36-EFT.WP.EDX.Current v1.0 | Chapter 3 — Axioms & Physical Picture (P10-*)

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Chapter 3 — Axioms & Physical Picture (P10-*)

I. Chapter Objectives & Prerequisites

1. Objective: Present the minimal axiom set P10-* for EDX.Current, establish the tension-mediated relay picture, and fix the testable framework that grounds S40-* / S50-* / Mx-* / Ixx-*.
2. Prerequisites & cross-volume dialect: Terminology/symbols follow Chapter 2. The shared time-of-arrival term admits two equivalent dialects, both with explicit path gamma(ell) and measure d ell and recorded delta_form:
   • Constant factored out: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
     Dimensional checks align with Core.Metrology v1.0. Example citation: See "EFT.WP.Core.Equations v1.1" Ch.2 S20-*; See "EFT.WP.Core.Tension v1.0" S72-*.

II. Physical Picture (tension-mediated relay)

• Basic image: Spatiotemporal perturbations of T_fil(x,t) form a relay chain in the medium that organizes J(x,t)—its routing, delay, and phase. Macroscopic observables I(t) / V(t) / Z(omega) retain the usual continuity and power-balance expressions; differences arise from explicit path dependence and time-varying media through the tension landscape.
• Path competition: Among candidate paths, dominant transport branches are selected by an effective relay capacity functional; the shared delay is set by T_arr. n_eff encapsulates medium/geometry/boundary influences on propagation.
• Coherence window: Within a specified temporal/frequency window, relay sub-paths superpose coherently; outside the window, contributions are suppressed or combined non-coherently.

III. Axiom Set (P10-*)

• P10-1 (Local relay & conservation)
  Statement: On any connected medium element, incremental current transfer is triggered by local tension difference ΔT_fil and modulated by its gradient ∇T_fil; macroscopic continuity and power conservation remain unchanged.
  Domain/Constraints: Medium non-singular; anisotropy and inhomogeneity allowed and measurable; T_fil differentiable; no propagation beyond c_ref.
  Falsifiability: On samples with fixed nominal properties, reshaping only the T_fil landscape (geometry/material unchanged) must reproducibly re-distribute J. Observation of continuity-violating flux jumps falsifies the axiom or its domain.
• P10-2 (Explicit paths & shared arrival term)
  Statement: Any propagation/delay observable must evaluate T_arr over explicit gamma(ell) with d ell. The two dialects
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell )
  are equivalent and interchangeable, with the choice recorded as delta_form.
  Domain/Constraints: n_eff ≥ 0 and measurable; piecewise-smooth paths; dimensional checks pass.
  Falsifiability: In a controllable multi-path apparatus, both dialects must predict identical delays for the same gamma(ell)/n_eff within metrological uncertainty.
• P10-3 (Relay capacity & path weighting)
  Statement: Candidate paths contribute under relay-capacity weights determined jointly by n_eff(x,t), local ∇T_fil, and boundary binding conditions; weights evolve with time and frequency.
  Domain/Constraints: Weights are invertible from observables or provided by S40-*; coherent summation within the window, energy-based composition outside.
  Falsifiability: By toggling geometry/boundaries to reshape n_eff, measured weight shifts must match predictions; otherwise the weighting model or its domain is rejected.
• P10-4 (Boundary continuity & binding)
  Statement: At material interfaces and geometric discontinuities, J and power flux satisfy boundary continuity and the binding layer mapping Ixx-*; differences appear only as re-allocation of path weights and T_arr.
  Domain/Constraints: Standard measurable boundary conditions; binding provides a bi-directional layout ↔ gamma(ell) map.
  Falsifiability: With identical netlist parameters but altered layout-path mapping, changes in I/V/Z must follow the gamma(ell) updates predicted by the binding; otherwise reject the axiom or its implementation.
• P10-5 (Response ceilings & saturation)
  Statement: Relay rate admits an upper bound set by available local tension resources together with c_ref and n_eff; when |∇T_fil| exceeds a threshold, saturation and non-linear corrections appear.
  Domain/Constraints: Ceiling parameters calibrated via M10-* / M20-*; non-linear regime governed by S40-*.
  Falsifiability: Under incrementally increasing drive/perturbation, failure to observe the predicted roll-off or knee falsifies the ceiling or its thresholds.

IV. Corollaries & Direct Engineering Rules

• Path engineering rule: For fixed materials and netlists, first optimize gamma(ell) (geometry/stacking/grounding) to reduce ∫ n_eff d ell, thereby lowering effective T_arr and phase delay without changing nominal component values.
• Coherence window partitioning: In designing/measuring Z(omega), estimate the coherence window; superpose paths coherently inside the window and apply energy-based composition/suppression outside.
• Binding priority at boundaries: During layout, lock the layout → gamma(ell) map and constraints early; Ixx-* bindings are early-stage layout checks.

V. Correspondence to the Classical Framework

• In the weak inhomogeneity/time-variation limit, P10-* reduces to parameter corrections absorbable by classical transmission-line/network models; discrepancies concentrate in path nonlocal corrections and coherence-window effects (formalized in S50-*).
• Local conservation laws and standard measurement definitions remain intact; the difference is the explicit, metrologically traceable modeling of where/how signals arrive via gamma(ell).

VI. Cross-Volume References & Anchor Guide

• This chapter relies on: See "EFT.WP.Core.Equations v1.1" Ch.2 S20-* (continuity/power), See "EFT.WP.Core.Tension v1.0" S72-* (tension ontology), See "EFT.WP.Core.Metrology v1.0" Mx-* (dimensions/uncertainty).
• Forthcoming chapters: S40-* (relay kernel & multi-scale), S50-* (impedance mapping), M10-* / M20-* (metrology & falsification), I30-* / I40-* (binding & circuit-level coupling).

VII. Chapter Summary
This chapter states P10-1…P10-5, establishes the T_fil-centric relay picture, fixes the unified T_arr dialect with explicit paths, and grounds coherence-window and boundary-binding practices—providing consistent, falsifiable premises for the minimal equations, impedance reinterpretation, metrology, and implementation binding that follow.