24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 13 — Atlas Comparison, Metrics, and Retrieval

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Chapter 13 — Atlas Comparison, Metrics, and Retrieval

One-Sentence Goal
Define similarity metrics and a retrieval pipeline for topological atlases, with exact metrics ↔ approximate embeddings running in parallel with a bias-closure loop, to support fast, auditable cross-source and cross-atlas comparison.

I. Scope and Objects

1. Inputs
   • Atlases & features: Atlas = { (U_a, phi_a, K_a(τ), D_a, Γ_a, G_{event,a}), T_ab, ψ_a } (Chs. 9–11).
   • Queries & database: Q (query atlas/subdomain/invariant set), DB = { Atlas_i }.
   • Metric policy: metric.policy = { on = { D, λ, β_seq, Inv, Γ, G_event }, align = { τ, trans }, norm, kernel, approx }.
2. Outputs
   • Similarity & ranking: score(Q, Atlas_i), top-K set TopK.
   • Two-form gap: delta_form_retr = | d_exact − d_approx | or a norm gap in landscape/kernel space.
   • Manifest: manifest.topo.search (metric forms, index summary, evidence, signatures).
3. Boundaries & constraints
   • Preserve non_decreasing(τ) and Σ_a ψ_a ≡ 1; cross-atlas comparisons must declare align and normalization norm.
   • All metric/similarity objects must carry explicit unit(•) and dim(•).

II. Terms and Variables

• Diagrams & landscapes: D = { (b_i, d_i) }, λ_D(t,k) (k-th largest slice height at t).
• Canonical distances: d_B (bottleneck), W_p (p-Wasserstein), || λ_D − λ_E ||_p.
• Kernels: k_PSS (persistence scale-space), k_SW (sliced Wasserstein).
• Atlas-level distance: d_Atlas (Ch. 11 S911-5), extended to aligned atlases as d_Atlas^align.
• Trajectory/curve distances: d_F (Fréchet), d_DTW (dynamic time warping); event-graph distance d_GED (graph edit).
• Embeddings & indexes: Φ(•) → R^m (landscape / kernel / atlas signatures), Z = Φ(Atlas), Index ∈ { HNSW, PQ, IVFPQ }.
• Two forms: form=exact (exact metric) and form=approx (embedded / indexed approximate).

III. Axioms P913-*

• P913-1 (Alignment first) — Before any cross-atlas metric, declare align={ τ, T_ab, norm }; unaligned distances are for coarse filtering only.
• P913-2 (Parallel two forms) — For every candidate, return both d_exact and d_approx; record delta_form_retr and gate on it.
• P913-3 (Decompose & aggregate) — Aggregate local distances with partition-of-unity weights ψ_a; correct for overlap double-counting.
• P913-4 (Dimensional compliance) — unit(d_B)=unit(τ), unit(W_p)=unit(τ), unit(d_F)=unit(x); kernel similarities are dimensionless.
• P913-5 (Auditability) — Persist query snapshot, index version, random seeds, and candidate evidence (paired features, shortest paths, matched items).
• P913-6 (Stability window) — Compare within stability windows (Ch. 8) to avoid scale-induced artifacts.

IV. Minimal Equations S913-*

1. Diagram distances & kernels

• S913-1 (bottleneck): d_B(D,E) = inf_γ sup_{p∈D} || p − γ(p) ||_∞, with bijections γ extended to the diagonal.
• S913-2 (Wasserstein): W_p(D,E) = ( inf_γ Σ_{p∈D} || p − γ(p) ||_∞^p )^{1/p}.
• S913-3 (landscape L_p): || λ_D − λ_E ||_p = ( ∬ | λ_D(t,k) − λ_E(t,k) |^p dt dk )^{1/p}.
• S913-4 (PSS kernel): k_PSS(D,E;σ) = ( Σ_{p∈D} Σ_{q∈E} exp( − ||p − q||_2^2 / (8σ^2) ) ) / (4πσ^2).
• S913-5 (SW kernel): slice by angle θ, aggregate W_1 across directions, and set k_SW = exp( − SW^2 / (2σ^2) ).

2. Atlas-level aggregation & alignment

• S913-6 (local→global):
  d_topo(Q, A) = ( Σ_a w_a * d_loc( D_Q^a, D_A^{π(a)} ) ), where
  w_a = ( ∫_{U_a} ψ_a dV ) / ( Σ_b ∫_{U_b} ψ_b dV ), and π(a) arises from transitions + nearest-neighbor matches.
• S913-7 (τ alignment): τ_Q = s τ_A + b, with (s,b) = argmin Σ_a d_loc( D_Q^a(τ_Q), D_A^a(τ_A) ), subject to s>0.
• S913-8 (aligned atlas distance): d_Atlas^align = max( d_topo, sup_x | Σ_a ψ_a − 1 | ) (inherits the consistency term from Ch. 11).

3. Trajectories & events

• S913-9 (Fréchet): d_F(Γ_1, Γ_2) = inf_{α,β} sup_t || Γ_1(α(t)) − Γ_2(β(t)) ||_2.
• S913-10 (Event graphs): d_GED(G_1,G_2) = min_{edit seq} cost(seq), with nodes weighted by event-topology labels and time windows.

