37-EFT.WP.EDX.HighSpeed v1.0 | Chapter 3 — Axioms & Physical Picture (P10-HF)

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

I. Chapter Objectives & Prerequisites

1. Objective: Present the high-frequency minimal axiom set P10-HF-*, provide the physical picture for dispersion and radiation, and ground subsequent sections S20-HF-* / S30-HF-* / S50-HF-* / M10-HF-* / I30-HF-* under a unified dialect.
2. Shared time-of-arrival dialect (equivalent; 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 )
3. Frequency variable is omega; key equalities must pass check_dim = pass; hard release gates: passivity and KK_consistency.

II. Physical Picture (High-Frequency Relay)

• Path-dominated phase: Within the coherence window, the port phase arg Z_eft(omega) is governed by the arrival times T_arr,p of the dominant path family {γ_p} plus a slowly varying extra phase φ_T(omega); the linear term omega·T_arr captures first-order dispersion.
• Weight-mediated modal competition: Geometry/return/boundary conditions shape w_p(omega) which varies slowly in-band; the dominant path/mode prevails, and switching appears primarily as a linear phase translation via ΔT_arr.
• Radiation as positive-real correction: Plane gaps, long stubs, and apertures activate a radiation channel recorded as a positive-real increment ΔZ_rad(omega), correlated with near/far-field observables.

III. Axioms (P10-HF-*)

• P10-HF-1 (Coherence-window linear phase)
  Statement: Within a pre-selected coherence window,
  arg Z_eft(omega) ≈ ( omega · T_arr ) + φ_T(omega),
  where φ_T(omega) varies slowly and second-order curvature is bounded.
  Domain/Constraints: Window set by the metrology chain; T_arr computed with one dialect and delta_form recorded; φ_T attributable to material/boundary dispersion.
  Falsifiability: If in-window linear-fit residuals systematically exceed the gate (see Chapter 7 E_phase), reject the window or the linear-phase hypothesis.
• P10-HF-2 (Positive-real radiation correction)
  Statement: When boundary discontinuities or open returns activate radiation,
  Z_eft(omega) = Z_ref(omega) + ΔZ_T(omega) + ΔZ_rad(omega),
  with Re{ΔZ_rad(omega)} ≥ 0 and K–K consistency.
  Domain/Constraints: Radiation activation determined by geometry/shielding continuity; ΔZ_rad recorded as an equivalent port-level correction.
  Falsifiability: If sealing gaps or shortening stubs does not reduce Re{ΔZ_rad} together with |E_rad|/|I_CM|, reject the axiom or the channel classification.
• P10-HF-3 (Slowly varying weights & path switching)
  Statement: In-band w_p(omega) responds smoothly to small perturbations; path/mode switching appears primarily as a linear phase shift driven by ΔT_arr, not as abrupt non-causal phase distortions.
  Domain/Constraints: Σ_p w_p ≤ 1; weights depend jointly on geometry/returns/boundaries and material dispersion.
  Falsifiability: Under controlled geometry/return toggles, if the slope of Δarg Z(omega) fails k_φ ≈ ΔT_arr (see Chapter 10, P1), reject the smooth-weight hypothesis or the path model.
• P10-HF-4 (Two-dialect agreement)
  Statement: For the same gamma(ell) and n_eff, the two T_arr writings agree within uncertainty:
  |T_arr^{(n_over_c)} − T_arr^{(n_over_c_times_n)}| ≤ u(T_arr).
  Domain/Constraints: Both evaluations must explicitly record gamma(ell), d ell, and delta_form.
  Falsifiability: If the difference exceeds the gate, reject release or return for path/unit/record audits.

IV. Engineering Corollaries & Minimal Rules

• Phase-linearity principle: Subject to impedance balance, minimize the in-band slope and ripple of
  Σ_p w_p · ∫_{γ_p} n_eff d ell (cf. Chapter 7 E_phase/GDR).
• Return-path priority: Provide local closed returns and stitching-via rings at layer/plane transitions to reduce high-frequency w_side(omega) and ΔT_arr.
• Radiation gating: When Re{ΔZ_rad} rises along with |E_rad|/|I_CM|, prioritize shielding continuity and stub mitigation.

V. Implementation & Records (snippets)

• Phase correction: arg Z_corr(omega) = arg Z_raw(omega) - ( omega · Δt_sync ).
• HF increments: alpha_per_m, beta_per_m, T_group_s, Z_c(omega), E_phase, GDR, ΔZ_rad(omega).
• Record template:
• highspeed:
• coherence_window: {w1: ω1, w2: ω2}
• arrival: {form:"n_over_c", gamma:"explicit", measure:"d_ell", c_ref:299792458.0, Tarr_s:..., u_Tarr_s:..., delta_form:"n_over_c"}
• dispersion: {alpha_per_m:[...], beta_per_m:[...], T_group_s:[...]}
• radiation_gate: {Re_Zrad_min: 0.0, KK_consistency: "pass"}
• weights: {w_main:[...], w_side:[...]}
• qa_gates: {check_dim:"pass", passivity:"pass", KK:"pass"}

VI. Baselines & Examples (reproducible)

• Differential microstrip A/B (equal length; guard/return differs): validate k_φ ≈ ΔT_arr and E_phase gate.
• Via / plane-transition board: assess w_p(omega) drift and GDR degradation.
• Gap/aperture board (seal-able): validate Re{ΔZ_rad} ≥ 0 and its reduction after sealing.
• Each case ships with dataset_card / pipeline_card / env_lock and reference outputs as regression baselines.

VII. Falsifiability Criteria (chapter-specific)

• J-HF-1: In-window linear fit of Δarg Z(omega) achieves R^2 ≥ 0.98 and slope k_φ ≈ ΔT_arr; otherwise reject P10-HF-3 or the path model.
• J-HF-2: If sealing does not reduce Re{ΔZ_rad} and |E_rad|/|I_CM|, reject P10-HF-2 or the radiation-channel assumption.
• J-HF-3: If the two T_arr dialects differ beyond u(T_arr), reject P10-HF-4 or the data/record dialect.

VIII. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Current Chapter 8 (Paths/Arrival), Chapter 9 (Metrology Chain).
• Downstream: Chapter 4 (S20-HF-* / S30-HF-* dispersion relations), Chapter 5 (S50-HF-* radiation correction), Chapter 7 (coherence-window KPIs), Chapter 12 (layout & process rules), Chapter 16 (design protocol).