36-EFT.WP.EDX.Current v1.0 | Chapter 6 — A New Interpretation of Impedance (S50-*)

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Chapter 6 — A New Interpretation of Impedance (S50-*)

I. Chapter Objectives & Structure

• Objective: Reinterpret Z(omega) through the tension landscape, composing Chapter 4 S20-* minimal conservation with Chapter 5 S40-* kernels and path weights into an engineering-ready impedance expression with falsifiable predictions and design rules.
• Structure: S50-1 Impedance–tension landscape mapping → S50-2 Frequency behavior & coherence window → S50-3 Path nonlocal corrections & design rules.
• Shared dialect (time-of-arrival): two equivalent forms; both must declare gamma(ell) and d ell, and the chosen form must be recorded as delta_form.
  • Constant factored out: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• Dependencies & interfaces: S20-2 (effective conduction), S20-5 (phase–delay), S40-* (kernels & weights), Ixx-* / Mx-* (binding & metrology).

S50-1 Impedance–Tension Landscape Mapping

Statement

• Define the tension-landscape impedance:
  Z_eft(omega) = Z_ref(omega) + ΔZ_T(omega)
  where Z_ref(omega) is the reference impedance in the no-tension limit (classical RLC/telegrapher baseline or a calibrated no-tension baseline), and ΔZ_T(omega) is the tension-induced correction.
• General form of the tension correction from kernels and path weights:
  ΔZ_T(omega) = Σ_p w_p(omega) · F_p[ K_s, K_t, T_fil, gamma_p ]
  F_p = ∫_{gamma_p} ( A_s(omega) · ( τ_p · ∇T_fil ) + A_t(omega) · ( ∂T_fil/∂t ) ) d ell
  with τ_p the unit tangent along path gamma_p, and A_s(omega), A_t(omega) the path response coefficients derived from K_s, K_t and geometry/boundaries, dimensioned so that ΔZ_T is in Ω.

Domain & constraints

• First-order (or low-order) approximation within the coherence window; w_p≥0, Σ_p w_p ≤ 1; causal/stable kernels and passivity gate Re{ Z_eft(omega) } ≥ 0 for publication.
• The choice of Z_ref must be fixed in Methods and recorded in data as baseline_id.

Falsifiability

• With netlist/materials fixed while reshaping only the T_fil landscape (via layout/boundary or external field control), measured amplitude/phase of Z_eft must match the prediction. Mismatch rejects the calibrated A_s/A_t or the completeness of the path set {gamma_p}.

S50-2 Frequency Behavior & Coherence Window

Statement

• Phase leading term with tension phase correction:
  arg Z_eft(omega) = arg Z_ref(omega) + ( omega · T_arr ) + φ_T(omega)
  where φ_T(omega) stems from ΔZ_T(omega) and obeys causality (consistent with Kramers–Kronig/Cauchy–Riemann).
• Magnitude composition (inside the coherence window):
  |Z_eft(omega)| = | Z_ref(omega) + Σ_p w_p(omega) · ΔZ_p(omega) |
  where ΔZ_p(omega) is the frequency image of F_p, carrying the path phase factor e^{ i·omega·ΔT_arr,p }.

Domain & constraints

• Use coherent summation inside the window; switch to energy composition outside to avoid spurious oscillations/overfit.
• Require check_dim and passivity/positive-real quick checks; for data-driven fits, forbid right-half-plane poles in kernel images.

Falsifiability

• By sweeping frequency and temperature (or stress) to slowly vary n_eff and T_fil, expect a shift in arg Z and smooth renormalization of |Z|. Causality/passivity violations (sharp spikes/negative real part) reject the kernel or weight model.

S50-3 Path Nonlocal Corrections & Design Rules

Statement

• First-order nonlocal path correction (dominant path + few side branches):
  ΔZ_path(omega) ≈ Σ_p w_p Σ_k C_{p,k}(omega) · ( e^{ i·omega·ΔT_arr,p,k } - 1 )
  with k indexing segments/branches of path p, and C_{p,k}(omega) effective coupling coefficients (Ω). The term captures amplitude/phase ripples from differential arrival times.

Design rules (actionable)

1. Phase linearization: Minimize ∑_p w_p · ∫_{gamma_p} n_eff d ell over the target band to reduce the slope of omega·T_arr. If compensation is required, adjust boundaries/guards to cancel ΔT_arr among main/side branches.
2. Branch suppression: For high-frequency-sensitive links, reduce capture in high-n_eff regions (limit the high-frequency rise of w_p) and enforce tighter layout → gamma(ell) consistency.
3. Kernel band-limiting: Apply band-limits or smoothing priors on K_s/K_t to avoid nonphysical feedback outside the window.

Domain & constraints

• Rules 1–3 hold in the passive linear regime; in the saturation regime (S40-3), include threshold gating in design acceptance.

Falsifiability

• Validate Rules 1–2 by toggling guards/grounds and inserting length-matched compensation. If arg Z and |Z| do not respond with ΔT_arr/w_p adjustments, reject the path model or its weighting implementation.

II. Variables & Units (new in this chapter)

• Z_eft(omega): tension-landscape impedance (Ω); Z_ref(omega): reference impedance (Ω).
• ΔZ_T(omega), ΔZ_p(omega): tension-induced corrections (Ω).
• A_s(omega), A_t(omega): path response coefficients ensuring F_p outputs Ω.
• φ_T(omega): tension-induced phase correction (dimensionless, rad).
• C_{p,k}(omega): nonlocal coupling coefficients (Ω).
• Other symbols per Chapters 2–5: n_eff, c_ref, T_fil, K_s, K_t, w_p, gamma(ell), d ell, T_arr.

III. Compliance Snippets (copy-ready)

• Minimal impedance mapping:
  Z_eft = Z_ref + Σ_p w_p · ∫_{gamma_p} ( A_s ( τ·∇T_fil ) + A_t ( ∂T_fil/∂t ) ) d ell
• Phase/time recording:
  arg Z_eft = arg Z_ref + ( omega · T_arr ) + φ_T ; T_arr, gamma(ell), d ell, delta_form → must record
• First-order nonlocal correction:
  ΔZ_path ≈ Σ_p Σ_k C_{p,k}(omega) · ( e^{ i·omega·ΔT_arr,p,k } - 1 )
• Publication gates (QA):
  check_dim{ S50-1, S50-2, S50-3 } = pass ; Re{ Z_eft } ≥ 0 (passivity)

IV. Correspondence & Degeneracy to Classical Framework

• With A_s = A_t = 0 or w_p=1 on the main path (others zero): Z_eft(omega) → Z_ref(omega), i.e., classical RLC/telegrapher behavior.
• In the weak-inhomogeneity/weak-time-variation limit: φ_T(omega) → 0, ΔZ_path becomes small, and S50-* aligns with the linear-phase approximation in S20-5.

V. Cross-Volume References & Anchor Guide

• This chapter pairs with: See S20-2 (effective conduction), See S20-5 (phase–delay), See S40-* (kernels & weights).
• For implementation: See I30-* / I40-* (layout binding & circuit-level coupling), See M10-* / M20-* (metrology chain & falsification workflow).

VI. Chapter Summary
This chapter reinterprets impedance via the tension landscape: Z_eft augments a reference impedance with kernel- and path-weighted tension corrections, clarifies the leading phase omega·T_arr and the additional phase φ_T(omega), and provides first-order path nonlocal corrections with actionable design rules—closing the loop among measurement, simulation, and layout under a single, falsifiable dialect.