26-EFT.WP.STG.Lensing v1.0 | Chapter 3 — Lens Primitives & Operator Families (Node / Edge / Path / Subgraph)

Home ／ Docs-Technical WhitePaper ／ 26-EFT.WP.STG.Lensing v1.0

Chapter 3 — Lens Primitives & Operator Families (Node / Edge / Path / Subgraph)

One-sentence goal: Define four lens primitives (node, edge, path, subgraph) and a composable operator family Φ_lens; provide minimal equations for both spectral and variational forms with implementation bindings, enabling subsequent composition, inversion, and streaming deployment.

I. Scope & Objects

1. Inputs
   • Graph & operators: G = (V, E, w), adjacency A, degree D, Laplacian L (type & scaling fixed in RefCond).
   • Runtime signal: x_in ∈ R^{|V|}; optional observation y = H x_true + v.
   • Primitive parameters: node weights w_node(v), edge gates m_ij, path set Π = { π_k }, subgraph set S ⊆ V.
   • Mode & window: mode ∈ { offline, streaming }, win = { Δt_win, Δt_slide }.
2. Outputs
   • Primitive lenses Φ_node, Φ_edge, Φ_path, Φ_subgraph, and their composition Φ_* = Φ_N ∘ Φ_E ∘ Φ_P ∘ Φ_S.
   • Lensed result x' = Φ_* ( x_in ) and the dual-form gap delta_form_lens (exact spectral vs approximation / offline vs streaming).
3. Boundary
   No change of dimensions: unit(x') = unit(x_in); physical propagation delays are out of scope (see …PathCorrection v1.0).

II. Terms & Variables

• Primitive weights & gates: W_node = diag( w_node(v) ), M = { m_ij } (edge gates), 0 ≤ w_node, m_ij ≤ g_cap.
• Paths & subgraphs: single path π: { v_0, …, v_L }; path weights ρ_k ≥ 0; subgraph indicator P_S = diag( 1_{v∈S} ).
• Anisotropic conductance: C_ani(e) (high along-path, low cross-path), B directed incidence matrix, L_C = B^T C B.
• Composition algebra: ∘ (operator composition), ⊕ (parallel stacking / residual merge), I (identity).
• Dual forms: spectral K(•) and variational argmin E(•) computed in parallel; record delta_form_lens.

III. Postulates P713-*

• P713-1 (Boundedness): Each primitive’s kernel K satisfies || K ||_2 ≤ 1 + ε (default ε ≤ 0.02), or provides spectral-bound evidence.
• P713-2 (Composability): The family Φ is closed under composition; if Φ_a, Φ_b are stable, the bound for Φ_b ∘ Φ_a is computable and persisted.
• P713-3 (View consistency): If a primitive depends on view/occlusion, it must use the unified π_view / vis (see Chapter 2) and weight them explicitly in the energy or kernel.
• P713-4 (Dual forms in parallel): For every primitive, provide at least one spectral and one variational implementation; run both on the same input and log delta_form_lens.
• P713-5 (Connectivity-preserving claim): If marked “preserve connectivity,” the modified L' remains connected on the main component (λ_2(L') > 0).

IV. Minimal Equations S713-*

1. S713-1 (Node lens)
   • Spectral: Φ_node(x) = W_node x, with W_node = diag( w_node(v) ).
   • Variational: x' = argmin_x ( (1/2) || x − x_in ||_2^2 + (β/2) || (I − W_node) x ||_2^2 ).
2. S713-2 (Edge lens — weight re-calibration)
   • Reweighting: A' = A ⊙ M, D'_{ii} = ∑_j A'_{ij}, L' = D' − A'.
   • Spectral / diffusion: x' = K_edge x_in = (I + β L')^{−1} x_in or x' = exp(−τ L') x_in.
   • Variational: x' = argmin_x ( (1/2) || x − x_in ||_2^2 + (β/2) x^T L' x ).
3. S713-3 (Path lens — along-path anisotropy)
   • Anisotropic Laplacian: L_ani = B^T C_ani B, with
     C_ani(e) = c_∥ if edge e aligns with the tangent estimate t_hat of Π, otherwise C_ani(e) = c_⊥, and c_∥ ≥ c_⊥ ≥ 0.
   • Spectral: x' = K_path x_in = exp(−τ L_ani) x_in or x' = ( I + β L_ani )^{−1} x_in.
   • Variational: x' = argmin_x ( (1/2) || H x − y ||_2^2 + (β/2) x^T L_ani x ).
4. S713-4 (Subgraph lens — local with boundary conditions)
   • Dirichlet style: fix x_{V\setminus S} = x_in and solve
     x'_S = ( I + β L_{SS} )^{−1} ( x_in )_S − β ( I + β L_{SS} )^{−1} L_{S,~S} ( x_in )_{~S};
     assemble x' by splicing.
   • Projection kernel: K_sub = ( I + β P_S L P_S )^{−1}, x' = K_sub x_in.
5. S713-5 (Composition & dual-form gap)
   • Composite kernel: K_* = K_node K_edge K_path K_sub (ordered by implementation), x'_spec = K_* x_in.
   • Gap: delta_form_lens = || x'_spec − x'_var ||_2, or the streaming-window version
     delta_form_stream = ( ∑_{t∈win} w_t || • ||_2^2 )^{1/2} .
6. S713-6 (Stability & spectral bounds)
   • If each K_i is diagonalizable in the same U, then ρ(K_*) ≤ ∏_i ρ(K_i); diffusion/Tikhonov types satisfy ρ(K_i) ≤ 1.
   • Subgraph leakage bound:
     leak = || x'_{~S} − x_in_{~S} ||_2 ≤ β || L_{~S,S} ||_2 || ( I + β L_{SS} )^{−1} ||_2 || x_in ||_2.

