24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 15 — Use Cases and Reference Implementations

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Chapter 15 — Use Cases and Reference Implementations

One-Sentence Goal
Deliver reproducible end-to-end pipelines—from raw data to a published Topological Atlas—across three representative scenarios: (A) phase singularities → 2D, (B) magnetic skyrmions → 2D/3D, and (C) worldline reconnections → spacetime. Bind each to concrete interfaces, contracts, and dashboard fields to form an engineering playbook.

I. Scope and Objects

1. Covered cases
   • Case A — Phase singularities (vortices): extract 2-D defects from interferometric/phase fields θ(x,y) and publish an atlas.
   • Case B — Magnetic skyrmions: compute topological densities of n(x,y) ∈ S^2 and construct a lattice atlas.
   • Case C — Worldline reconnections: from n(x,y,t), extract defect worldlines Γ, braids, and reconnection events in 3-D or spacetime.
2. Inputs: cleaned or raw data (Ch. 12) f, θ, n, Γ0; filtration strategy τ_policy; base atlas { U_a, φ_a, ψ_a, T_ab }; runtime policy and resource budgets.
3. Outputs:
   • Atlas_case, D_case, key invariants and evidence for each case;
   • two-form gaps delta_form_case and compliance reports;
   • manifest.topo.case.* (reproducible manifests with signatures).
4. Boundaries & constraints: non_decreasing(τ); Σ_a ψ_a ≡ 1; all metrics pass check_dim; randomness fixed via seed; cross-domain comparisons declare norm.

II. Terms and Variables

• Fields/orientation: θ(x,y), n(x,y) ∈ S^2, n(x,y,z,t); worldlines Γ = { γ_i(s) }.
• Topological densities & charges: q_vortex, q_sky, Q = ∫ q dA or ∫ q dV; linking/self-geometry Lk, Tw, Wr.
• Complex / filtration: K(τ), diagrams D, landscapes λ_D; thresholds/windows τ, τ_pers.
• Two forms: formA = clean→build→measure and formB = build→simplify→measure.
• Compliance: delta_form_case, tol_case, u(•), U = k * u_c.

III. Axioms P915-*

• P915-1 (Physics → topology conservation) — For A/B, net charge Q respects conservation within ±u(Q); for C, topological type is invariant on non-reconnection segments.
• P915-2 (Parallel forms) — Compute both formA/B for every case; record delta_form_case and evidence.
• P915-3 (Atlas coherence) — Align overlaps via T_ab; strictly maintain Σ_a ψ_a ≡ 1.
• P915-4 (Units & dimensions) — unit(Q) = "[1]"; unit(d_B) = unit(τ); unit(Lk) = "[1]"; all check_dim must pass.
• P915-5 (Reproducibility) — Persist data hashes, versions, seeds, and hyperparameters with signatures.

IV. Minimal Equations S915-*

1. Case A — Phase singularities (2-D)

• S915-1 (circulation): q_vortex(x0) = ( 1 / (2*pi) ) * ( ∮_C ∇θ • dl ), with C a pixel ring around x0.
• S915-2 (discrete winding): q_vortex = round( (1/(2*pi)) * Σ_{edges} wrap(Δθ) ), with wrap mapping phase differences to (-pi, pi].
• S915-3 (net charge): Q = Σ_i q_vortex(i), unit(Q) = "[1]", dim(Q) = [1].

2. Case B — Skyrmions (2-D / 3-D slices)

• S915-4 (density): q_sky(x,y) = ( 1 / (4*pi) ) * n • ( ∂_x n × ∂_y n ).
• S915-5 (total charge): Q = ( ∫ q_sky dA ) (or volumetric sum on slices); use Jacobian/area weights discretely.

3. Case C — Worldlines & reconnection (spacetime)

• S915-6 (Gauss linking):
  Lk(Γ_i, Γ_j) = ( 1 / (4*pi) ) * ∬ ( ( dℓ_i × dℓ_j ) • ( r_i − r_j ) / || r_i − r_j ||^3 ).
• S915-7 (reconnection event): within t0 ± Δt, a step in Lk or component count with energy-conservation evidence defines E_reconn.

4. Atlas-level aggregation

• S915-8 (local → global): Inv_global = Σ_a ( ∫_{U_a} ψ_a * inv_a dV ).
• S915-9 (two-form gap): delta_form_case = d_B( D_{formA}, D_{formB} ) or | Q_{formA} − Q_{formB} |.

