24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 8 — Persistent Homology and Stability

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Chapter 8 — Persistent Homology and Stability

One-Sentence Goal
Establish stability conventions from filtrations to persistent-homology invariants; provide reproducible vectorizations (landscapes / images / kernel embeddings) and metrics; and ensure cross-noise / sampling consistency via parallel forms and bootstrap evaluation.

I. Scope and Objects

1. Inputs
   • Filtration F = { K(τ) } with homology cap k_max (see Chapter 7).
   • Diagram set D = { D_k | k = 0..k_max } computed via two forms: form=bdry and form=coho/morse.
   • Counterfactual / perturbed data {F'}, {D'}, noise bounds η_noise, grid step Δτ.
   • Vectorization & kernel parameters: grid, σ, weight(•), dirs (collection of slicing directions).
2. Outputs
   • Stability metrics: d_B, d_W,p, d_I (via slicing or single-parameter approximation), ρ_rank (rank correlation).
   • Vectorizations: Λ (persistence landscapes), PI_σ (persistence images), Φ_K (diagram-kernel embeddings).
   • Report & manifest: manifest.topo.stability (gates, distributions, delta_form_*, signature).
3. Boundaries & constraints
   • Single-parameter stability requires tame f or finite complexes K; multi-parameter cases use slicing or matching-distance approximations.
   • All distances/kernels must share units with the filtration scale; declare scaling/normalization strategies.

II. Terms and Variables

• Persistence diagram: D_k = { (b_i, d_i) } (b_i ≤ d_i), diagonal Δ = { (t,t) }.
• Distances: d_B(D,D') (bottleneck), d_W,p(D,D') (p-Wasserstein).
• Modules & interleavings: M_f, M_g (persistence modules), d_I(M_f,M_g) (interleaving distance).
• Landscapes: Λ = { λ_r(t) }_{r≥1}; images: PI_σ(u,v) on (birth, persistence) grids.
• Kernel embeddings: Φ_K(D), common K ∈ { PSSK, SWK }.
• Two-form discrepancies: delta_form_PH (diagram-computation routes), delta_form_vec (landscape/image alignment).
• Statistics: u(x), U = k * u_c, nu_eff, bootstrap size B.

III. Axioms P908-*

• P908-1 (Tameness/finite) — F arises from finite complexes with monotone grade(σ) or from tame/PL functions; each D_k is locally finite.
• P908-2 (Comparable settings) — Cross-dataset comparisons require consistent type / metric / coeff-field / k_max / grade.policy; otherwise comparisons are invalid.
• P908-3 (Parallel two forms) — Compute D via boundary and cohomology/Morse routes and record delta_form_PH.
• P908-4 (Scale declaration) — Every distance or kernel declares unit(τ) and scale_τ; hidden unit changes are forbidden.
• P908-5 (Reproducible vectorization) — Fix and persist grid / σ / weight / dirs, plus interpolation/discretization policies; seed any randomness.
• P908-6 (Stability assessment) — With B ≥ 20 bootstrap/perturbation samples, estimate distributions and confidence; publish p95 and U.
• P908-7 (Slicing sufficiency) — For multi-parameter filtrations, use at least |dirs| ≥ 32 uniform directions to approximate matching/interleaving distances and annotate the error model.

IV. Minimal Equations S908-*

1. Single-parameter stability and interleavings

• S908-1 (Bottleneck stability): for tame f,g,
  d_B( D(f), D(g) ) ≤ ( ∥ f − g ∥_∞ ).
• S908-2 (Interleaving distance):
  d_I(M_f, M_g) = inf { ε | M_f, M_g are ε-interleaved }, and for single-parameter cases,
  d_B( D(f), D(g) ) = d_I(M_f, M_g).

2. Diagram distances

• S908-3 (bottleneck):
  d_B(D,D') = inf_{γ} ( sup_{x∈D} ( ∥ x − γ(x) ∥_∞ ) ), allowing matches to Δ.
• S908-4 (p-Wasserstein):
  d_W,p(D,D') = ( inf_{γ} ( Σ_{x∈D} ∥ x − γ(x) ∥_∞^p ) )^{1/p}.

3. Landscapes and images

• S908-5 (landscape): for each (b,d), define λ_{(b,d)}(t) = max( 0, min(t−b, d−t) ); then λ_r(t) is the r-th largest over all pairs, assembling Λ = { λ_r }.
• S908-6 (image): with transform ψ: (b,d) → (u=b, v=d−b), weights w(u,v), kernel ρ_σ,
  ( PI_σ )(ξ) = ( Σ_{p∈D} w(ψ(p)) * ρ_σ( ξ − ψ(p) ) ), with grid center ξ.

4. Kernel embeddings & similarity

• S908-7 (PSSK example): view diagrams as signed measures; diffuse by a heat kernel to time σ^2 and take the L_2 inner product:
  K_{PSS}(D,D'; σ) = ⟨ Φ_{σ}(D), Φ_{σ}(D') ⟩_{L_2}.
• S908-8 (Sliced-Wasserstein kernel):
  K_{SW}(D,D') = exp( − ( ( 1/|dirs| ) * Σ_{θ∈dirs} W_2^2( π_θ D, π_θ D' ) ) / (2σ^2) ).

5. Vectorization stability (Lipschitz-type)

• S908-9: there exist constants L_Λ, L_PI such that
  ∥ Λ(D) − Λ(D') ∥_{L_q} ≤ L_Λ * d_B(D,D'),
  ∥ PI_σ(D) − PI_σ(D') ∥_{L_1} ≤ L_PI(σ,weight) * d_B(D,D').

