26-EFT.WP.STG.Lensing v1.0 | Chapter 14 — Use Cases & Reference Implementations

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Chapter 14 — Use Cases & Reference Implementations

One-sentence goal: Provide deployable reference implementations and end-to-end use cases of lenses K = g(L_*) across offline and streaming modes, with contract- and manifest-driven release.

I. Scope & Objects

1. Inputs
   • Graph & measure: G = (V, E, w), L_* ∈ { L, L^vis, L_ani }, M, boundary B.
   • Lens specification: kernel family g(•; θ), polynomial order order, estimator for λ_max, and the hierarchical composition K_eff = compose({ K_i }).
   • Data modalities: batch x_batch and streaming x(t), optional reference y.
   • Runtime conditions: RefCond (version / device / resources / cache policy).
2. Outputs
   • Results & metrics: x_out = K_eff x_in, Q_*, delta_form, T_trans, ρ(K_eff), runtime logs and resource usage.
   • Persistence: manifest.lens.* (specs, parameters, thresholds, metrics, signatures).
3. Applicability
   • Passive by default (T_trans ≤ 1 + ε); active cases require allow-listing and explicit P_inj.
   • Dual forms (spectral / variational) must run in parallel.

II. Terms & Variables

• Use case IDs: U14-A / B / C / ....
• Lens layers: K_i = g_i(L_*; θ_i); composite lens K_eff = K_L ∘ ... ∘ K_2 ∘ K_1.
• Key metrics: Q_fidelity, Q_consistency (delta_form), Q_energy (T_trans), Q_latency, ρ(K_eff).
• Runtime fields: window, stride, cache.ttl, passthrough_on_fail ∈ { true, false }.

III. Postulates P714-*

• P714-1 (Parallel dual forms): Every published use case computes x_spec and x_var in parallel and records delta_form.
• P714-2 (Passivity & conservation): Enforce T_trans ≤ 1 + ε and ρ(K_eff) ≤ 1 + ε by default (ε scenario-configurable).
• P714-3 (Traceability): Persist RefCond / method / seed and kernel coefficients θ, order / λ_max.
• P714-4 (Units / dimensions): Run check_dim before publishing.
• P714-5 (Rollback loop): On any critical assertion failure, trigger the fallback chain (lower-order → damping → bypass → rollback).

IV. Minimal Equations S714-*

• S714-1 (Stacked lens output)
  x_out = K_eff x_in = ( K_L ∘ ... ∘ K_1 ) x_in.
  Spectral form: if K_i ≈ U g_i(Λ; θ_i) U^T, then
  x_spec = U ( ∏_i g_i(Λ) ) U^T x_in.
  Variational form: x_var = prox_{ϕ_L} ∘ ... ∘ prox_{ϕ_1}( x_in ).
• S714-2 (Dual-form gap)
  delta_form = || x_spec − x_var ||_2, with go-live assertions in C71-14x.
• S714-3 (Transmittance & spectral radius)
  T_trans = E( x_out ; M ) / E( x_in ; M ), and ρ(K_eff) ≤ ∏_i sup_λ | g_i(λ) |.
• S714-4 (Streaming window stitching)
  For window W, hop H:
  x_out[t : t + W) = K_eff( x[t : t + W) ),
  stitched with Hann overlap w_hann and ∑ w = 1.

V. Metrology Pipeline M71-14 (Ready → Implement → Validate → Assert → Release)

• Ready: freeze G / L_* / M / B / RefCond, and dataset splits (train / validation / online samples); set contractual thresholds and fallback strategy.
• Implement: build K_eff per use-case spec (kernel family, order, λ_max, multi-layer / gated / residual); produce both spectral and variational implementations.
• Validate: execute offline batches, log Q_* / delta_form / T_trans / ρ / resources; replay streaming to verify windowing and cache consistency.
• Assert: enforce C71-14x; on failure, trigger fallback (lower-order → damping → bypass).
• Release: emit manifest.lens and dashboard cards; enable canary and rollback switches.

VI. Contracts & Assertions C71-14x (unified thresholds; scenario overrides allowed)

• C71-1401: delta_form_p99 ≤ 1e−3 * || x_in ||_2 (up to 3e−3 for non-smooth prox stacks).
• C71-1402: T_trans_p95 ≤ 1.01 (passive); active requires allow-list and E_balance_p95 ≤ 1e−6.
• C71-1403: ρ(K_eff) ≤ 1.02.
• C71-1404: lat_p95 ≤ 0.5 W (streaming); cpu_p95 ≤ 80%, mem_p95 ≤ cap − 10%.
• C71-1405: check_dim must pass; completeness of RefCond / method / seed = 100%.

