26-EFT.WP.STG.Lensing v1.0 | Chapter 10 — Physical Consistency and Conservation (Transmittance / Energy / Boundary)

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Chapter 10 — Physical Consistency and Conservation (Transmittance / Energy / Boundary)

One-sentence goal: Establish a unified, contractual framework for energy conservation, transmittance, and boundary conditions for lens operators so that learned or hand-crafted K remains consistent, passive, auditable, and persistable under both spectral and variational formulations.

I. Scope & Objects

1. Inputs
   • Graph & operators: G = (V, E, w); Laplacian or variants L_* ∈ { L, L^vis, L_ani }; optional mass matrix M (discrete measure).
   • Lens kernel: K = g(L_*) or a multi-layer composite K_eff (see Chapter 8).
   • Boundary information: B.type ∈ { Dirichlet, Neumann, Robin, Absorbing }; boundary subset ∂Ω ⊂ V.
   • Input signal: x_in (amplitude/feature, unit(x_in) = u_x), optional reference y.
2. Outputs
   • Physical quantities: T_trans, R, A, energy checks E_in / E_out, boundary flux Φ_boundary.
   • Reports: passivity/stability indices, dual-form gap delta_form_phys, contract assertions.
3. Constraints
   unit(T_trans) = 1; 0 ≤ T_trans ≤ 1. If the system is active, an explicit power-injection term must be provided.

II. Terms & Variables

1. Energy (discrete, weighted): E(x; M) = ( 1 / 2 ) * ( x^T M x ), with unit(E) = u_x^2 * unit(M).
2. Transmittance: T_trans = ( E( K x_in ; M ) / E( x_in ; M ) ).
3. Reflection / absorption: R, A ≥ 0, with T_trans + R + A = 1 for a passive closed domain.
4. Boundary flux (graph form): Φ_boundary = x^T B_n x, where B_n is a normal-difference or cut-set operator.
5. Passivity: K is non-amplifying, ||K||_2 ≤ 1 + ε; spectral form sup_λ | g(λ) | ≤ 1 + ε.
6. Boundary conditions:
   • Dirichlet: x|_{∂Ω} = c;
   • Neumann: ∂_n x|_{∂Ω} = 0 (on graphs: zero cut flow);
   • Robin: α x + β ∂_n x = γ.
7. Dual-form gap: delta_form_phys = || x_spec − x_var ||_2.

III. Postulates P710-*

• P710-1 (Parallel dual forms): Every physical check must produce both the spectral result x_spec = ( U g(Λ) U^T ) x_in and the variational result x_var = argmin_x J(x), and record delta_form_phys.
• P710-2 (Passivity & conservation): Default assumption is passive, hence T_trans ≤ 1. If T_trans > 1 + ε is detected, the artifact must declare an active term and the location of power injection.
• P710-3 (Explicit boundaries): All boundary conditions are persisted in structured B fields; any equivalent elimination (e.g., Schur) must retain a recovery map.
• P710-4 (Unit consistency): check_dim( E_out − E_in + Φ_boundary + A_diss − P_inj ) = "[E]".
• P710-5 (Spectral–energy alignment): Ensure orthogonality consistency between M and U. If U^T M U = I, Parseval enables direct energy audits.
• P710-6 (Name disambiguation): T_trans denotes transmittance only; do not confuse with T_fil (tension).

IV. Minimal Equations S710-*

• S710-1 (Spectral energy conservation check)
  E_out = ( 1 / 2 ) * || g(Λ) U^T x_in ||_2^2 when U is an M-orthogonal basis and U^T M U = I.
  T_trans = ( || g(Λ) \hat{x} ||_2^2 / || \hat{x} ||_2^2 ), where \hat{x} = U^T x_in.
• S710-2 (Variational equivalence)
  If g(λ) = 1 / ( 1 + μ λ^p ), then K ≡ ( I + μ L_*^p )^{−1} equals
  x_var = prox_{(μ/2)|| L_*^{p/2} • ||_2^2}( x_in ).
  In general, approximate K by a proximal cascade (see Chapters 8/9), and set delta_form_phys = || K x_in − x_var ||_2.
• S710-3 (Boundary flux and absorption)
  Discrete power balance: E_out − E_in = − A_diss + P_inj − Φ_boundary.
  For symmetric diffusion kernels K = g(L_*),
  A_diss = (1/2) * x_in^T ( M − K^T M K ) x_in.
• S710-4 (Transmittance of multi-layer compositions)
  Series: T_trans^series = ∏_{l} T_trans^{(l)} = ∏_{l} ( sup_λ | g_l(λ) |^2 ) (upper bound).
  Parallel: T_trans^parallel ≤ ( ∑_l w_l sup_λ | g_l(λ) | )^2.
  Residual: T_trans^res ≤ ( 1 + ∑_l β_l sup_λ | g_l(λ) | )^2.
• S710-5 (Consistency of absorption–reflection–transmittance)
  Closed passive domain with Neumann: A = 1 − T_trans.
  With boundary leakage: A = 1 − T_trans − R, with R = Φ_boundary / E_in.

