25-EFT.WP.STG.Dynamics v1.0 | Appendix E — Error and Uncertainty Propagation (Dynamics Edition)

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Appendix E — Error and Uncertainty Propagation (Dynamics Edition)

One-sentence goal: Establish, for STG dynamics, a unified methodology that decomposes error sources from graph structure/operators through propagation/assimilation, runs Linearized Propagation of Uncertainty (LPU) and Monte Carlo (MC) in parallel for cross-checking, and produces guardbands and compliant persisted artifacts.

I. Scope and Objects

• Objects: graph(V, E), L/A/H, kernel K = g(L), continuous/discrete dynamics f(x, θ, Δt), numerical integration & approximations, assimilation (KF/UKF/PF), uncertainty composition and guardbanding.
• Inputs: RefCond, units.*, baseline.metrics, graph/L/H/K.hash, solver/meta, noise priors Q/R, parameter prior θ ~ N(θ0, Σ_θ).
• Outputs: uncertainty.* ( u_num, u_model, u_c, g, nu_eff, α ), dual-form gap delta_form_unc, verification report and signature.
• Boundary: H may be time-varying; L may be symmetric positive semidefinite (undirected) or general (directed); process stationarity is declared via RefCond.

II. Terms and Variables

• Dynamics: x_{k+1} = f(x_k, θ, Δt) + w_k, y_k = h(x_k) + v_k; linearization F_k = ( ∂f / ∂x )|_{x̂_k}, H_k = ( ∂h / ∂x )|_{x̂_k}.
• Graph operators & kernel: L, A, H (observation matrix), K = g(L), spectral decomposition L = U Λ U^T.
• Error components: u_num (numerical integration/approximation), u_model (model structure/parameters), u_obs (observation noise), u_proc (process noise), u_init (initial condition).
• Covariances & propagation: P_k, Q_k, R_k, Σ_θ, J_* (Jacobians), ⊕ (covariance composition).
• Coverage & guardband: u_c (combined standard uncertainty), g = k * u_c, α (tail probability), nu_eff (effective degrees of freedom).
• Dual-form gap: delta_form_unc = | u_c^{LPU} - u_c^{MC} |.

III. Postulates P70E-*

• P70E-1: Error decomposition follows an additive approximation: u_total^2 ≈ u_proc^2 ⊕ u_obs^2 ⊕ u_model^2 ⊕ u_num^2 ⊕ u_init^2; sources and estimation methods must be persisted.
• P70E-2: LPU and MC run in parallel; record delta_form_unc and enforce contractual thresholds.
• P70E-3: All matrix-valued quantities must declare unit(field) and dim(field) and pass check_dim.
• P70E-4: If L is orthogonally diagonalizable, prefer spectral-domain bounds to derive kernel approximation and propagation upper limits.
• P70E-5: Numerical integration error must expose an auditable upper bound via local truncation error (LTE) and a stepsize controller.

IV. Minimal Equations S70E-*

• Continuous → discrete (linear approximation): ẋ = A x + B u + w_c, F = exp( Δt A ), Q_d = ∫_0^{Δt} exp(τ A) Q_c exp(τ A)^T dτ.
• Nonlinear linearized propagation: P_{k+1|k} = F_k P_{k|k} F_k^T + Q_k + U_num,k + U_model,k.
• Measurement update (KF/UKF): S_k = H_k P_{k+1|k} H_k^T + R_k, K_k = P_{k+1|k} H_k^T S_k^{-1}, P_{k+1|k+1} = (I - K_k H_k) P_{k+1|k}.
• Kernel approximation error (spectral, symmetric L): with K = U g(Λ) U^T, K_approx = U g_m(Λ) U^T,
  ε_approx = ( || g(Λ) - g_m(Λ) ||_2 / || g(Λ) ||_2 ),
  U_kern ≈ ε_approx^2 * K P_k K^T (first-order bound).
• Numerical integration error (order p, step Δt): ||e_{LTE}||_2 = O( Δt^{p+1} ),
  U_num ≈ J_f Σ_num J_f^T, Σ_num = σ_num^2 I, σ_num ∝ Δt^{p+1} (stepsize controller back-fills the constant).
• Parameter-uncertainty propagation: U_model ≈ J_θ Σ_θ J_θ^T, J_θ = ( ∂f / ∂θ )|_{x̂, θ }.
• Combined standard uncertainty: u_c = sqrt( tr( W P_out W^T ) ) ( W maps to the quantity of interest ); g = k * u_c (k determined by α, nu_eff).
• Welch–Satterthwaite: nu_eff ≈ ( ∑ w_i u_i^2 )^2 / ∑ ( w_i^2 u_i^4 / ν_i ) (approximate dof composition across components).
• Dual-form gap: delta_form_unc = | u_c^{LPU} - u_c^{MC} |.