V. Metrology Pipeline M71-3 (Primitive Selection → Assembly → Verification → Persist)

1. Select primitives & parameters: choose among node/edge/path/subgraph and their order; set θ = { w_node, M, Π, S, β, τ }; lock RefCond.
2. Build operators: produce W_node, L', L_ani, K_sub; if dependent on π_view / vis, inject visibility weights (see Chapter 2).
3. Solve in dual forms:
   • Spectral: compute x'_spec via exact spectra or Chebyshev approximation;
   • Variational: solve by primal–dual / ADMM to obtain x'_var.
4. Checks & stability: evaluate ρ(K_i) and ρ(K_*), leak, delta_form_lens, || y − H x' ||_2; propagate u_c and guardband U.
5. Persist & publish: record in manifest.lens: Φ.hash, θ, impl, spectral_bounds, delta_form_lens, contracts.*, signature.

VI. Contracts & Assertions C71-3x (suggested thresholds)

• C71-31 (Bounds on weights): 0 ≤ w_node(v) ≤ g_cap, 0 ≤ m_ij ≤ g_cap, default g_cap = 2.
• C71-32 (Spectral stability): ρ(K_i) ≤ 1 + ε and ρ(K_*) ≤ 1 + ε_* (recommend ε, ε_* ≤ 0.02).
• C71-33 (Connectivity-preserving): if claimed, verify λ_2(L') > 0.
• C71-34 (Subgraph leakage): leak ≤ tol_leak (recommend tol_leak = 1e−3 * || x_in ||_2).
• C71-35 (Path alignment): direction consistency
  align = ( ∑_{e∈E_Π} cos^2⟨e, t_hat⟩ / |E_Π| ) ≥ α_align (recommend α_align ≥ 0.8).
• C71-36 (Dual-form gap): delta_form_lens ≤ tol_lens (default 1e−3 * || x_in ||_2).
• C71-37 (Unit consistency): check_dim( x' − x_in ) = "[same]".

VII. Implementation Bindings I71-3* (interfaces, I/O, invariants)

• build_node_lens(w_node, caps) -> Φ_node, meta
  inv: 0 ≤ w_node ≤ g_cap; returns W_node and spectral-bound evidence.
• build_edge_lens(graph, M, variant, params) -> Φ_edge, meta
  in: variant ∈ { tikhonov, heat }, params ∈ { β | τ }; out: L', K_edge.
• build_path_lens(graph, paths, anisotropy, params) -> Φ_path, meta
  in: anisotropy = { c_∥, c_⊥ } or a direction field; out: L_ani, K_path, align.
• build_subgraph_lens(graph, S, params, bc) -> Φ_subgraph, meta
  in: bc ∈ { dirichlet, proj }; out: K_sub, leakage-bound estimate.
• compose_lenses([Φ_i], order) -> Φ_*, spectral_bounds
  inv: compute & persist ρ(K_*) and approximation-error bounds.
• apply_lens(Φ, x_in, mode, win) -> x', metrics
• eval_delta_form_lens(x'_spec, x'_var) -> delta_form_lens
• assert_lens_contracts(ds, rules) -> report
• emit_lens_manifest(results, policy) -> manifest.lens
  Invariants: Δt_win > 0; hash(*) is traceable; ρ(K_*) and delta_form_lens must be persisted; unit consistency holds.

VIII. Cross-References

• Spectral kernels & Chebyshev approximations: EFT.WP.STG.Dynamics v1.0, Chapter 4; numerical stability & rigidity: Chapter 9.
• View/visibility weighting: EFT.WP.STG.Lensing v1.0, Chapter 2.
• Path directions & worldline fragments: EFT.WP.Particle.TopologyAtlas v1.0, Chapters 5 and 7.
• Runtime SLOs & dashboard fields: EFT.WP.STG.Dynamics v1.0, Chapter 14.

IX. Quality & Risk Control

• SLI/SLO: track focus_gain, stability_margin, leak, delta_form_lens_p99, latency_p95, coverage.
• Fallbacks: if spectral bounds are violated → reduce β/τ or raise approximation order; if leakage exceeds threshold → switch boundary conditions or shrink S; if alignment is weak → re-estimate direction field or the path set.
• Audit: retain construction evidence for W_node / L' / L_ani / K_sub, computations for align / λ_2 / leak, contract pass rates, and anomalous samples.

Summary

• This chapter decomposes graph lensing into four composable primitives—node weighting, edge gating, path alignment, and subgraph localization.
• It provides computable equations for both spectral and variational forms with stability bounds, plus unified implementation bindings and contracts.
• manifest.lens must add:
  primitive = { node, edge, path, subgraph }, θ_primitive, spectral_bounds, align, leak, delta_form_lens, contracts.*.