36-EFT.WP.EDX.Current v1.0 | Chapter 11 — Circuit-Level Modeling & Parameter Inference (I40- / Mx-)

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Chapter 11 — Circuit-Level Modeling & Parameter Inference (I40- / Mx-)

I. Chapter Objectives & Structure

1. Objective: Build a circuit-level joint modeling and inversion framework that maps from netlists/layouts to tension-landscape parameters, returning design- and diagnostics-ready posteriors, evidences, and model-comparison metrics; close the loop with S20-* / S40-* / S50-* / I30-* / M10-* / M20-* / Chapter 8 under one metrological and recording dialect.
2. Structure: I40-1 Circuit-level binding API → Mx-1 Parameter table & default priors → Mx-2 Likelihood & evidence → Mx-3 Model comparison & selection → Mx-4 Inference & uncertainty propagation → Data structure & records → Compliance templates → Correspondence & degeneracy → Cross-chapter pointers & summary.
3. Shared dialect (time-of-arrival; path/measure explicit; record delta_form):
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

II. Variables & Units (new in this chapter)

• netlist: circuit netlist (ports/devices/topology/component values).
• layout: routing/stackup/guards and dielectric stack.
• binding: layout→physical-path record with path = { γ_p(ell) }, segment-level n_eff, and weights w_p.
• θ: parameter vector; subsets θ_T (tension-related: K_s, K_t, w_p, etc.), θ_E (equivalent electrical: σ_eff or Σ_eff, etc.), θ_N (nuisance: noise/alignment/de-embedding).
• Z_model(omega; θ): modeled impedance; Z_ref(omega): reference impedance; Z_eft(omega; θ) = Z_ref + ΔZ_T(θ).
• π(θ): prior; L(data|θ): likelihood; p(θ|data): posterior; Z_evid: evidence.
• Units follow SI; all key equalities must pass check_dim.

I40-1 Circuit Binding (API)

Statement
Compile netlist + layout + binding into a differentiable circuit model with explicit paths gamma(ell), arrival terms T_arr, and weights w_p, with unified ports, de-embedding, and timebase.

API prototypes (minimum executable)

I40-1:

- id: "bind.compile"

proto: "compile(netlist, layout, binding, bc, options) -> model_handle"

- id: "bind.predict"

proto: "predict(model_handle, theta, freq_grid) -> {Z_model(ω), T_arr_p(ω), w_p(ω)}"

- id: "bind.jacobian"

proto: "jacobian(model_handle, theta, freq_grid) -> ∂Z/∂θ"

- id: "bind.update_paths"

proto: "update_paths(model_handle, binding_ref) -> model_handle'"

requirements:

arrival_record: {form, gamma, measure, c_ref, delta_form}

qa_gates: ["check_dim","passivity(Re{Z}≥0)","KK_consistency"]

Domain & constraints
Ports/probes aligned via anchors; deemb and sync consistent; binding versioned in step with I30-*.

Falsifiability
If predict() fails passivity/K–K on calibrated reference fixtures or contradicts recorded T_arr, reject the binding setup or its domain.

Mx-1 Parameters & Default Priors

Groups & symbols

• Conduction/medium (equivalent): σ_eff (or anisotropic Σ_eff), constrained σ_eff ≥ 0.
• Tension kernels/paths: amplitudes and time constants (or spectral parameters) of K_s, K_t; weights w_p (w_p ≥ 0, Σ_p w_p ≤ 1); segment n_eff[i].
• Nuisance: Δt_sync, noise_scale, deemb correction factors (if estimated).

Recommended priors (release defaults)

• σ_eff/Σ_eff: lognormal(μ, s) or half-normal halfnormal(σ).
• K_s, K_t amplitudes: halfnormal; time constants: loguniform over engineering ranges.
• w_p: dirichlet(α⃗) (use α⃗ < 1 for sparsity).
• n_eff[i]: Gaussian or folded Gaussian around calibrated n̂_eff[i], convergence gates from M10-*.
• Δt_sync: zero-mean Gaussian with variance from the M10-* sync budget.

Parameter card (example)

params:

sigma_eff: {prior: "lognormal", mu: -12.0, s: 0.8, unit: "S/m"}

Ks_amp: {prior: "halfnormal", sigma: 0.5, unit: "A·m^-2"}

Ks_tau: {prior: "loguniform", low: 1e-12, high: 1e-7, unit: "s"}

w: {prior: "dirichlet", alpha: [0.3,0.3,0.3]}

n_eff_seg: {prior: "normal", mu: n_eff_hat, sigma: 0.05*mu}

dt_sync: {prior: "normal", mu: 0.0, sigma: 2.0e-12, unit: "s"}

Mx-2 Likelihood & Evidence

Likelihood (complex-impedance domain; whitened)

• Joint Gaussian on real/imag parts:
  L(data|θ) ∝ exp( -½ Σ_ω [ (Re ε_ω)^T W_ω (Re ε_ω) + (Im ε_ω)^T W_ω (Im ε_ω) ] )
  with ε_ω = Z_meas(ω) - Z_model(ω; θ) and W_ω the inverse covariance from noise and de-embedding uncertainty.
• Magnitude/phase form (optional): independent Gaussians on |Z| and arg Z; for phase, restrict to the coherence window and subtract the sync term ω · Δt_sync.

