sx	Forgetting rate, transfer learning, self-reference, phase	Meta-gradient recovers near-optimal \(\lambda^*\)	Conjectures 6.6–6.10
exp_liquid_hive.sx	Liquid neural network within hive architecture	Dynamic agent reconfiguration converges	Theorems 6, 7 (Hive extension)

Cross-Domain

Applications of the theorem framework to chaos theory, game theory, GANs, ODE solvers, number theory, and multi-domain I-ratio validation.

File	Validates	Result	Theorems / Conjectures
exp_chaos_boundary.sx	Convergence score \(S\) detects chaos onset in logistic map	Feigenbaum point located at \(r \approx 3.57\)	Theorem 12 (Chaos Detection)
exp_s_vs_lyapunov.sx	\(S\) and Lyapunov exponent \(\lambda\) are complementary	\(S\)-\(\lambda\) complementarity confirmed	Proposition 12.1
exp_nash_equilibrium.sx	Skeptical desire escapes Nash to reach Pareto	83.5% Pareto-optimal vs 33% Nash	Theorem 11 (Game Theory)
exp_gan_convergence.sx	GAN training stabilisation via projection	Generator-discriminator balance achieved	Theorem 2, 11 (GANs)
exp_ode_solvers.sx	Learned blending of ODE solver methods	Adaptive solver outperforms fixed methods	Theorems 1, 5 (ODE Application)
exp_prime_gaps.sx	Prime gap derivative series analysis	Gap structure aligns with convergence framework	Exploratory (Number Theory)
exp_iratio_applications.sx	\(I = -\tfrac{1}{2}\) across 5 distinct domains	5/5 domains validated	Theorem 13 (I-Ratio Universality)
exp_collatz_analysis.sx	Collatz sequence convergence diagnostics	Convergence score tracks trajectory collapse	Exploratory (Number Theory)

Code / Compiler

Applying the convergence framework to compiler optimisation passes, code structure discovery, and equilibrium mapping in program analysis.

File	Validates	Result	Theorems / Conjectures
exp_code_gates.sx	Code structure convergence via gating mechanisms	\(S \to 0\) at step 50	Theorem 8 (Code Gates)
exp_compiler_passes.sx	Per-program compiler pass interaction and adaptation	Per-program adaptation confirmed	Theorems 9, 10 (Compiler)
exp_structure_discovery.sx	Gradient topology as structural probe	Constraint graph discovered from gradients	Theorem 4 (Topology)
exp_equilibrium_mapping.sx	B-flow equilibrium location in optimisation landscape	B-flow equilibrium matches theoretical prediction	Theorem 14 (B-Flow)

Stress / Robustness

Stress tests, adversarial conditions, high-dimensional landscapes, stochastic projections, and learnable projection variants. These experiments push the theorem to its limits.

File	Validates	Result	Theorems / Conjectures
exp_sensitivity.sx	Stability across 3 orders of magnitude in learning rate	Stable over 3 OOM	Propositions 7.1–7.4
exp_stress_test.sx	Combined Rosenbrock + Rastrigin stress	Converges under combined stress	Theorems 1–3 (Robustness)
exp_stress_rosenbrock.sx	Rosenbrock banana valley at 4–10 dimensions	Convergence in high-D banana valley	Theorems 1, 2 (High-D)
exp_stress_adversarial.sx	Anti-parallel gradient objectives	Projection resolves adversarial conflicts	Theorem 2 (Adversarial)
exp_symmetry_breaking.sx	Group discovery, perturbation recovery, phase transition	Groups discovered, recovery within \(O(10)\) cycles	Conjectures 6.2, 6.8, 6.9
exp_convergence_ratios.sx	Ratio series, entropy, and dominant pair analysis	Ratios converge but are not universal	Conjecture 6.1
exp_stochastic_projection.sx	Noise injection unnecessary — implicit exploration suffices	Deterministic projection matches stochastic	Theorem 2 (Implicit Exploration)
exp_stochastic_rastrigin.sx	Stochastic projection on multimodal Rastrigin landscape	Finds global region despite local minima	Theorem 2 (Multimodal)
exp_pcgrad_refinement.sx	PCGrad vs cosine-scaled projection comparison	Cosine projection outperforms PCGrad (100% vs 66.5%)	Theorem 2 (Projection Comparison)
exp_lyapunov_refinement.sx	Refined Lyapunov construction and tighter bounds	Tighter Lyapunov bounds achieved	Theorem 3, Conjecture 6.11
exp_learnable_projection.sx	Learnable projection scale parameter	Learned \(\alpha\) converges to stable value	Theorem 2 (Learnable Extension)
exp_learnable_projection2.sx	Extended learnable projection with momentum	Momentum-augmented projection improves convergence rate	Theorem 2 (Learnable Extension v2)

Verification

Four independent verification points confirming the structural foundations of the scaffold framework. All 4 passing.

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

Compiler Math Validation

These tests validate that the Simplex compiler produces correct numerical results for every mathematical operation used by the experiments above. 188 tests total.

File	Tests	Coverage	Operations
test_math_arithmetic.sx	75	f64/i64 arithmetic, casts, edge cases	+, -, *, /, %, int↔float casts
test_math_comparisons.sx	23	All 6 comparison operators, both types	<, ≤, >, ≥, ==, != for f64 and i64
test_math_transcendental.sx	66	Transcendental functions and identities	sqrt, sin, cos, tan, exp, ln, pow, tanh, Pythagorean identity
test_math_loops.sx	10	Loop-based numerical computation	Accumulation, Newton's method, series, convergence
test_math_functions.sx	14	Function composition and recursion	Composition, recursion, dot product, nested calls

Total: 188/188 pass.