V. Metrology Pipeline M71-10 (Ready → Apply → Audit → Fallback → Persist)

• Ready: choose L_* and M, normalize λ ∈ [0, λ_max]; declare B and RefCond = { λ_max, M.kind, B.type }.
• Apply: compute x_spec = ( U g(Λ) U^T ) x_in or use polynomial/rational matvec approximations; compute x_var in parallel.
• Audit: evaluate E_in / E_out / T_trans / Φ_boundary / A_diss, ||K||_2, ρ(K), delta_form_phys.
• Fallback: if T_trans > 1 + ε or ρ(K) > 1 + ε, perform order reduction / weight scaling / damping (see §IX). If boundaries mismatch, switch B or add absorbing layers.
• Persist:
  manifest.lens.phys.* = { L_*.hash, M.hash, g.hash / order, B, T_trans, A, R, rho, delta_form_phys, contracts.*, signature }.

VI. Contracts & Assertions C71-10x (suggested thresholds)

• C71-101 (Passive radius): ρ(K) ≤ 1.02, sup_λ | g(λ) | ≤ 1.02.
• C71-102 (Transmittance bounds): 0 ≤ T_trans ≤ 1.00 + ε (default ε = 0.01 for numerical slack).
• C71-103 (Energy-balance tolerance):
  | E_out − ( E_in − A_diss − Φ_boundary + P_inj ) | ≤ tol_E, with tol_E = 1e−6 * E_in.
• C71-104 (Dual-form consistency): delta_form_phys ≤ 1e−3 || x_in ||_2 (quadratic priors); relax to 3e−3 with non-smooth priors.
• C71-105 (Boundary consistency):
  Neumann: | Φ_boundary | ≤ 1e−6 * E_in;
  Dirichlet: boundary residual ≤ 1e−6 || x_in ||_2.
• C71-106 (Unit checks): check_dim( T_trans ) = "[1]", check_dim( E(•; M) ) = "[u_x^2]".

VII. Implementation Bindings I71-10* (interfaces, I/O, invariants)

• measure_transmittance( L_*, M, g, x_in ) -> { T_trans, E_in, E_out }
• measure_boundary_flux( G, B, x ) -> Φ_boundary
• enforce_passivity( g, policy ) -> g_tilde, report (policy ∈ { coef_clamp, rational_shrink, spectral_proj })
• apply_boundary_condition( G, B, x ) -> x_bc, meta
• energy_balance_report( L_*, M, g, B, x_in[, P_inj] ) -> { A_diss, R, balance_error }
• eval_delta_form_phys( L_*, g, prox_stack, x_in ) -> delta_form_phys
• assert_physical_contracts( ds, rules ) -> report
• emit_lens_manifest( results, policy ) -> manifest.lens
  Invariants: λ_max > 0; M symmetric positive definite; B traceable; active scenarios must provide P_inj.location / hash.

VIII. Cross-References

• Kernels & spectral realizations: Chapter 5; multi-layer composition: Chapter 8; learning & invariance: Chapter 9.
• Graph operators & stability: EFT.WP.STG.Dynamics v1.0, Chapters 4 and 9.
• Runtime publication & dashboards: EFT.WP.STG.Dynamics v1.0, Chapter 14; manifests & contracts: appendices of this volume.

IX. Quality & Risk Control

1. SLI/SLO: rho_p95, T_trans_p50/p95, balance_error_p99, delta_form_phys_p99, latency_p95, fail_rate_boundary.
2. Fallbacks:
   • ρ(K) out of bounds → coef_clamp or spectral_proj: g ← clip(g, [-1, 1]) or scale K ← K / α;
   • T_trans > 1 + ε → add damping g_damp(λ) = g(λ) / ( 1 + τ λ ) or lower approximation order;
   • Boundary leakage → switch Neumann→Robin and add absorbing layer (increase β on ∂Ω);
   • Excessive dual-form gap → raise spectral approximation order or add proximal layers.
3. Audit: store sampled g(λ), energy/flux balance tables, boundary residuals, fallback triggers and impact scope.

Summary

• Provides transmittance/energy/boundary consistency definitions and checks for K = g(L_*).
• Builds energy balance and passivity criteria under both spectral and variational forms, and unifies transmittance upper bounds for multi-layer compositions.
• With interfaces I71-10* and contracts C71-10x, enables auditable, reversible, and persistable physically consistent deployments.