VII. Implementation Bindings I71-14* (interfaces, I/O, invariants)

• build_lens_from_spec( spec, L_*, RefCond ) -> { K_eff, meta }
• run_lens_batch( K_eff, x_batch, mode ∈ { spec, var } ) -> { y, metrics }
• run_lens_stream( K_eff, x_stream, window, stride, cache ) -> { y_stream, metrics }
• validate_dual_forms( K_eff, x ) -> { x_spec, x_var, delta_form }
• assert_and_plan_fallback( metrics, rules ) -> { pass, actions }
• emit_lens_manifest( results, policy ) -> manifest.lens
  Invariants: L_* symmetry / (semi-)positive-definite checks; order ≥ 1; λ_max > 0; dual forms in parallel; logs reproducible.

VIII. Cross-References

• Kernels & spectral realization: Chapter 5; features & saliency: Chapter 6; inversion: Chapter 7; multi-layer & gating: Chapter 8; learned kernels: Chapter 9.
• Physical consistency: Chapter 10; runtime: Chapter 11; contracts & metrics: Chapter 12; uncertainty & guardbands: Chapter 13.
• Graph dynamics / integration stability: EFT.WP.STG.Dynamics v1.0, Chapters 9 & 14.

IX. Quality & Risk Control

• Panel: delta_form_p99, T_trans_p95, ρ_p95, lat_p95, cpu / mem / gpu, error_rate, fallback_rate.
• Fallback sequence: lower_order → damping → gate_off(residual) → passthrough → rollback.
• Audit: version signatures, parameter diffs, blast-radius assessment, canary / rollback logs, quarantined outliers.

Use-Case Suite (Reference Implementations)

U14-A: Occlusion Compensation & Sharpening for Metropolitan Camera Arrays (LOS/NLOS)

1. Scenario: Multi-view frames are embedded on G; occlusions and shadows create NLOS artifacts.
2. Implementation
   • Build L^vis (view & occlusion constraints; see Chapter 4).
   • Multi-layer lens K_eff = K_focus ∘ K_ani ∘ K_deblur:
     1. K_focus: g(λ) = exp(−β λ) (diffusion-style focusing);
     2. K_ani: anisotropic kernel suppressing cross-edge diffusion along tangent directions;
     3. K_deblur: rational deconvolution with gated residuals (see Chapter 8).
   • Dual forms: spectral (Chebyshev / rational factorization) and variational (TV + data-fidelity proximal).
   • Contracts: delta_form_p99 ≤ 1e−3 || x ||_2; T_trans_p95 ≤ 1.01; ρ ≤ 1.02.
   • Release: manifest.lens.case = "U14-A", including θ / order / λ_max / RefCond / Q_*.

U14-B: Multipath Suppression & Salient Trajectory Enhancement for mmWave / UWB Localization

1. Scenario: Spatio-temporal graph signal x(t) encodes ranges / AoA intensities; highlight LOS while suppressing mirror paths.
2. Implementation
   • Build L_ani: geometry-weighted (Ch. 3/4), increasing damping along likely reflection directions.
   • Lens K_eff = K_band ∘ K_gate ∘ K_sparse:
     1. K_band: spectral band-selection preserving LOS-matched bands;
     2. K_gate: visibility / geometric gating (Ch. 4);
     3. K_sparse: variational ℓ1 sparsity to isolate unique paths.
   • Streaming: window = W, stride = H, cache.ttl = 2W.
   • Metrics: Q_fidelity vs. ground-truth trajectory, false-positive delta, lat_p95.
   • Contracts: as in C71-14x, plus false-positive rate ≤ 2%.

U14-C: Salient Anomaly Highlighting in Irregular Sensor Networks (Industrial / Power Grid)

1. Scenario: Node time series exhibit jumps / leaks; need salient anomaly highlighting on the graph.
2. Implementation
   • Construct L from physical / pipeline topology and flow weights.
   • Lens K_eff = K_hp ∘ K_resid:
     1. K_hp: high-pass lens g(λ) = λ / ( λ + α ) to accentuate non-smooth components;
     2. K_resid: residual gating that amplifies only on subgraphs with high Expl_gain.
   • Metrics: alert recall / false alarms, Expl_gain, energy conservation.
   • Contracts: Expl_gain ≥ 0.6 (Ch. 12), others per C71-14x.

U14-D: Deconvolution with Energy Conservation for Sparse-Array Imaging (Astronomy / Ultrasound)

1. Scenario: Observations are sampled by sparse arrays with spatially varying PSF.
2. Implementation
   • Build L_* and measure M on the sampling graph.
   • Rational kernel g(λ) ≈ ∑ α_i / ( 1 + β_i λ ) with domain-wise adaptive parameters.
   • Variational form uses nonnegativity + TV proximals.
   • Conservation: strict T_trans_p95 ≤ 1.005.
   • Uncertainty: propagate via MC (Ch. 13) to report U_ρ and U(T_trans).

Summary

• Delivers four representative use cases and reusable implementation skeletons spanning spectral/variational dual forms and offline/streaming modes.
• Ensures measurability and rollback through M71-14 pipeline and C71-14x contracts.
• Realizes auditable, reproducible, and operable lens deployments via I71-14* interfaces and manifest.lens.