24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 3 — Topological Particles and Defect Typology

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Chapter 3 — Topological Particles and Defect Typology

One-Sentence Goal
Establish the object hierarchy and naming conventions for particles / defects / braids / knots, and provide the companion set of computable invariants and minimal criteria that act as label sources for atlas construction and persistent structures.

I. Scope and Objects

1. Families of objects
   • Point defects (codim = d): 2-D phase vortices; 3-D monopoles / “hedgehogs.”
   • Line defects (codim = d−1): 3-D vortex filaments; dislocation/displacement lines.
   • Surface defects (codim = 1): domain walls; interfaces.
   • Textures / dissolved defects: field configurations with nontrivial π_k but no explicit holes (e.g., skyrmions, Hopf textures).
   • Braids / knots: spacetime interweavings of worldlines Gamma_i(t) and link–knot structures of closed curves.
2. Inputs
   • Fields & order parameters: phi(x,t) (complex/phase/vector/spin), Xi(x,t) (tensor/framing field).
   • Discrete carriers: point clouds / voxels / meshes data; geometric/topological domain M with boundary ∂M; coefficient field coeffs.
3. Outputs
   • Defect instances: defects = { (type, support, invariants, u/U, tags) }.
   • Measured invariants: Q, w, Lk, Sl, β_k, plus domain-specific derivations (e.g., Burgers vector).
   • Atlas labels: per-chart inv_i on each U_i, to be stitched in Chapter 9.

II. Terms and Variables

• Support & neighborhoods: supp(D) (defect support), N_eps(supp) (ε-neighborhood).
• Maps & target space: phi: M \setminus supp(D) → N, where N is the order-parameter target; deg(phi; S) denotes the degree.
• Dimensional shorthand: d = dim(M), p = dim(supp(D)), codim = d − p.
• Invariants: w (winding), Q (topological charge), Lk (linking), Sl (self-linking), Tw/Wr (twist/writhe).
• Linear-defect kinematics: displacement field u(x) with Burgers vector b = ( ∮_{gamma} ( ∇u • d l ) ).
• Diagnostics: u(x), U = k * u_c, nu_eff; event and processing time ts, tau_mono.

III. Axioms P903-*

• P903-1 (Codimension governs class) — In d-dimensional space, a p-dimensional defect is classified by pi_{codim-1}(N); the classification is independent of chart choices.
• P903-2 (Integrality) — The integer-valued invariants w, Q, Lk, Sl are integers in the ideal continuum limit; numerical realizations admit deviations within tol_int.
• P903-3 (Local–global coherence) — Invariants computed locally (small spheres/tubes) must agree with global computations (enclosing curves/surfaces/volumes).
• P903-4 (Boundaries & relative homology) — If ∂M ≠ ∅, use relative homology or add boundary terms so invariants are well-defined in relative classes.
• P903-5 (Computability) — Each category maps to an executable I90-* interface with a stated complexity bound; defect localization and invariant estimation are independent yet cross-checkable.
• P903-6 (Robustness) — Classifications are stable within small neighborhoods under d_B (bottleneck distance of persistence) and under energy/noise perturbations.

IV. Minimal Equations S903-*

• S903-1 (Homotopy classification criterion)
  class(D_p) ↔ pi_{d-p-1}(N), where D_p is a p-dimensional defect and N is the order-parameter target space.
• S903-2 (2-D point-vortex winding)
  w = ( 1 / ( 2*pi ) ) * ( ∮_{gamma} d θ ), with gamma a closed loop encircling the defect and θ = arg( phi ).
• S903-3 (3-D monopole topological charge)
  Q = ( 1 / ( 4*pi ) ) * ( ∫_{S^2} ( n_hat • ( ∂_θ n_hat × ∂_φ n_hat ) d Ω ) ), where n_hat = phi / |phi|.
• S903-4 (Skyrmion number, 2-D O(3) texture)
  Q_sk = ( 1 / ( 4*pi ) ) * ( ∫_{A ⊂ R^2} ( n_hat • ( ∂_x n_hat × ∂_y n_hat ) dA ) ).
• S903-5 (Gauss linking number)
  Lk(Γ1, Γ2) = ( 1 / ( 4*pi ) ) * ( ∬_{Γ1×Γ2} ( ( (r1 - r2) / |r1 - r2|^3 ) • ( d r1 × d r2 ) ) ).
• S903-6 (White’s formula — self-linking decomposition)
  Sl(Γ) = Tw(Γ, f_hat) + Wr(Γ), where f_hat is a reference framing in a tubular neighborhood.
• S903-7 (Dislocation Burgers vector)
  b = ( ∮_{gamma} ( ∇u • d l ) ), with u the displacement field and gamma encircling a dislocation line.
• S903-8 (Hopf invariant, schematic)
  Q_hopf = ( 1 / ( 4*pi ) ) * ( ∫_{R^3} ( A • B dV ) ), with B = ∇ × A the “magnetic” field associated with the pullback connection of n_hat; gauge and boundary decay must be declared.
• S903-9 (Local Jacobian counting)
  For isolated zeros of phi: R^d → R^d, Q = sign( det( ∂ phi / ∂ x ) ); a volume-integral form is
  Q = ( 1 / C_d ) * ( ∫_{∂V} J(phi) • dS ), where the constant C_d and the chosen topological-current gauge are specified.

