26-EFT.WP.STG.Lensing v1.0 | Chapter 1 — Lensing Domain: Definition & Scope

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Chapter 1 — Lensing Domain: Definition & Scope

One-sentence goal: Delimit the objects, I/O, and engineering boundaries of Graph Lensing (STG Lensing), and establish a unified, composable, invertible, and measurable convention together with the published artifact manifest.lens.

I. Scope & Objects

1. Inputs
   • Graph & view: G = (V, E), L, A, H, Ω_view, π_view: V → Ω_view, vis(v, Ω).
   • Signal & modes: x ∈ R^{|V|}, mode ∈ { offline, streaming }, window parameters Δt_win / Δt_slide.
   • Lens & parameters: operator/kernel Φ_lens, K_lens = g(L; θ), composition algebra ⊕ / ∘, regularization & constraints reg / constraints.
   • Reference & environment: RefCond (sampling rate, timezone, hardware version, numerical precision).
2. Outputs
   • Lensed result & inversion: x' = Φ_lens(x), and x_est solving y = H Φ_lens x + v.
   • Quality & uncertainty: metrics.*, u(x'), u_c, U = k * u_c, guardband g.
   • Dual-form gap: delta_form_lens (exact spectral vs. polynomial approximation / offline vs. streaming).
   • Publication: manifest.lens (see Summary).
3. Applicable scenarios
   Focusing / de-lensing / visibility orchestration; multi-layer lens composition and runtime execution; STG tasks across transport/power grids/social/IoT, etc.
4. Out of scope
   • Physical medium propagation delay and EM/optical device specifics (see EFT.WP.Metrology.PathCorrection v1.0).
   • Rendering and vision-pipeline details outside graph structures.

II. Terms & Variables

• Graph & view: G, L, A, H, Ω_view, π_view, vis.
• Lens & kernels: Φ_lens, K_lens, K_focus, K_defocus, K_aniso(∇_G), θ (kernel parameters).
• Signal & noise: x, y = H x, v (noise).
• Runtime: wm(t) (watermark), latency_p95, coverage.
• Dimensions & units: declare unit(x) and dim(x) (often dimensionless or engineering units for amplitude/power). The check check_dim(y - H x') must pass.
• Conflict names (mandatory): T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index).

III. Postulates P711-*

• P711-1 (Composability): Lenses form a bounded family of linear/nonlinear operators supporting ∘ / ⊕ / residual composition, with decidable spectral bounds after composition.
• P711-2 (Explicit view domain): Any use of visibility/occlusion must explicitly define the domains and measures of π_view and vis.
• P711-3 (Dual forms in parallel): For the same input, compute both exact spectral and polynomial approximations (or offline and streaming) in parallel; record delta_form_lens with thresholds.
• P711-4 (Invertibility annotation): If invertibility is claimed, publish reg, condition-number or spectral-bound evidence concurrently.
• P711-5 (Traceability): graph.hash / L.hash / Φ.hash / θ / version must reproduce the result; RefCond is mandatory.

IV. Minimal Equations S711-*

• S711-1 (Lens action): x' = Φ_lens(x), where Φ_lens admits a spectral-function approximation via the kernel K_lens = g(L; θ):
  x' ≈ ( ∑_{k=0}^m c_k T_k(ĤL) ) x, with ĤL the scaled Laplacian.
• S711-2 (Observation & de-lensing): y = H x' + v = H Φ_lens x + v; de-lensing solves
  x_est = argmin_x ( || H Φ_lens x - y ||_2^2 + reg(x) ).
• S711-3 (Dual-form gap): delta_form_lens = || Φ_exact(x) - Φ_approx(x) ||_2; the streaming counterpart delta_form_stream is defined by a window-weighted norm.
• S711-4 (Stability bound): ρ(K_lens) ≤ 1 + ε or || K_lens ||_2 ≤ 1 + ε serves as the runtime contractual upper bound.
• S711-5 (Uncertainty composition): u_c^2(x') = J_x V_x J_x^T + u_num^2 + u_model^2, with J_x the Jacobian of the lens w.r.t. the input.