4. Multi-channel fusion & scoring

• S913-11 (fusion): score = σ( − Σ_c λ_c * d_c + Σ_k μ_k * k_k ), where σ is a monotone squashing (e.g., softplus/sigmoid).
• S913-12 (two-form gap): delta_form_retr = | d_exact − d_approx | or || Φ_exact − Φ_approx ||_2.

5. Units & dimensions

• S913-13: check_dim(d_B) = unit(τ), check_dim(W_p) = unit(τ), check_dim(d_F) = unit(x), check_dim(k_*) = "[1]".

V. Metrology Workflow M90-13

• Ready: parse Q and DB; lock metric.policy, alignment and RefCond = { τ_window, σ, p, weights }.
• Align: estimate (s,b) for τ rescaling; construct cross-atlas local matches π(a) and weights w_a.
• Exact distances: compute d_exact (combined d_B / W_p / landscape / trajectory / event) with channel-wise breakdown.
• Embedding & index: compute Z_Q = Φ(Q), Z_i = Φ(Atlas_i); search candidates Cand on Index.
• Re-rank & two-form check: for Cand, compute d_exact and d_approx, re-rank, and record delta_form_retr.
• Gates & backoff: if gates fail (Sec. VI), adjust align / weights / σ / p or widen τ_window and recompute.
• Persist:
  manifest.topo.search = { Q.hash, Index.ver, Φ.ver, policy, align, TopK, d_breakdown, delta_form_retr_stats, RefCond, evidence, signature }.

VI. Contracts & Assertions C90-131x (suggested gates)

• C90-13101 (alignment): | s − 1 | ≤ tol_s and | b | ≤ tol_b (suggest tol_s ≤ 0.1, tol_b ≤ 0.05 * range(τ)).
• C90-13102 (two-form): delta_form_retr_p95 ≤ tol_retr (suggest tol_retr = 0.03 * scale_τ).
• C90-13103 (consistency): sup_x | Σ_a ψ_a − 1 | ≤ 1e-6, and no degradation in Δ_cyc (Ch. 9).
• C90-13104 (evidence): every TopK item carries ≥3 evidence types (paired diagram bars, matching maps, trajectory comparisons, kernel similarities).
• C90-13105 (performance): approximate-retrieval recall R@K ≥ R_min (suggest R_min = 0.95) with P95 latency within budget.
• C90-13106 (dimensions): all d/k pass check_dim; cross-domain comparisons must declare norm.

VII. Implementation Bindings I90-13*

• I90-131 build_persistence_distance(D1, D2, policy) -> { d_B, W_p, landscape_p }
• I90-132 atlas_align_tau(Q, A) -> { s, b }
• I90-133 atlas_local_match(Q, A, T_ab, ψ) -> { π, w }
• I90-134 atlas_distance(Q, A, π, w, policy) -> d_exact, breakdown
• I90-135 embed_atlas(Atlas, Φ_policy) -> Z (landscape / kernel features / graph-kernel fusion)
• I90-136 build_index({ Z_i }, index_policy) -> Index
• I90-137 query_index(Index, Z_Q, K) -> Cand
• I90-138 rerank_with_exact(Q, Cand, policy) -> TopK, delta_form_retr_stats
• I90-139 compare_tracks_events(Q, A) -> { d_F, d_DTW, d_GED }
• I90-13A assert_search_contracts(results, rules) -> report
• I90-13B emit_search_manifest(results, policy) -> manifest.topo.search

Invariants: non_decreasing(τ); Σ ψ ≡ 1; delta_form_retr ≤ tol_retr; all check_dim pass; index/embedding versions persisted for traceability.

VIII. Cross-References

• Filtration & stability windows: Chs. 7–8.
• Atlas construction & transitions: Ch. 9 (T_ab, ψ_a, Δ_cyc).
• Invariants & metrology: Ch. 10 (Inv as a retrieval channel).
• Incremental maintenance: Ch. 11 (incremental index updates and patch triggers).
• Cleaning robustness: Ch. 12 (metric stability and τ rescaling under noise).
• Error propagation: Appendix E (uncertainty composition for metrics & embeddings).

IX. Quality & Risk Control

• SLOs: R@K ≥ R_min, delta_form_retr_p95 ≤ tol_retr, TopK evidence completeness ≥ 99%.
• Fallback: approximate retrieval → enlarge candidate set → local-only metrics → fast H0/β_seq filters → human review, progressively relaxing and inflating U.
• Audit: retain candidate pairings, cost matrices, kernel-similarity heatmaps, trajectory-comparison curves, index dynamics, and seeds; publish signed, privacy-preserving summaries.

Summary
This chapter provides the engineering loop Atlas Comparison & Retrieval as P913 / S913 / M90-13 / C90-131x / I90-13*. By aligning across atlases, running exact vs approximate in parallel, and re-ranking with evidence and gates, retrieval is interpretable, reproducible, and auditable; results are uniformly persisted as manifest.topo.search.*.