26-EFT.WP.STG.Lensing v1.0 | Chapter 2 — Mathematical Baselines (View Mapping / Visibility / Graph Variational)

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Chapter 2 — Mathematical Baselines (View Mapping / Visibility / Graph Variational)

One-sentence goal: Provide a unified mathematical convention for the view mapping π_view, visibility vis, and graph variational energies; establish the equivalence (and measurable gap) between the spectral and variational forms of the lens operator Φ_lens, together with stability bounds.

I. Scope & Objects

1. Inputs
   • Graphs & operators: G = (V, E, w), A, D, L (make type ∈ { unnormalized, normalized } explicit), ∇_G, div_G.
   • View & occlusion: Ω_view, π_view: V → Ω_view, occlusion set O ⊆ Ω_view, visibility vis: V × Ω_view → [0, 1].
   • Data & observation: x_in ∈ R^{|V|}, y = H x_true + v, with H possibly sparse/subsampled.
   • Trade-offs & references: λ, β, τ ≥ 0, RefCond (sampling rate, precision, Laplacian variant, time scale).
2. Outputs
   • Dual-form solutions: variational-lens solution x' and spectral-lens output K_lens x_in, reported in parallel with the dual-form gap delta_form_lens.
   • Visibility-weighted energies, stability/spectral bounds, and unit/dimension checks.
3. Boundaries & constraints
   • Work on connected graphs (or per connected component); vis may be binary or probabilistic; measures for π_view and vis are explicit.
   • Excludes physical optics hardware modeling and EM propagation (see EFT.WP.Metrology.PathCorrection v1.0).

II. Terms & Variables

• Graph & spectrum: A (adjacency), D (degree), L = D − A or L = I − D^{−1/2} A D^{−1/2}, spectral factorization U Λ U^T.
• Discrete calculus pair: ∇_G: R^{|V|} → R^{|E|}, div_G = − ∇_G^T (inner product ⟨x, y⟩ = x^T y).
• View & visibility: π_view(v) ∈ Ω_view, vis(v, Ω) ∈ [0, 1], W_vis(Ω) = diag(vis(v, Ω)).
• Variational energies:
  E_dir(x) = (1/2) x^T L x,
  TV_G(x) = ( ∑_{(i,j)∈E} w_{ij} | x_i − x_j | ),
  TV_iso(x) = ( ∑_{i} || (∇_G x)_i ||_2 ).
• Lens & kernels: Φ_lens, K_lens = g(L; θ), K_focus = (I + β L)^{−1}, K_diff(τ) = exp(−τ L).
• Dimensions: enforce unit(x_in) ≡ unit(y); unit(L) = "[1]" (dimensionless adjacency weights) or as implied by weighting; check_dim(y − H x') must pass.
• Time semantics: compute at tau_mono, publish at ts; record offset/skew/J (see …STG.Dynamics v1.0 and …TimeBase).

III. Postulates P712-*

• P712-1 (Measured view domain): π_view is measurable, with a clear discrete–continuous mapping between ( ∑_{v∈V} • ) and ( ∫_{Ω_view} • dμ ); optionally normalize with μ(Ω_view) = 1.
• P712-2 (Visibility consistency): vis is monotone under occlusion: if Ω1 ⊆ Ω2 then vis(v, Ω1) ≤ vis(v, Ω2); probabilistic vis must be a well-defined marginal.
• P712-3 (Operator duality): div_G = − ∇_G^T holds; under unweighted inner products E_dir(x) = (1/2) || ∇_G x ||_2^2; any weighting choice must be persisted.
• P712-4 (Traceable spectra): Laplacian type, scaling, and normalization are hashed in RefCond to avoid cross-implementation drift.
• P712-5 (Dual forms in parallel): compute both the spectral form (K_lens) and the variational form (x' = argmin E(x)) and record delta_form_lens.
• P712-6 (Convexity & stability): if invertibility/stability is claimed, publish smoothness/strong-convexity parameters of the energy or equivalent spectral evidence with results.

IV. Minimal Equations S712-*

1. S712-1 (View-weighted Dirichlet):
   E_dir^{vis}(x; Ω) = (1/2) x^T W_vis(Ω) L W_vis(Ω) x;
   if edge masking is used, update weights w_{ij} ← w_{ij} * m_{ij}(Ω).
2. S712-2 (Graph TV):
   • Anisotropic: TV_G(x) = ( ∑_{(i,j)∈E} w_{ij} | x_i − x_j | ).
   • Isotropic: TV_iso(x) = ( ∑_{i} || (∇_G x)_i ||_2 ).
3. S712-3 (ROF on Graphs):
   x' = argmin_x ( (1/2) || H x − y ||_2^2 + λ TV_G(x) + β E_dir^{vis}(x; Ω) ).
4. S712-4 (Euler–Lagrange / first-order condition):
   0 ∈ H^T ( H x' − y ) + β L^{vis} x' + λ div_G( ψ( ∇_G x' ) ),
   with ψ(z) = z / max(ε, |z|) (anisotropic) or ψ(z) = z / max(ε, ||z||_2) (isotropic).
5. S712-5 (Spectral equivalence: quadratic case): when λ = 0,
   x' = (I + β L^{vis})^{−1} x_in = K_focus x_in.
6. S712-6 (Diffusion lens):
   x' = exp(−τ L^{vis}) x_in = K_diff(τ) x_in, and ρ(K_diff) ≤ 1.
7. S712-7 (Approximation & dual-form gap): with Chebyshev approximation g(L) ≈ ∑_{k=0}^m c_k T_k(ĤL),
   delta_form_lens = || K_exact x_in − K_cheb x_in ||_2;
   the streaming-window version delta_form_stream = ( ∑_{t∈win} w_t || • ||_2^2 )^{1/2}.
8. S712-8 (Stability bounds): in quadratic energies || x' ||_2 ≤ || x_in ||_2; for diffusion kernels || K_diff ||_2 = 1, and K_focus satisfies || K_focus ||_2 ≤ 1.

