36-EFT.WP.EDX.Current v1.0 | Chapter 5 — Tension-Mediated Relay Mechanism (S40-*)

Home ／ Docs-Technical WhitePaper ／ 36-EFT.WP.EDX.Current v1.0

Chapter 5 — Tension-Mediated Relay Mechanism (S40-*)

I. Chapter Objectives & Structure

1. Objective: Provide a computable mechanism and kernel formulation for tension-mediated relay, consistent with Chapter 4 S20-*, and define calibratable intermediates for the impedance reinterpretation (Chapter 6), metrology, and implementation binding.
2. Structure: S40-1 Local response kernel → S40-2 Multi-scale relay chain → S40-3 Saturation and response ceilings.
3. 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 )
4. Dependencies & interfaces: S20-* (continuity/energy & minimal conduction), Ixx-* (binding & measurement stack), Mx-* (metrology checks).

S40-1 Local Response Kernel (minimal form)

Statement

1. Write the tension-coupled terms in S20-2 as causal-kernel driven spatiotemporal convolutions:
   J_T(x,t) = ( K_s ⊛ ∇T_fil )(x,t) + ( K_t ⊛ ( ∂T_fil/∂t ) )(x,t)
   where ⊛ denotes convolution over (x,t); K_s(x,t), K_t(x,t) are causal (K_{·}(t<0)=0) and dominate within the coherence window.
2. Relation to Chapter 4:
   chi_T = ∫ K_s(x,t) dx dt and psi_T = ∫ K_t(x,t) dx dt in the local/slowly-varying limit, hence S20-2 is the local approximation of S40-1.

Structure & parameters

1. Anisotropy via tensor kernels: K_s → 𝕂_s(x,t), K_t → 𝕂_t(x,t); prioritize components along the local principal path tangent τ(x):
   J_T ≈ ( 𝕂_s : ( τ ⊗ ∇T_fil ) ) + ( 𝕂_t : ( τ ⊗ ∂T_fil/∂t ) ).
2. Causality & stability: Laplace/Fourier images of K_{·} have no right-half-plane poles; L¹/L² integrable to keep energy bounded.

Domain & constraints

1. Linearization valid within the coherence window; kernel support no larger than the object’s characteristic scales; check_dim passes with [J_T]=A·m⁻².

Falsifiability

1. With geometry and nominal properties fixed while only reshaping the T_fil landscape, inverted K_s,K_t must be compatible (within uncertainty) with predicted frequency/path trends; otherwise reject the kernel form or its domain.

S40-2 Multi-Scale Relay Chain (path weighting; series/parallel)

Statement

1. Express relay as a weighted chained composition along an explicit family of paths {γ_p(ell)}:
   J_T(x,t) = Σ_p w_p(x,t;omega) · R_p[T_fil](x,t)
   where R_p is the first- (or low-) order kernel response along γ_p, and w_p are relay-capacity weights jointly determined by n_eff(x,t), local ∇T_fil, and boundary binding (aligned with P10-3 in Chapter 3).

Structure & parameters

1. Multi-scale decomposition: K_{·}(x,t) = Σ_{s∈Scales} a_s · K_{·}^{(s)}(x,t), each scale s carrying τ_s (delay) and ℓ_s (path scale);
   w_p = g(n_eff, ∇T_fil, BC; s) and evolves with omega and t.
2. Series/parallel systematics: coherent summation inside the coherence window; energy composition outside:
   R_p^{coh} = |Σ_{segments} r_k · e^{ i·omega·ΔT_arr,k }| ,
   R_p^{incoh} = ( Σ_k |r_k|² )^{1/2}.

Domain & constraints

1. {γ_p} must be provided by the binding layer layout ↔ gamma(ell) (see I30-*); w_p≥0 and Σ_p w_p ≤ 1.

Falsifiability

1. In switchable-path/boundary apparatus, adjusting controls that set w_p (geometry/grounding/guards) must change the phase and magnitude of I/V/Z in agreement with Σ_p w_p · ΔT_arr,p; otherwise reject the weighting model or the path set.

