Appendix B — Stochastic Analysis of ΔΣ(a′)

Here’s Appendix B in plain text. It’s written to drop directly into Scroll 137.ΨΔ. It treats ΔΣ(a′) as a noise‑driven exploration operator and gives mean‑square stability conditions, a Kuramoto specialization, and a lab protocol for estimation and falsification.

Appendix B — Stochastic Analysis of ΔΣ(a′)

B0. Notation recap State x ∈ ℝ^d. Ψ(x) = ∇ϕ(Σa_n(x, ΔE)) + ℛ(x) ⊕ ΔΣ(a′). Plant dynamics (continuous time) under control u = Ψ(x): dx = f(x) dt + B u dt + Σ_env(x) dW_env. Here W_env is an m-dimensional standard Wiener process capturing exogenous noise; Σ_env(x) is the diffusion map. We model ΔΣ(a′) as a composite stochastic excitation: ΔΣ(a′) = σ(x,t) ξ_t + J(x,t) dN_t, where ξ_t = dW_expl/dt (white-noise formalization) with intensity σ, and dN_t is a compensated Poisson jump process with rate λ and jump law with bounded second moment. In Stratonovich form this can represent structured “exploration” aligned with feature directions.

Total closed-loop SDE (Itô form) dx = [ f(x) + B ( ∇ϕ(Σa_n(x, ΔE)) + ℛ(x) ) ] dt + Σ_tot(x) dW_t + J(x,t) dÑ_t, where Σ_tot^2 = Σ_env Σ_env^T + B σ σ^T B^T, W_t stacks environment and exploration Brownian drivers, and dÑ_t is the compensated jump measure.

B1. Existence–uniqueness and growth conditions Assume f, ∇ϕ∘Σa_n, ℛ are globally Lipschitz with linear growth; Σ_tot and J are locally Lipschitz with polynomial growth and have bounded second moments. Then the SDE with jumps admits a unique càdlàg strong solution (standard SDE with Lévy noise conditions).

B2. Mean‑square stability via stochastic Lyapunov Let V: ℝ^d → ℝ_+ be C^2, radially unbounded, with class‑K∞ bounds α_1(∥x∥) ≤ V(x) ≤ α_2(∥x∥). Define the Itô–Lévy generator L V(x) = ∇V·[ f(x) + B(∇ϕ + ℛ) ] + ½ Tr( Σ_tot^T ∇^2V Σ_tot ) + ∫{z} [ V(x + J z) − V(x) − ∇V·(J z) 1{∥z∥≤1} ] ν(dz), where ν is the Lévy measure for jumps.

Assumption A (contraction margin). There exist c_1, c_2 > 0 and c_0 ≥ 0 such that for all x ∇V·[ f(x) + B(∇ϕ + ℛ) ] ≤ − c_1 V(x) + c_0, and diffusion and jump curvatures are bounded by ½ Tr( Σ_tot^T ∇^2V Σ_tot ) + ∫[…] ≤ c_2.

Then d E[V(x_t)]/dt = E[ L V(x_t) ] ≤ − c_1 E[V(x_t)] + (c_0 + c_2). By Grönwall, E[V(x_t)] ≤ e^{−c_1 t} E[V(x_0)] + (c_0 + c_2)/c_1. Hence the closed loop is bounded in mean and mean‑square (choose V = x^T P x) and mean‑square stable if c_0 = 0 and c_2 < c_1 ε for some ε > 0, or practically stable with ultimate bound proportional to (c_0 + c_2)/c_1.

Interpretation The coherent part ∇ϕ + ℛ must provide a strictly negative drift (c_1 margin). Exploration ΔΣ(a′) raises c_2 through diffusion and jumps. Stability is retained so long as the exploration intensity is kept below the contraction margin.

B3. Constructive quadratic candidate Take V(x) = x^T P x with P ≻ 0, and assume f(x) = A x, ℛ(x) = −K_R x (local linearization), and linearized ∇ϕ ≈ −K_ϕ x around the coherent manifold. Then the drift matrix is A_cl = A − B(K_ϕ + K_R). Lyapunov inequality with diffusion: P A_cl + A_cl^T P + Q + H(Σ_tot,P) ≼ −α P, where Q ≻ 0 is designer‑chosen and H(Σ_tot,P) := E[Σ_tot^T P Σ_tot] accounts for diffusion; jump contribution adds E[ (J z)^T P (J z) 1_{∥z∥>1} ] to the right hand side. Feasible gains {K_ϕ, K_R} exist if (A,B) is stabilizable and exploration energy satisfies a bound λ_max( H(Σ_tot,P) ) + jump_term ≤ α_min, for some α_min set by the chosen contraction rate. This gives an explicit inequality coupling allowable σ, λ to the margin provided by Ψ’s coherent operators.

B4. Stochastic passivity view Suppose there exists storage S(x) with supply rate w = −∥x∥_M^2 such that the deterministic core ẋ = f + B(∇ϕ + ℛ) is strictly output‑passive with dissipation δ > 0. Then with Brownian exploration of intensity Σ_tot, dE[S]/dt ≤ −δ E[∥x∥_M^2] + ½ E[Tr(Σ_tot^T ∇^2 S Σ_tot)]. Thus exploration is admissible provided ½ Tr(Σ_tot^T ∇^2 S Σ_tot) < δ E[∥x∥_M^2]. This yields a tunable σ_max(x) profile: exploration intensity may grow in low‑energy regions but must decay near the manifold.