Evidence & regularization

• Z_evid = ∫ L · π dθ; use Laplace or nested sampling in engineering practice; passivity and K–K are hard gates.
• Soft constraints: Re{Z_model} ≥ 0, no right-half-plane poles in kernel spectra (via priors or penalties).

Mx-3 Model Comparison & Selection

Candidate models

• M_classic: Z_ref(ω; θ_E) only.
• M_eft-min: Z_ref + ΔZ_T with minimal K_s/K_t.
• M_eft-ms: M_eft-min plus multi-scale/multi-path weights w_p(ω).

Metrics

• Δlog Z = log Z_evid(M_i) − log Z_evid(M_j); thresholds: > 5 strong evidence, 2–5 moderate.
• Alternatives: AIC/BIC and leave-one-out error when evidence is hard to stabilize.

Selection rule
Evidence first, then parsimony; on ties, favor models with stronger extrapolation consistency and lower parameter entanglement.

Mx-4 Inference & Uncertainty Propagation (UQ)

Workflow

1. Initialize: center priors at M10-* calibrations; compile model_handle.
2. MAP/ML: obtain θ_MAP via GN/LM or quasi-Newton.
3. Posterior sampling: HMC/NUTS or SMC to draw p(θ|data).
4. UQ propagation: push posteriors to T_arr,p(ω), arg Z(ω), w_p(ω), and design KPIs (phase slope, in-band ripple, etc.).
5. PPC: posterior predictive checks (residual whitening, spectral correlation, secondary passivity/K–K).

Outputs
θ_MAP, θ_mean, CI/HPD, corr(θ), log Z_evid, posterior_predictives; diagnostics: R̂, effective sample sizes, residual spectra, gate outcomes.

VII. Data Structure & Records (minimal template)

inversion:

model_id: "EDX-Current-eft-ms"

freq_grid_Hz: [...]

priors: {...} # see Mx-1

results:

theta_map: {...}

theta_mean: {...}

CI_95: {...}

logZ: -1234.5

diagnostics: {Rhat: {...}, ess: {...}}

arrival:

form: "n_over_c"

gamma: "explicit"

measure: "d_ell"

qa_gates: {check_dim:"pass", passivity:"pass", KK:"pass"}

VIII. Compliance Snippets (copy-ready)

• Minimal impedance with mapping (aligned to S50-*):
  Z_eft(ω; θ) = Z_ref(ω; θ_E) + Σ_p w_p(ω; θ_T) · ∫_{γ_p} ( A_s(ω; θ_T) · ( τ·∇T_fil ) + A_t(ω; θ_T) · ( ∂T_fil/∂t ) ) d ell
• Whitened phase with sync correction:
  arg Z_corr(ω) = arg Z_meas(ω) − ( ω · Δt_sync )
• Evidence gates:
  check_dim = pass ; Re{Z_eft} ≥ 0 ; KK_consistency = pass

IX. Falsifiability Criteria (inference side)

• PPC failure: systematic deviations of posterior predictives within the pre-registered band (non-white residuals) invalidate the current kernel or weighting model.
• Dialect agreement: the two T_arr dialects must yield consistent key derived quantities (e.g., effective T_arr, phase slope) within u(T_arr); otherwise inspect delta_form and path records.
• Path/temperature extrapolation: under path switching or small n_eff perturbations, posterior extrapolations must respond linearly with ΔT_arr/Δn_eff; otherwise reject the model or priors.

X. Correspondence & Degeneracy to Classical Framework
With K_s = K_t = 0 or w_p = 1 on the main path (others zero), Z_eft → Z_ref; inversion reduces to classical equivalent-parameter fitting, while explicit gamma(ell) and T_arr records retain traceability and falsifiability.

XI. Cross-Chapter Pointers & Summary

• Dependencies & links: I30-* (binding/alignment), S20-* (minimal equations), S40-* (kernels & multi-path), S50-* (impedance mapping), M10-* (metrology chain), M20-* (falsification workflow).
• Summary: This chapter fixes circuit-level modeling and parameter inversion as the I40-* / Mx-* execution dialect. By unifying the prior→likelihood→evidence→PPC→design-KPI chain, it closes the loop from data to design for tension landscapes, paths, and impedance within a single reproducible and falsifiable engineering process.