V. Metrology Pipeline M71-1 (Domain Ready → Modeling → Verification → Persist)

• Ready: load G / L / H and RefCond; validate check_dim(y - H x) and unit consistency.
• Model: select Φ_lens type and θ; if occlusions apply, build π_view / vis; generate both spectral-exact and approximate implementations.
• Estimate: apply x' = Φ_lens(x); if needed, solve de-lensing x_est and evaluate residual r = y - H x_est.
• Verify: compute delta_form_lens, ρ(K_lens), inv_residual, coverage, latency_p95; propagate uncertainty u_c and provide guardband U.
• Persist: produce manifest.lens, a contract report, and signature; update dashboards and audit logs.

VI. Contracts & Assertions C71-* (suggested thresholds)

• C71-11: delta_form_lens ≤ tol_lens (default tol_lens = 1e−3 * ||x||_2).
• C71-12: ρ(K_lens) ≤ 1 + ε (recommend ε ≤ 0.02), or || K_lens ||_2 ≤ 1 + ε.
• C71-13: inversion residual || y − H x_est ||_2 ≤ r_cap (set by noise variance).
• C71-14: coverage ≥ 1 − α (window coverage; recommend α ≤ 0.01).
• C71-15: latency_p95 ≤ B_lens (per SLO).
• C71-16: check_dim(y − H x') = "[same]"; reject publication on unit mismatch.

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

1. build_lens(graph, type, params) -> Φ_lens
   • in: G / L, type ∈ { focus, defocus, aniso, visibility, learned }, θ / approx_order.
   • inv: returns an operator with declaration or evidence handle for ρ(K_lens) ≤ 1 + ε.
2. apply_lens(Φ_lens, x, mode, win) -> x'
   • in: mode ∈ { offline, streaming }, win = { Δt_win, Δt_slide }.
   • out: x' plus a snapshot of runtime metrics.
3. invert_lens(y, Φ_lens, H, reg) -> x_est, report
   out: x_est, residuals, and condition-number estimates.
4. compose_lenses([Φ_i], algebra) -> Φ_*
   inv: evaluate and persist ρ(K_*) after composition.
5. solve_visibility(graph, Ω_view, obstacles) -> vis_map
6. assert_lens_contracts(ds, rules) -> report
7. emit_lens_manifest(results, policy) -> manifest.lens
   Invariants: non_decreasing(tau_mono); Δt_win > 0; hash(*) is traceable; delta_form_lens ≤ tol_lens.

VIII. Cross-References

• Graph kernels & spectral realizations: EFT.WP.STG.Dynamics v1.0, Chapters 4 and 9 (stability/rigidity/approximation order).
• Runtime windows & dashboards: …STG.Dynamics v1.0, Chapter 14.
• LOS/NLOS & occlusion semantics: EFT.WP.Metrology.PathCorrection v1.0, Chapters 3 and 7.
• Subgraphs / worldline fragments: EFT.WP.Particle.TopologyAtlas v1.0, Chapters 5 and 7.

IX. Quality & Risk Control

• SLO/SLI: track focus_gain, stability_margin, inv_residual, latency_p95, coverage.
• Drift & fallback: monitor spectral_drift and delta_form_lens_p99; if violated, raise approximation order / switch to spectral-exact / increase regularization / tighten windows.
• Audit: record approximation-error proofs, view-domain change logs, contract pass rates, and anomaly alerts.

Summary

1. This chapter defines the objects, I/O, and boundaries of the STG lensing domain and establishes the minimal closed loop P711 / S711 / M71 / I71 / C71.
2. Minimal keys for manifest.lens:
   • TraceID, graph.hash, L.hash, H.hash, Φ.hash, θ, impl = { exact, approx, mode }
   • RefCond, win = { Δt_win, Δt_slide }
   • metrics = { focus_gain, stability_margin, inv_residual, latency_p95, coverage }
   • delta_form_lens, tol_lens, u / U, contracts.*, signature