S40-3 Saturation & Response Ceilings (nonlinear corrections)

Statement

1. Relay rate and amplitude admit ceilings:
   J_T^{sat} = g_sat( J_T^{lin} ; J_* , R_* )
   where J_T^{lin} is the linear output of S40-1/2, J_* is the amplitude ceiling, and R_* the slew ceiling (e.g., |∂J_T/∂t| ≤ R_*).
2. Generic saturations (replaceable by empirics):
   g_sat(ξ) = ( ξ / ( 1 + |ξ| / J_* ) ) or g_sat(ξ) = J_* · tanh( ξ / J_* ).
3. Threshold triggering: when |∇T_fil| > θ_* or |∂T_fil/∂t| > ϑ_*, enable nonlinear corrections to avoid nonphysical growth in Z(omega).

Domain & constraints

1. J_* , R_* , θ_* , ϑ_* calibrated via M10-* / M20-*; nonlinear mode only above thresholds, preserving S20-* continuity and energy constraints.

Falsifiability

1. Under incrementally increasing drive/perturbation, absence of the predicted knee/roll-off in arg Z(omega) or saturation in |J| contradicts threshold settings or the chosen saturation form.

II. Variables & Units (new in this chapter)

• K_s(x,t), K_t(x,t): spatiotemporal response kernels, units chosen so that J_T is in A·m⁻²; anisotropic forms as tensors 𝕂_s, 𝕂_t.
• w_p(x,t;omega): path weights (dimensionless, 0…1).
• R_p[T_fil]: kernel response along path γ_p (same units as J or normalized).
• J_* , R_* , θ_* , ϑ_*: amplitude ceiling, slew ceiling, and gradient/time-derivative thresholds (SI per definition).

III. Compliance Snippets (copy-ready)

• Local kernel (minimal):
  J_T = ( K_s ⊛ ∇T_fil ) + ( K_t ⊛ ( ∂T_fil/∂t ) )
• Path weighting:
  J_T = Σ_p w_p · R_p , where w_p = g(n_eff, ∇T_fil, BC; omega,t)
• Saturation map:
  J_T^{sat} = ( J_T / ( 1 + |J_T| / J_* ) ) ; |∂J_T/∂t| ≤ R_*
• Report/data recording:
  delta_form = "n_over_c" or "one_over_c_times_n" ; path: {γ_p(ell)} ; measure: d ell ; weights: {w_p}
• Dimensional checks:
  check_dim{ S40-1, S40-2, S40-3 } = pass

IV. Counterexamples & Failure Modes (engineering cautions)

• Out-of-window coherence: applying coherent summation beyond the coherence window yields spurious oscillations in Z(omega).
• Unmodeled path leakage: omission of minor branches γ_q whose w_q rises at high frequency causes residual phase shifts.
• Non-causal kernels: inverted K_{·} with support at t<0 violates causality/stability.
• Mis-set thresholds: J_* , θ_* too high/low, producing saturation predictions inconsistent with measurements.

V. Correspondence & Degeneracy to Classical Framework

• With K_s(x,t) → chi_T·δ(x)·δ(t) and K_t(x,t) → psi_T·δ(x)·δ(t), and w_p=1 on the main path (others zero), S40-* reduces to the local linear coupling of S20-2; further setting chi_T=psi_T=0 recovers classical Ohmic/transmission-line forms.
• In the skin-effect and diffusion limits, the multi-scale weighting corresponds to traditional dispersive/diffusive kernels; the distinctive explicit path and weight evolution remain experimentally accessible.

VI. Cross-Volume References & Anchor Guide

• This chapter pairs with: See "EFT.WP.EDX.Current v1.0" S20-2 (effective conduction), See "EFT.WP.Core.Tension v1.0" S72-* (tension ontology), See "EFT.WP.Core.Metrology v1.0" Mx-* (dimensions/uncertainty).
• Forward links: See "EFT.WP.EDX.Current v1.0" S50-* (impedance mapping), See "EFT.WP.Methods.SimStack v1.0" (kernel inversion & simulation), See "EFT.WP.Methods.Falsification v1.0" (falsification workflow).

VII. Chapter Summary
This chapter models tension-mediated relay via causal kernels and path weighting, provides the linear approximation aligned with S20-* and its nonlinear saturation extension, and fixes calibration, recording, and validation dialects. It underwrites the tension-landscape mapping of Z(omega), the metrology chain, and implementation binding in the chapters that follow.