36-EFT.WP.EDX.Current v1.0 | Chapter 8 — Paths & Time-of-Arrival Dialects

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Chapter 8 — Paths & Time-of-Arrival Dialects

I. Chapter Objectives & Structure

• Objective: Unify the two dialects of T_arr, and specify the engineering expressions, records, and validation of the path gamma(ell) and the measure d ell, so that S20-* / S40-* / S50-* / I30-* interoperate and are verifiable under a single arrival-time framework.
• Structure: Unified dialects & definitions → Path representation & measure → Engineering computation & records → Multi-path & composition → Metrology & uncertainty → Falsifiability → Compliance templates → Correspondence & degeneracy → Cross-chapter pointers & summary.

II. Unified Dialects & Definitions (equivalent; path/measure explicit)

1. Constant-factored dialect: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ).
2. General dialect: T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
3. Constraints:
   • gamma(ell) is the physical propagation path; ell is the arclength; d ell the associated measure.
   • n_eff(x,t) ≥ 0 is measurable and bound to medium/geometry/boundaries; c_ref is the SI reference propagation limit.
   • The chosen dialect must be recorded as delta_form ∈ {"n_over_c","one_over_c_times_n"} in data/reports; with the same gamma(ell) and n_eff the two forms are equivalent.

III. Path Representation & Measure (from layout to physical paths)

• Source of paths: The binding layer layout ↔ gamma(ell) (see I30-2) produces the path family {γ_p}, consistent with ports/probes anchors.
• Segmentation & smoothness: Each γ_p is piecewise smooth; segment attributes include {material, geometry, nearby boundaries, n_eff(segment), constraints}.
• Measure convention: d ell is the arclength element; path reversal does not change T_arr. Unmeasurable spans are merged as equivalent segments and flagged as mask in data.
• Record dialect: For every path, record id / segments {ell0, ell1, n_eff, …} / weight w_p and global arrival{form,gamma,measure}.

IV. Engineering Computation & Records (discretization, units, precision)

1. Discrete estimates:
   • Constant-factored: T_arr ≈ ( 1 / c_ref ) * Σ_i n_eff(i) · Δell_i.
   • General: T_arr ≈ Σ_i ( n_eff(i) / c_ref ) · Δell_i.
2. Units & precision: Report T_arr in seconds (s); keep ≥ 6 significant digits in reports; store as double-precision with unit metadata.
3. Selection rule: Prefer the constant-factored form when c_ref is a shared experimental constant across paths/devices; otherwise either form is acceptable, but must be written to delta_form.
4. QA gates: check_dim = pass for T_arr and its drivers; n_eff definition and traceability per M10-*.

V. Multi-Path & Composition (inside/outside the coherence window)

1. Main expressions: For {γ_p},
   • Inside the coherence window: phase(ω) ≈ ω · T_arr,p and coherent sum
     R^{coh} = | Σ_p w_p · r_p · e^{ i·ω·T_arr,p } |.
   • Outside the window: energy composition/suppression
     R^{inc} = ( Σ_p w_p · |r_p|^2 )^{1/2}.
2. Effective arrival time (diagnostic): T_arr,eff = Σ_p w_p T_arr,p (diagnostic only; does not replace per-path computation).
3. Constraints: w_p ≥ 0, Σ_p w_p ≤ 1; w_p is set by n_eff, ∇T_fil, BC (see P10-3 / S40-2).

VI. Metrology Chain & Uncertainty (M10- / M20-)**

• Sync & timebase: Record sync and Δt_sync, propagate into u(T_arr); alignment per I30-3.
• Uncertainty propagation (discrete):
  u^2(T_arr) ≈ Σ_i ( (∂T_arr/∂n_eff(i))^2 u^2(n_eff(i)) + (∂T_arr/∂Δell_i)^2 u^2(Δell_i) ) + u^2(Δt_sync) ,
  with ∂T_arr/∂n_eff(i) = Δell_i / c_ref for either dialect.
• Metrology gates: Publish u(T_arr) and u(n_eff) with sources/values/combination rules, and pass K–K/passivity consistency checks (verified in S50-*).

VII. Falsifiability Criteria (apparatus & operations)

• Path-switching test: With materials and netlist fixed, switch layout → gamma(ell) (e.g., guards/ground return) to vary T_arr,p; if arg Z(ω) does not translate linearly with ΔT_arr, reject the path model or the coherence-window assumption.
• Medium-perturbation test: Under small perturbations of n_eff (temperature/stress/bias), T_arr must renormalize smoothly; non-causal spikes or negative real parts reject kernel/weighting models (S40-* / S50-*).
• Dialect consistency: On the same data, both dialects must agree within metrological uncertainty; otherwise check delta_form, units, and path records.

VIII. Compliance Templates (copy-ready)

• Arrival record (report/data):
• arrival:
• form: "n_over_c" # or "one_over_c_times_n"
• gamma: "explicit" # path must be stored
• measure: "d_ell"
• c_ref: 299792458.0 # m/s (example)
• Tarr_s: 1.234e-09 # result in seconds
• u_Tarr_s: 6.0e-12 # combined uncertainty (s)
• check_dim: pass
• Discrete computation (pseudocode):
• Tarr = (1/c_ref) * sum_i( n_eff[i] * Δell[i] ) # n_over_c
• # or
• Tarr = sum_i( (n_eff[i]/c_ref) * Δell[i] ) # one_over_c_times_n
• Multi-path coherent composition (diagnostic):
• R_coh(ω) = abs( Σ_p w_p * r_p * exp( 1j * ω * T_arr_p ) )
• R_inc = sqrt( Σ_p w_p * abs(r_p)**2 )

IX. Correspondence & Degeneracy to Classical Framework

• With n_eff → constant and negligible path differences, T_arr reduces to a constant delay and arg Z(ω) ≈ ω·T_arr, matching the classical linear-phase approximation (S20-5).
• With a single path and w_p = 1, this chapter reduces to traditional single-path delay modeling; the distinctive additions are explicit gamma(ell) records and traceable metrology.

X. Cross-Chapter Pointers & Execution Guidance

• Upstream: S20-5 (first-order phase–delay link); I30-2/3 (binding & alignment).
• Parallel: S40-2 (path weights); S50-2/3 (phase term & nonlocal corrections).
• Downstream: In the design code (Chapter XVI), adopt minimize ∑ w_p ∫ n_eff d ell as the principle for phase linearization.

XI. Chapter Summary
This chapter unifies the two T_arr dialects and the engineering treatment of paths and measures—covering discrete computation, records, and metrological validation. Any published data/model must explicitly provide gamma(ell), d ell, and delta_form and pass check_dim and consistency gates, so that S20-* / S40-* / S50-* / I30-* derivations, mappings, and validations close under a reproducible, falsifiable semantics.