V. Metrology Workflow M90-3

1. Ready: choose an order parameter phi and target N; declare coeffs, working dimension d, and boundary strategy (closed / relative).
2. Candidate support detection:
   • Detect point/line/surface candidates via thresholds and morphology: |phi| < τ_amp, peaks in |∇θ|, sign-change(phi), etc.
   • Produce a candidate set S0 = { cells / edges / faces }.
3. Neighborhood construction: for each candidate, build gamma (loops), S^2 (small spheres), tubular neighborhoods Tube(Γ, ρ), or cross-domain surfaces Σ.
4. Invariant estimation:
   • Apply the relevant S903-* formulas to compute w / Q / Lk / Sl / b / … together with u_c / U / nu_eff.
   • In parallel, perform local–global coherence checks (P903-3).
5. Robustness checks: compare local PD_k across windows/perturbations and gate with d_B ≤ tol_PD.
6. Component merging and de-duplication: merge connected components; decompose or recombine via Sl = Tw + Wr; remove numerical duplicates.
7. Persistence & labeling:
   • Emit the defects list (type, support, invariants, u/U, tags, quality).
   • Write local inv_i into the metadata for chart U_i to support stitching in Chapter 9.

VI. Contracts & Assertions C90-31x (suggested thresholds)

• C90-3101 Integrality gate: dist_to_Z(w) ≤ tol_int, dist_to_Z(Q) ≤ tol_int, dist_to_Z(Lk) ≤ tol_int, dist_to_Z(Sl) ≤ tol_int.
• C90-3102 Neighborhood sufficiency: radius(neighborhood) ≥ κ * grid_spacing (suggest κ ≥ 2.5).
• C90-3103 Consistency: | Q_local − Q_global | ≤ tol_glue; | Sl − (Tw + Wr) | ≤ tol_glue.
• C90-3104 Separation: minimum spacing between distinct supports ≥ sep_min; otherwise tag merge_or_ambiguous.
• C90-3105 Freshness & coverage: ts_now − ts_data ≤ T_max and coverage(M \setminus N_eps(supp)) ≥ cov_min.
• C90-3106 Complexity bound: per frame/window, compute_invariants runtime ≤ T_budget; exceeding triggers down-sampling or sparsification.
• C90-3107 Boundary compliance: if ∂M ≠ ∅, use relative homology / boundary compensation explicitly; absence blocks publication.

VII. Implementation Bindings I90-3*

• I90-31 detect_defect_candidates(field, rules) -> S0 (thresholding / morphology / gradient circulation / zero-set tracing)
• I90-32 build_neighborhood(S0, kind, params) -> { loops, spheres, tubes }
• I90-33 compute_winding(phi, loops) -> { w_i, u/U } (includes phase unwrapping and loop orientation alignment)
• I90-34 compute_skyrmion(n_field, patch) -> { Q_sk, u/U }
• I90-35 gauss_linking_lines(lines) -> { Lk, Sl, Tw, Wr } (discrete line integrals and White’s formula)
• I90-36 burgers_vector(u_field, loops) -> { b_i }
• I90-37 consolidate_defects(cands, invariants, policy) -> defects
• I90-38 assert_defect_contracts(defects, rules) -> report
• I90-39 emit_defect_manifest(defects, policy) -> manifest.topo.defects

Invariants: domains/paths and measures are explicit; dist_to_Z(•) passes; local–global consistency holds; boundary strategy is recorded.

VIII. Cross-References

• Field → topological density and topological currents: Chapter 4.
• Worldline tracking, braid words, and knots: Chapter 5 (I90-51/52).
• Persistence stability and d_B / d_W: Chapter 8.
• Atlas stitching and transition coherence: Chapter 9.
• Uncertainty and bias correction: Chapter 10 and Appendix E.
• Runtime and dashboard publication: Chapter 14; manifest keys: Appendix C.

IX. Quality & Risk Control

• SLOs: integrality pass-rate ≥ 99%; d_B stability p95 ≤ tol_PD; runtime p99 ≤ T_budget.
• Fallback path: full-res → decimated → coarse-neighborhood → summary-tags, inflating U at each step.
• Audit: persist candidate masks, neighborhood parameters, domain/path discretization summaries, integrality/consistency reports, and signatures; guarantee replay via recorded seeds and unwrapping strategies.

Summary
This chapter forges the math–engineering bridge for defect typology and its companion invariants as P903 / S903 / M90-3 / C90-31x / I90-3*. Under explicit neighborhood construction, integrality gates, and robustness metrics, any dataset can extract and publish defect labels that power atlas stitching (Ch. 9) and retrieval (Ch. 13).