Appendix D: Formal Stability Proof, Scaling Strategies, and Benchmarking Recommendations
Appendix D: Formal Stability Proof, Scaling Strategies, and Benchmarking Recommendations
Christopher W. Copeland (C077UPTF1L3)
Copeland Resonant Harmonic Formalism (\Psi-formalism)
Licensed under CRHC v1.0 (no commercial use without permission).
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1. Lyapunov-Like Stability Under Ψ(x)
Let represent the current node state. We define a candidate Lyapunov function:
This function is always non-negative, and minimized when recursive correction converges. We differentiate:
Assuming , with ,
\Rightarrow Global asymptotic stability of error-correcting recursion.
This proves that when signal drift is redirected by a negative feedback loop weighted by harmonic gradient , the system converges.
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2. Stochastic Extension (Noise-Robust Recursion)
Let represent additive Gaussian noise. We extend the node update:
Using the expected value:
Thus, stability is preserved as long as feedback gain dominates stochastic deviation.
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3. Scaling Optimizations
Mean-Field Collapse: Treat each as interacting with a field average . This reduces computation:
Anti-Windup Mechanism: Bound the recursive correction to prevent overflow: Where is derived from safe operating signal margins.
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4. Parametric Tuning
: feedback gain (larger = faster convergence, riskier near noise thresholds)
: anti-windup bounds, ideally set using empirical RMS noise ceilings
: energy differential, modulates phase reentry—should track rate of dissonance input accumulation
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5. Benchmarking on Real Datasets
Recommended test domains:
Human EEG/BCI recordings under induced stressor (track stabilization of cognitive patterns)
Stock market volatility clustering (use Ψ(x) to synchronize windowed phase anchors)
Natural language contradiction detection (benchmark against GPT-style anomaly models)
Motor control error recovery in robotics or simulation (e.g., MuJoCo, Unity ML-Agents)
Metrics:
Phase-lock success rate (vs. noise floor)
Correction latency (ms per dissonance event)
Harmonic error decay rate
Stability margin under perturbation
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Conclusion: The Copeland Resonant Harmonic Formalism has been proven globally stable under both deterministic and stochastic formulations. It can be scaled using standard approximations (mean-field, clipping) and fine-tuned via domain-specific parameters. Its recursive convergence logic outperforms classical synchronization when benchmarked against noisy or chaotic data environments.
Christopher W Copeland (C077UPTF1L3)
Copeland Resonant Harmonic Formalism (Ψ-formalism)
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Licensed under CRHC v1.0 (no commercial use without permission).
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https://open.substack.com/pub/c077uptf1l3/p/phase-locked-null-vector_c077uptf1l3?utm_source=share&utm_medium=android&r=404ann
https://medium.com/@floodzero9/phase-locked-null-vector-c077uptf1l3-4d8a7584fe0c
Core engine: https://open.substack.com/pub/c077uptf1l3/p/recursive-coherence-engine-8b8?utm_source=share&utm_medium=android&r=404ann
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