V. Metrology Workflow M90-15 (common to all cases)

• Ready: load data and RefCond; estimate noise σ and stability windows (Chs. 12 & 8).
• Cleaning & orientation preservation: TV/anisotropic for scalars; circular mean for phases; projected mean for S^2 fields (see S912-6/7).
• Complex & filtration: build K(τ) (Rips/Alpha/graph-Laplacian), compute D and landscapes; enforce non_decreasing(τ).
• Topological measures: compute q, Q, Lk, E_reconn via S915-1/5/6; produce local evidence slices and matchings.
• Atlas synthesis: aggregate inv_a with weights ψ_a; repair T_ab; verify cocycle Δ_cyc.
• Parallel forms: run formA/B concurrently; compute delta_form_case and u(Inv); if gates fail, back off and recompute.
• Publish & sign: output Atlas_case, D_case, Inv, evidence and manifest.topo.case.*; sign and archive for audit.

VI. Contracts & Assertions C90-151x (suggested)

• C90-15101 (charge conservation): for A/B, | Q_out − Q_in | ≤ tol_Q (suggest tol_Q = 0.5 unit charges).
• C90-15102 (two-form): delta_form_case_p95 ≤ tol_case (suggest 0.02 * scale_τ).
• C90-15103 (Atlas coherence): sup_x | Σ_a ψ_a − 1 | ≤ 1e-6, Δ_cyc_p95 ≤ tol_cyc.
• C90-15104 (evidence sufficiency): each topological object has ≥3 evidence types: local slices, persistent pairings, geometric contour/orientation fields.
• C90-15105 (dimension gate): check_dim(Q) = "[1]", check_dim(d_B) = unit(τ), check_dim(Lk) = "[1]".
• C90-15106 (stability): under ε-perturbations, d_B(D, D+δ) ≤ ε + 0.05 * scale_τ (per Ch. 8).

VII. Implementation Bindings I90-15* (reference interfaces)

• I90-151 detect_vortices_phase(theta, policy) -> { points, q_vortex, Q, evidence }
• I90-152 measure_skyrmion_charge(n_field, scheme) -> { q_map, Q, lattice, evidence }
• I90-153 trace_worldlines(n_field_t, seed_policy) -> { Γ, tags }
• I90-154 detect_reconnection(Γ, window) -> { E_reconn, Lk_series, evidence }
• I90-155 build_case_complex(data, τ_policy) -> { K(τ), D, λ_D }
• I90-156 compose_case_atlas(tiles, ψ, T_ab) -> Atlas_case, Δ_cyc
• I90-157 compare_case_forms(resultsA, resultsB, metrics) -> delta_form_case
• I90-158 assert_case_contracts(results, rules) -> report
• I90-159 emit_case_manifest(results, policy) -> manifest.topo.case

Invariants: non_decreasing(τ); Σ ψ ≡ 1; delta_form_case ≤ tol_case; all check_dim pass; reproducible with fixed seed.

VIII. Cross-References

• Mathematical baseline & topological objects: Chs. 2–3.
• Field → topological density: Ch. 4 (q_vortex / q_sky discretizations, Jacobians).
• Worldlines & braids: Chs. 5–6 (Lk / Tw / Wr and event graphs).
• Data → complex & filtration: Ch. 7; persistent stability: Ch. 8.
• Atlas & transitions: Chs. 9–11; cleaning & robustness: Ch. 12.
• Retrieval & comparison: Ch. 13; runtime & dashboards: Ch. 14.
• Error propagation & uncertainty: Appendix E (compose u(Q), u(Lk)).

IX. Quality & Risk Control

• SLO/SLI: delta_form_case_p95, Q_bias, R@K (if retrieval involved), panel.runtime.*, reconnection detection latency lat_reconn.
• Fallback: higher-order differencing → lower-order differencing → coarser grid → H0/H1 only, or expand τ_window for larger persistence gates.
• Audit: persist input hashes, discrete Jacobians/grids, evidence slices, pairing lists, worldline segments, and index version; publish signed, privacy-preserving summaries.

Summary
This chapter delivers three reference implementations for Data → Complex → Invariants → Atlas, formalized as P915 / S915 / M90-15 / C90-151x / I90-15*. With parallel forms, contract gates, and evidence-backed publication, the Topological Atlas remains reproducible, auditable, and integrable across scenarios—published uniformly as manifest.topo.case.* to serve downstream retrieval (Ch. 13) and runtime dashboards (Ch. 14).