V. Metrology Pipeline M70-5 (Ready → Decompose → Propagate → Verify → Persist)

1. Ready: load RefCond/units.*, graph/L/H/K.hash, solver/meta, Q/R/Σ_θ/x0/P0; determine the mapping W for the quantity of interest.
2. Decompose: tag error sources proc/obs/model/num/init; set MC sample size N_mc and step-size strategy.
3. Propagate (parallel):
   • LPU: build F_k, H_k, J_f, J_θ; advance P per S70E-*, accumulate U_num, U_model;
   • MC: sample {w_k, v_k, θ} with the stepper to obtain output sample variance u_c^{MC}.
4. Verify: compute delta_form_unc, coverage, nu_eff; enforce contracts C70E-*; on failure, generate an audit bundle.
5. Persist: write uncertainty.*, component shares, test statistics and signature; synchronize the dashboard.

VI. Contracts & Assertions C70E-* (threshold guidelines)

• C70E-01: coverage ≥ 1 - α - ε_cov (estimated via replay or MC).
• C70E-02: delta_form_unc ≤ tol_unc (recommend ≤ 0.15 * u_c).
• C70E-03: ε_approx ≤ ε_cap (kernel approximation), σ_num ≤ σ_cap(Δt, p) (integrator).
• C70E-04: P is positive definite and cond(P) ≤ κ_cap.
• C70E-05: Spectral stability: ρ(F) < 1 (discrete) or max Re(eig(A)) < 0 (continuous).
• C70E-06: Unit/dimension checks pass: check_dim(P) = "[x]^2", check_dim(g) = unit(W x).

VII. Implementation Bindings I70-* (interface prototypes & invariants)

• linearize_dynamics(f, x_hat, θ, Δt) -> { F, J_f }
• propagate_cov(P, F, Q, U_num, U_model) -> P_next
• estimate_num_uncertainty(stepper_state, order_p, Δt) -> U_num
• estimate_kernel_uncertainty(L, g, g_m, P) -> U_kern, ε_approx
• propagate_param_uncertainty(J_θ, Σ_θ) -> U_model
• uncertainty_to_guardband(P, W, k, α) -> { u_c, g, nu_eff }
• mc_propagate(stepper, priors, N) -> { u_c_MC, samples_digest }
• assert_uncertainty_contracts(stats, rules) -> report
  Invariants: non_decreasing(wm); inputs are hashable and traceable to RefCond; P ≻ 0; on numerical failure, return tagged NaN and block publication.

VIII. Cross-References

• Numerical integration & stability bounds: see Chapter 9.
• Assimilation consistency & residual statistics: see Chapter 12.
• Uncertainty metrics & guardband dashboard: see Chapter 13 and Appendix D.
• Manifest embedding & signatures: see Appendix C.

IX. Quality & Risk Control

• SLO: uncertainty.pipeline.latency_p95 ≤ B_mod; adapt MC budget N_mc until se(u_c^MC) ≤ r_mc * u_c.
• Fallback strategies: if ε_approx exceeds limits → raise approximation order or switch to exact spectral; if σ_num exceeds limits → reduce Δt or switch to a stiff integrator; if delta_form_unc exceeds limits → increase MC samples or re-center the linearization point.
• Audit & reproducibility: persist jacobians.hash, samples.digest, lte.trace, spectral.bounds, and rules.version.

Summary

• This appendix establishes a unified STG dynamics framework for error decomposition and propagation: parallel LPU and MC, explicit kernel and integrator errors, matrix-form parameter uncertainty, and publication of guardbands g = k * u_c.
• Deliverables are written to uncertainty.* and contracts.report, serving runtime monitoring, comparison, and compliance modules.