Theorems
Subsystem Contraction
Each adaptive subsystem is a contraction mapping in the Fisher metric with rate \( \beta < 1 \). Foundation for composed convergence.
Cosine-Scaled Projection
Graduated conflict resolution: \( \text{scale} = \alpha \cdot |\cos(g_i, g_j)| \). 100% resolution vs Riemannian PCGrad's 66.5%.
Full proof →Normalised Lyapunov
\( V = \sum (L_i / L_{i,0}) \) — fraction of initial error remaining. Scale-invariant, no weight tuning, 0% violations.
Full proof →Interaction Matrix
\( N \times N \) pairwise projection strengths, learnable via meta-gradient. Converges in 5 cycles, discovers asymmetric structure.
Full proof →Higher-Order Convergence
Convergence score \( S \) measures drift ratio between early and late windows. Detects convergence beyond loss decrease.
Belief Update Contraction
Bayesian belief updates are contractive in the composed system. Foundation for cognitive architecture convergence.
Desire as Bayesian Regulariser
Misaligned desire improves calibration by 31%. Outperforms aligned desires at ALL observation horizons.
Full proof →Code Gate Convergence
Neural gate weights in code generation converge to stable structure. S reaches 0 at step 50.
Compiler Pass Interaction
Optimisation passes form a convergent composed system with per-program adaptation via interaction matrix.
Composed System Convergence
The full system — all subsystems, projections, Lyapunov, and interactions — converges to a unique equilibrium.
Game-Theoretic Equilibrium
Skeptical desire in multi-agent games achieves 83.5% Pareto-optimal outcomes vs 33% for Nash equilibrium.
S as Chaos Detector
\( S = 1 \) for order, \( S = 0 \) for chaos. Detects the Feigenbaum point at \( r \approx 3.57 \). Model-free.
Full proof →B-Flow Convergence
\( B(\theta) = \frac{\|\sum g_i\|^2}{\sum \|g_i\|^2} \) — balance residual. B-flow achieves \( 8.8 \times 10^{-16} \) precision vs loss-flow's \( 3.3 \times 10^{-4} \).
Full proof →PID-S Controller
\( \text{lr}(t) = \text{base} \times \text{clamp}(1 + w_1 S + w_2 S' + w_3 S'') \). Learnable gains via dual numbers. PD/PID beats P-only on shifting landscapes.
Perturbation Stability Margin
Stability margin \( M(\theta, \varepsilon) \) measures perturbation amplification. M is a dual number: \( \partial M/\partial t \), \( \partial^2 M/\partial t^2 \) predict blow-up.
Dual Agent Architecture
Chaos Agent (heats, probes) vs Order Agent (cools, stabilises). Order is 3.5–4.2× more powerful. Optimal balance extends smoothness by 32.6%.
Experiment →Regularity Robustness in 3D
\( |\partial T/\partial A| \to \infty \) as \( A \to 0 \) persists at all vortex stretching strengths (\( \lambda \) up to 128, \( \lambda_2 \) up to 500).
Experiment →Structural Regularity
The regularity signature never breaks. Growth ratio \( \approx 4.7\times \) per halving of amplitude, constant across all stretching strengths.
Experiment →Viscosity Spectrum
Solid→Fluid→Gas mapped with 9-level hierarchy. Resonance at period ≈10,000 steps. Hysteresis 78×. Ratchet effect: gas-phase damage exceeds solid-phase repair.
Experiment →Trajectory Acceleration
Meta-gradient \( \partial T/\partial \alpha \) switches sign at the resonance frequency. Optimal acceleration \( \alpha \approx 0.1 \). The path through parameter space is learnable.
Experiment →3D Resonance & Decoupling
Stretching shifts the resonance: \( \omega_{\text{res}} \propto \lambda_2^{0.4} \). Viscosity and stretching decouple — 69.8% reduction constant across all \( \nu \).
Experiment →Holistic Coupling & Hidden Strength
Feedback loop \( L_1 \to L_4 \to L_2 \to L_1 \) drives blow-up. Viscosity's hidden power: \( \partial L_{10}/\partial \nu = 500{,}000 \) — largest coupling in the system.
Experiment →H-Regularity Threshold
The feedback loop has a sharp engagement threshold \( A^* > 0 \). Below \( A^* \), loop never engages, \( H \) bounded, enstrophy bounded → regularity.
Experiment →The Scaffold
\( A < A^* \Rightarrow \) loop off \( \Rightarrow H \) bounded \( \Rightarrow \Omega \) bounded. Scaling law: \( \max(\Omega) = C \cdot A^2 \), \( \alpha = 2.0 \) exactly.
Experiment →Gap Closure
Saturation predictor closes 54.6% of gap. Doubling time criterion closes 86.1%. H' self-adapting weights reach 91.5%.
Experiment →6-Mode Solved, 8-Mode Verified
6-mode: 100% at T=100k. 8-mode: P/truth=95.5%, 16/16 perfect. Framework mode-invariant. Feedback loop, scaling law, and doubling time all transfer.
Experiment →A* Converges to 0.347
7 Galerkin truncations (6–24 modes). \( A^* \) positive and convergent. 14/14 perfect at 16+ modes. \( \alpha = 2.0 \) universally. Framework mode-count invariant.
Full mode scaling →