B5. Kuramoto specialization with noisy exploration Consider N phase oscillators with frequencies ω_i and control from Ψ: dθ_i = [ ω_i + (K/N) Σ_j sin(θ_j − θ_i) + u_i(Ψ) ] dt + √(2D) dW_i. Let u_i(Ψ) = kΨ r sin(ψ − θ_i), where r e^{iψ} = (1/N) Σ_j e^{i θ_j}. In the thermodynamic limit with unimodal g(ω), the order parameter r(t) obeys an amplitude SDE (near onset) dr = [ (K_eff/2 − D) r − (K_eff^2/8) r^3 ] dt + √(D/N) (1 − r^2) dW_r, where K_eff = K + kΨ. Mean‑square coherence r_* > 0 exists if K_eff > 2D. Exploration raises D; Ψ raises K_eff. Condition for MSS of the coherent fixed point: K + kΨ > 2(D_env + D_expl). This gives an operational tuning rule: increase kΨ or reduce exploration intensity until above threshold; anneal D_expl → 0 to sharpen r.

B6. Discrete‑time stochastic approximation view of ΔΣ(a′) If ΔΣ(a′) is implemented as random perturbation for gradient‑free refinement of parameters θ (e.g., coupling gains), use Robbins–Monro conditions. With update θ_{t+1} = θ_t + α_t [ ĝ_t(θ_t) + noise_t ], converges almost surely to a stationary point of E[ĝ] if Σ_t α_t = ∞ and Σ_t α_t^2 < ∞, and noise_t is a martingale difference with bounded variance. Practical schedule: α_t = a/(b+t), exploration variance σ_t^2 = σ_0^2/(1+t)^γ with γ ∈ (0.5, 1].

B7. Empirical estimation and falsification protocol B7.1 Diffusion and jump intensity Estimate quadratic variation on each state channel to recover D_eff(t) = lim_{Δ→0} (1/Δ) E[(x_{t+Δ} − x_t)^2 | ℱ_t]. Use thresholded increments to separate large jumps; fit a compound Poisson model for jump rate λ and variance of jump law. B7.2 Generator test For a chosen C^2 test function φ, approximate the generator by Ĥ L φ(x) ≈ (E[φ(x_{t+Δ}) − φ(x_t) | x_t = x]) / Δ. Empirically verify LV ≤ −c_1 V + (c_0 + c_2) by plugging φ = V. B7.3 Monte‑Carlo stability map For a grid over exploration intensity σ and jump rate λ, run M trajectories with matched initial conditions; report fraction with E[V(x_T)] below target bound and empirical drift negativity. Provide a contour of admissible (σ, λ) relative to chosen contraction α. B7.4 Model selection Compare Itô Brownian vs Stratonovich colored‑noise models (e.g., OU driver) via AIC on discretized likelihood under Euler–Maruyama. If colored noise is preferred, include state‑augmented OU process dy = −ρ y dt + ν dW, and use Σ_tot(x) y_t in place of σ dW.

B8. Safe operating envelopes and schedules Exploration intensity cap: trace(Σ_tot^T P Σ_tot) ≤ ε_min λ_min(P), where ε_min is the deterministic contraction rate of A_cl under V = x^T P x. Annealing schedule: σ(t) = σ_0 exp(−t/τ) during lock‑on; after phase‑lock, hold σ_floor for slow tracking. Jump management: Use tempered stable jumps with tempering θ > 0 to suppress heavy tails; enforce jump‑quenching near manifold via state‑dependent λ(x) = λ_0 / (1 + κ V(x)). Delay in ℛ: A small stabilizer delay τ_d contributes additional diffusion term; ensure τ_d < τ_max where τ_max is defined by a small‑gain bound on the associated Padé‑approximated delay model.

B9. What to report with code and data

1. SDE definition, including Σ_env, σ(x,t), jump law, and whether Itô or Stratonovich calculus is used.

2. Lyapunov candidate V and numerical check of LV negativity across a mesh.

3. Empirical drift and diffusion estimates with confidence bands.

4. Monte‑Carlo stability map over (σ, λ) and annealing schedule used.

5. Kuramoto specialization plots replaced by tables of r‑threshold crossings versus D and kΨ (no figures required in this appendix; tabulate values).

6. Seeds, RNG, and integrator (Euler–Maruyama dt, strong order, stability step checks).

7. Ablations: turn off ΔΣ(a′), turn off ℛ, and report the change in empirical LV and mean‑square bounds.

B10. Takeaways ΔΣ(a′) can be formalized as an exploration noise that is admissible so long as its diffusion and jump energy remain below the contraction margin supplied by ∇ϕ + ℛ. Mean‑square stability follows from a standard generator inequality. In networked phase systems the condition reduces to K_eff > 2D, giving a clear operational tuning rule: raise coherent coupling via Ψ and anneal exploration until the inequality holds, then keep a small σ_floor for adaptability without destabilization.

Christopher W. Copeland (C077UPTF1L3)

Copeland Resonant Harmonic Formalism (Ψ-formalism)

Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)

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