V. Metrology Pipeline M71-2 (Baseline → Solve → Verify → Persist)

1. Baseline setup: choose the Laplacian variant and scaling; construct π_view / vis and/or edge masks m_{ij}(Ω); finalize RefCond and units.
2. Energy assembly: instantiate E_dir^{vis}, TV_*, and the observation term; set priors/search ranges for λ, β, τ.
3. Dual-form solving:
   • Variational: solve S712-3 by primal–dual / ADMM to obtain x'_var;
   • Spectral: build K_lens (exact or Chebyshev) to produce x'_spec.
4. Checks & stability: evaluate delta_form_lens, ρ(K_lens), || H x' − y ||_2, monotonic energy descent E(x'_k); propagate u_c.
5. Persist: emit Φ.hash / L.hash / π_view.hash / vis.hash, impl = { exact, approx }, delta_form_lens, u / U, contracts.* into manifest.lens.

VI. Contracts & Assertions C71-2x (suggested thresholds)

• C71-21: adjoint_check = || div_G + ∇_G^T ||_2 ≤ 1e−10.
• C71-22: ρ(K_lens) ≤ 1 + ε (recommend ε ≤ 0.02), or || K_lens ||_2 ≤ 1 + ε.
• C71-23: delta_form_lens ≤ tol_lens (default tol_lens = 1e−3 * || x_in ||_2).
• C71-24: energy descent: E(x'_{k+1}) ≤ E(x'_k); otherwise force rollback or increase β / τ.
• C71-25: observation consistency: || y − H x' ||_2 ≤ r_cap (calibrated by var(v)).
• C71-26: check_dim(y − H x') = "[same]"; block publication on unit mismatch.

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

• make_laplacian(graph, variant, scale) -> L, meta
  inv: meta.hash locks type & scaling; spectral radius measured.
• build_view_map(G, Ω_view, method) -> π_view, W_vis
  out: W_vis(Ω) or edge mask m_{ij}(Ω); includes RefCond and hashes.
• solve_graph_rof(H, y, λ, β, W_vis, opts) -> x'_var, report
  report: energy curve, KKT residuals, stopping criteria.
• cheb_filter(L, coeffs, x_in) -> x'_spec, approx_report
  report: approx_order, approximation error bounds.
• eval_delta_form_lens(x'_var, x'_spec) -> delta_form_lens
• assert_lens_contracts(ds, rules) -> report
• emit_lens_manifest(results, policy) -> manifest.lens
  Invariants: Δt_win > 0; hash(*) is traceable; div_G = − ∇_G^T; delta_form_lens ≤ tol_lens.

VIII. Cross-References

• Graph operators/spectra & stability: EFT.WP.STG.Dynamics v1.0, Chapter 4 (kernels & diffusion) and Chapter 9 (numerical stability).
• Runtime windows / lateness & dashboards: …STG.Dynamics v1.0, Chapter 14.
• Occlusion and LOS/NLOS semantics: EFT.WP.Metrology.PathCorrection v1.0, Chapters 3 and 7.
• Subgraphs & event fragments: EFT.WP.Particle.TopologyAtlas v1.0, Chapters 7 and 8 (complexes & stability).

IX. Quality & Risk Control

• SLO/SLI: track stability_margin = 1 − max( ρ(K_lens) − 1, 0 ), inv_residual, delta_form_lens_p99, energy_drop_rate.
• Fallbacks: if δ_form breaches threshold → raise Chebyshev order or switch to exact spectra; if energy fails to decrease → increase β or strengthen convexity; if spectral bounds are exceeded → rescale L / reduce τ.
• Auditables: L.hash / π_view.hash / vis.hash, approximation-error evidence, energy-convergence curves, contract pass rates.

Summary

1. This chapter formalizes measurable definitions for π_view / vis, graph variational energies, and the equivalence/contrast between spectral and variational lens formulations—with a quantifiable gap.
2. Key manifest.lens.* fields:
   • L.hash, π_view.hash, vis.hash, Φ.hash, impl = { exact, approx }
   • params = { λ, β, τ, approx_order }, metrics = { ρ(K), inv_residual, energy_drop }
   • delta_form_lens, tol_lens, u / U, contracts.*, signature