47-PTN Template v1.0 | Chapter 3 — Control Equations & Core Assumptions

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Chapter 3 — Control Equations & Core Assumptions

I. Control Equations (Sxx-?)

• S20-1 | Arrival time (explicit path & measure)
  T_arr = ( ∫ ( n_eff / c_ref ) d ell ) = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  Requirements: record delta_form in metadata; declare gamma(ell) and d ell explicitly; n_eff is dimensionless, c_ref in m/s.
• S21-2 | Phase accumulation (scalar coherence window)
  Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )
  Constraint: λ_ref matches the observation band; state the coherence window on the results page.
• S30-1 | Transport & continuity (conservative form)
  ∂_t ρ + ∇·( ρ v ) = 0
  Constraint: ρ denotes density or occupancy; if using n, state units and distinguish from n_eff.
• S31-1 | Paraxial propagation with index gradient (geometric-optics limit)
  ∂_z A + (1/2k_ref) ∇_⊥^2 A + i k_ref ( n_eff - 1 ) A = 0
  Conditions: paraxial, small-angle, slowly varying medium; k_ref = 2π/λ_ref.
• S40-1 | Observation noise model (wideband, additive, weakly correlated)
  y = H[x] + η, with η ~ 𝒩(0, Σ) or a specified robust distribution; H[·] is the system operator and must be bound to an implementation Ixx-*.
• S50-1 | Metrology mapping & dimensional closure
  u_meas = G( u_true ; θ ) with dim( u_meas ) = dim( u_true ); calibration of θ aligns with Mx-*.

Items 1–6 constitute the minimal control-equation set; domain-specific Sxx-* may be added, provided they remain consistent with this chapter’s assumptions and metrology closure.

II. Core Assumptions (Pxx-?)

• P10-1 | Measurable paths & slowly varying medium — gamma(ell) is piecewise smooth; n_eff varies slowly within the coherence window; measure d ell is consistent with the path.
• P10-2 | Local Lorentz invariance (zero-order) — within the local experimental/observational domain, use c_ref as the propagation limit and timing yardstick.
• P11-1 | Zero-order independence + first-order weak coupling — principal quantities are modeled independently at zero order; first-order background drift is allowed without breaking dimensional closure or conservation.
• P12-3 | Linear additivity (within window) — within the calibrated coherence window and dynamic range, the system operator H[·] is linear or piecewise linear.
• P13-2 | Statistical regularity — the second moment of noise η exists; when using robust losses, provide an equivalent second-order surrogate for error propagation.
• P14-1 | Metrological consistency — all expressions pass check_dim; units, dimensions, and symbols follow the unified conventions (inline backticks; parentheses for any division/integral/composite operator).

III. Asymptotics & Approximations

• Slow-variation (WKB/eikonal): if |∇ n_eff| · L_coh ≪ 1, use the geometric-optics limit in S31-1; outside the threshold, revert to full-wave or numerical methods.
• Paraxial/small-angle expansion: |θ| ≪ 1, |∇_⊥ A| ≪ |∂_z A|; provide an error bound O(θ^2).
• Thin-screen approximation: if medium perturbations are localized, use a phase screen exp( i k_ref ΔOPL ); free-space between screens uses the Fresnel kernel.
• First-order coupling expansion: for slowly drifting background b(·), treat as O(ε): n_eff = n_0 + ε n_1, and state the working range of ε.
• Discretization & numerical stability: approximate integrals by segmented sums ∑ w_i f_i; state Δell and window function; compare with analytic forms and give convergence criteria.
• Log-domain computation: for multiplicative/convolutional forms, provide log-sum-exp equivalents to avoid under/overflow while preserving dimensions and baselines.

IV. Stability, Well-Posedness & Conservation

• Well-posedness: S20-1 and S21-2 are well-posed under n_eff ∈ L^∞ and finite path length; S30-1 conserves mass with bounded flow v and appropriate boundary conditions.
• L² stability (propagation): if ‖n_eff - 1‖_∞ ≤ ε with sufficiently small ε and bounded ∇_⊥, solutions of S31-1 exhibit non-increasing or controlled growth in energy norms w.r.t. initial perturbations.
• Sensitivity bound: first-order sensitivity of arrival time to medium perturbations
  δT_arr = ( 1 / c_ref ) * ( ∫ δ n_eff d ell ), usable as a lower-bound estimate for noise propagation and power analysis.
• Error propagation: for y = H[x] + η, if H is linear, Cov[x̂] = (Hᵀ Σ⁻¹ H)⁻¹; for nonlinear H, use first-order linearization or bootstrap and report confidence intervals.
• Invariants & conserved quantities: with source-free, lossless, stationary boundaries, S30-1 preserves total quantity; paraxial propagation conserves cross-sectional flux up to O(θ^2).
• Metrology closed loop: provide a check_dim report and unit table with results; exports include references[] and version to support reproducibility.

Appendix: Control-Equation Registry (drop-in)

version: "1.0.0"

equations:

- code: S20-1

name: arrival_time

form: "T_arr = ( ∫ ( n_eff / c_ref ) d ell )"

requires:

path: "gamma(ell)"

measure: "d ell"

units:

T_arr: "s"

c_ref: "m/s"

see:

- "EFT.WP.Core.Equations v1.1:S20-1"

- "EFT.WP.Core.Metrology v1.0:check_dim"

- "EFT.WP.Core.DataSpec v1.0:TARR"

- code: S21-2

name: phase_accumulation

form: "Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )"

units:

Phi: "rad"

lambda_ref: "m"

- code: S30-1

name: continuity

form: "∂_t ρ + ∇·( ρ v ) = 0"

units:

rho: "<per context>"

v: "m/s"

- code: S31-1

name: paraxial_propagation

form: "∂_z A + (1/2k_ref) ∇_⊥^2 A + i k_ref ( n_eff - 1 ) A = 0"

units:

k_ref: "1/m"

A: "<field>"

- code: S40-1

name: noise_model

form: "y = H[x] + η"

noise: "Gaussian_or_Robust"

outputs: ["Σ", "residuals"]

- code: S50-1

name: metrology_mapping

form: "u_meas = G( u_true ; θ )"

constraint: "dim(u_meas)=dim(u_true)"

interfaces:

bind_to:

- "Methods.SimStack v1.0"

- "Methods.Repro v1.0"

exports:

must_include: ["references", "version", "check_dim_report"]

Implementation Bindings (directive)

• Bind I20-compute_arrival_time(path, n_eff, c_ref) -> T_arr (record delta_form).
• Bind I31-propagate_paraxial(A0, n_eff, k_ref, grid) -> A (support thin-screen/free-space composition).
• Bind I40-fit_noise(y, H, model) -> {Σ, residuals} (robust/Gaussian options).
• Bind I50-calibrate(u_true, θ) -> u_meas and connect with Mx-* metrology workflows.