6. Two-form discrepancies and alignment

• S908-10: delta_form_PH = dist_intervals( D_bdry , D_coho );
  delta_form_vec = ( ∥ Λ(D_bdry) − Λ(D_coho) ∥_{L_2(grid)} + ∥ PI_σ(D_bdry) − PI_σ(D_coho) \|_{L_1} ) / 2.

7. Multi-parameter approximations

• S908-11: matching-distance proxy
  d_match(D, D') ≈ max_{θ∈dirs} ( d_B( π_θ D , π_θ D' ) );
  sliced interleaving
  d_I^{slice}(M,N) = max_{θ∈dirs} d_I( M_θ, N_θ ).

V. Metrology Workflow M90-8

1. Ready: choose k_max, form ∈ { bdry, coho/morse }, p ∈ {1,2}, dirs, σ / grid / weight; set bootstrap B and gates tol_*.
2. Diagram computation (two forms): from F, compute D_bdry, D_coho, and delta_form_PH; if over-threshold, revisit Chapter 7 construction.
3. Distance evaluation:
   • Compute d_B, d_W,p (single-parameter).
   • For multi-parameter, slice to obtain d_match, d_I^{slice}.
4. Vectorization: build Λ and PI_σ; produce Φ_K (K ∈ { PSSK, SWK }); compute delta_form_vec and ρ_rank (rank correlation with d_B).
5. Stability estimation:
   • Generate { D^{(b)} }_{b=1..B} via bootstrap/perturb;
   • Summarize distributions of d_B(D, D^{(b)}) and vectorization gaps; report u/U and p95.
6. Consistency & gates: enforce d_B ≤ η_noise + tol_stab, delta_form_PH ≤ tol_PH, delta_form_vec ≤ tol_vec, ρ_rank ≥ ρ_min.
7. Persist:
   manifest.topo.stability = { k_max, forms, dists, kernels, Λ, PI_σ, delta_form_PH, delta_form_vec, dirs, B, p95, U, params, algo.ver, seed }, with signature.

VI. Contracts & Assertions C90-81x (suggested thresholds)

• C90-8101 Comparable computation: type / metric / coeff-field / k_max / grade.policy must match exactly to compare.
• C90-8102 Diagram validity: ∀(b,d)∈D, b ≤ d; diagonal pairing sufficient; |D| < ∞.
• C90-8103 Bottleneck stability: with declared η_noise, require d_B_p95 ≤ η_noise + tol_stab (suggest tol_stab = 0.05 * scale_τ).
• C90-8104 Two-form coherence: delta_form_PH_p95 ≤ tol_PH (suggest tol_PH = 1e-3 for Z_2).
• C90-8105 Vectorization coherence: delta_form_vec_p95 ≤ tol_vec (suggest tol_vec = 0.02 * scale_Λ); ρ_rank ≥ 0.9.
• C90-8106 Slicing coverage: |dirs| ≥ 32 and uniform; if not, tag slice.sparse=true and inflate U.
• C90-8107 Resources & reproducibility: quota overruns trigger fallbacks; persist seed and grid/hash.
• C90-8108 Units/dimensions: check_dim(d_B) = unit(τ); check_dim(Λ, PI, Φ_K) = "[1]".

VII. Implementation Bindings I90-8*

• I90-81 persistence_diagram(F, k_max, form) -> D
• I90-82 bottleneck_distance(D1, D2) -> d_B
• I90-83 wasserstein_distance(D1, D2, p) -> d_Wp
• I90-84 interleaving_distance_sliced(F1, F2, dirs) -> d_I_slice
• I90-85 persistence_landscape(D, grid?) -> Λ
• I90-86 persistence_image(D, grid, σ, weight) -> PI_σ
• I90-87 diagram_kernel(D1, D2, kind, params) -> K_val
• I90-88 bootstrap_stability(D, B, noise_policy) -> { dist_stats, u/U }
• I90-89 compare_vectorizations(Λ, PI_σ) -> { delta_form_vec, ρ_rank }
• I90-8A assert_stability_contracts(results, rules) -> report
• I90-8B emit_stability_manifest(results, policy) -> manifest.topo.stability

Invariants: non_decreasing(τ); delta_form_PH ≤ tol_PH; seeds/params persisted; all check_dim pass.

VIII. Cross-References

• Complexes & filtrations: Chapter 7 (K(τ), grade).
• Worldlines/events: Chapters 5–6 (stability thresholds inform event robustness).
• Atlas stitching & transition harmonization: Chapter 9 (cross-domain stability & shared gates).
• Query & indexing: Chapter 10 (landscape/image/kernel indexing & retrieval).
• Runtime & dashboards: Chapter 14; manifest keys: Appendix C; error propagation: Appendix E.

IX. Quality & Risk Control

• SLOs: d_B success ≥ 99.5%; delta_form_PH_p95 ≤ tol_PH; ρ_rank ≥ 0.9; bootstrap B completion rate 100%.
• Fallback path: full diagrams → sliced-matching → H0-only (MST diffs) → thresholded features, inflating U and tagging fallback.stage progressively.
• Audit: persist optimal matchings γ*, weights paired to the diagonal, slicing directions, kernel/landscape/image parameters, random seeds, and resource traces—with signatures & hashes.

Summary
This chapter provides the engineering convention for persistence stability as P908 / S908 / M90-8 / C90-81x / I90-8*. With parallel-form computation, dual stability checks (distances & vectorizations), and bootstrap reports, persistent features remain reproducible and auditable under noise and sampling perturbations; results are published uniformly as manifest.topo.stability.*.