24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 4 — Constructing Field → Topological Densities (Phase / Gradient / Jacobian)

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Chapter 4 — Constructing Field → Topological Densities (Phase / Gradient / Jacobian)

One-Sentence Goal
Stably map raw fields phi(x,t) and their derivatives (phase, gradients, Jacobians) into local topological densities and currents q(x), j_topo(x), supplying integrable, audit-ready invariant sources for defect identification (Chapter 3) and subsequent atlas stitching.

I. Scope and Objects

1. Inputs: gridded/meshed fields phi(x,t) (complex/phase/vector/spin); optional unit field n_hat(x); discrete carrier K (cubical/simplicial); optional metric g; sampling windows and filtering policies.
2. Outputs:
   • Phase density & vorticity: θ(x), ∇θ, q_wind_2D(x), j_topo^θ(x).
   • Jacobian / triple-product densities: q_sk_2D(x), q_mono_3D(x).
   • Quality payloads: u(q), U, masks and quality tags.
3. Boundaries & constraints: explicitly declare integration domains and measures—( ∮_{γ} • d l ), ( ∫_{A} • dA ), ( ∫_{V} • dV ); separate probabilistic from geometric/physical densities; persist boundary choices (closed vs relative).

II. Terms and Variables

• Phase & gradient: θ = arg(phi), ∇θ; unit(θ) = rad, dim(θ) = "[1]", dim(∇θ) = "[L^-1]".
• Topological densities: q_wind_2D, q_sk_2D, q_mono_3D; dim(q_d) = "[L^-d]" so their integrals over d-domains are "[1]".
• Jacobian & derivatives: J = ∂ phi / ∂ x; for n_hat: R^d → S^m use component derivatives and Levi–Civita tensors ε_{ijk}, ε_{abc}.
• Topological current: j_topo (distribution/sparse vector field), dim(j_topo) = "[L^-2]" in the 2-D phase setting.
• Metrology & time: u(•), U = k * u_c, nu_eff, ts / tau_mono.

III. Axioms P904-*

• P904-1 (Continuous phase representatives) — Phase arithmetic lives on S^1; numerically apply unwrap(•). Any ∇θ/circulation computation must be path-independent or executed on closed loops.
• P904-2 (Explicit measures) — Each density is constructed on an explicit domain with a declared measure, and a cell-level discrete approximation.
• P904-3 (Orientation consistency) — Surface/volume elements used for q_sk_2D / q_mono_3D must share a consistent orientation; for non-orientable carriers, transition corrections are applied in the atlas.
• P904-4 (Regularize first) — Derivatives/Jacobians are evaluated at a specified scale ℓ_reg, with default ℓ_reg ≥ 2.5 * grid_spacing.
• P904-5 (Parallel two-form routes) — Provide at least two equivalent routes per density (e.g., circulation vs curl; triangulated area vs finite differences) and record delta_form_topo ≤ tol_topo.
• P904-6 (Integrable integrality) — Surface/volume integrals must approximate integers within tol_int; failing gates are degraded or blocked.

IV. Minimal Equations S904-*

1. Phase and vorticity (2-D)

• S904-1 (circulation form): w(γ) = ( 1 / ( 2*pi ) ) * ( ∮_{γ} dθ ).
• S904-2 (curl form): q_wind_2D = ( 1 / ( 2*pi ) ) * ( ∇ × ∇θ )_z (in the sense of distributions).
• Discrete cell rule: for grid cell P,
  q_wind_2D(P) = ( 1 / ( 2*pi*area(P) ) ) * ( Σ_{e∈∂P} wrap(Δθ_e) ).

2. Skyrmion density (2-D, n_hat: R^2 → S^2)

• S904-3: q_sk_2D = ( 1 / ( 4*pi ) ) * ( n_hat • ( ∂_x n_hat × ∂_y n_hat ) ).
• Berg–Lüscher triangulation: for oriented image-space triangles Δ(n1,n2,n3) with solid angle Ω_Δ,
  q_sk_2D(cell) = ( 1 / ( 4*pi ) ) * ( Σ_Δ Ω_Δ ) / area(cell).

3. Monopole density (3-D, n_hat: R^3 → S^2)

• S904-4: q_mono_3D = ( 1 / ( 8*pi ) ) * ( ε_{ijk} ε_{abc} ∂_i n_a ∂_j n_b ∂_k n_c ).
• Tetrahedral discretization:
  q_mono_3D(tet) = ( 1 / ( 12*pi^2 ) ) * Vol_S^3( n_hat(v0),…,n_hat(v3) ) / vol(tet) (solid-angle approximation; orientation required).

4. Topological current (phase fields)

• S904-5: j_topo = ( 1 / ( 2*pi ) ) * ( ∇θ )^{⊥} in the distributional sense concentrated on defect cores; discretely locate w(cell) at centroids and smooth with kernel K_ρ.

5. Consistency and two-form discrepancy

• S904-6: delta_form_topo = | w(∂A) - ( ∫_{A} ( 2*pi * q_wind_2D dA ) ) | with gate delta_form_topo ≤ tol_topo.
• S904-7: check_dim( ∫_{A} q_sk_2D dA ) = "[1]", check_dim( ∫_{V} q_mono_3D dV ) = "[1]".

V. Metrology Workflow M90-4

1. Ready: confirm field type and the target density set; set ℓ_reg, choose method_fd ∈ { central, sobel, spectral }, select triangulation/tetrahedralization strategies, and define tol_topo / tol_int.
2. Preprocess:
   • Phase unwrapping: θ ← unwrap(arg(phi)), repairing jumps and micro-holes.
   • Normalization: n_hat = phi / |phi| with |phi| ≥ τ_amp thresholding, else mask.
3. Derivative evaluation: compute ∇θ and ∂_i n_a at scale ℓ_reg (consistent convolution / polynomial fitting / spectral differentiation).
4. Density construction: compute q_wind_2D / q_sk_2D / q_mono_3D / j_topo per S904-*, in parallel via two equivalent routes, and record delta_form_topo.
5. Quality & integrality checks: apply integrality gates, boundary/relative-homology corrections, and local d_B stability sampling; on failure, degrade (increase ℓ_reg, coarsen grid).
6. Persist: export density layers, masks, u/U, form discrepancies, parameters, and diagnostics to manifest.topo.density.*.

VI. Contracts & Assertions C90-41x (suggested thresholds)

• C90-4101 Two-form difference: delta_form_topo_p95 ≤ tol_topo (suggest tol_topo = 1e-2).
• C90-4102 Integrality gate: dist_to_Z( ∫_{A} q_sk_2D dA ) ≤ tol_int, dist_to_Z( ∫_{V} q_mono_3D dV ) ≤ tol_int.
• C90-4103 Regularization scale: ℓ_reg ≥ 2.5 * grid_spacing; otherwise tag under_resolved and inflate U.
• C90-4104 Orientation consistency: triangle/tetra orientation conflict rate ≤ 0.1%; else block.
• C90-4105 Ringing control: under spectral differentiation, ringing_metric ≤ tol_ring (default tol_ring = 0.05).
• C90-4106 Boundary compliance: when ∂M ≠ ∅, use relative classes or add boundary compensation; absence blocks publication.
• C90-4107 Units/dimensions: all fields must pass check_dim.

VII. Implementation Bindings I90-4*

• I90-41 unwrap_phase(phi, policy) -> θ, mask, diag
• I90-42 smooth_and_diff(field, ℓ_reg, method_fd) -> { grads, diag }
• I90-43 wind_density_from_loops(θ, grid) -> q_wind_2D (circulation route)
• I90-44 wind_density_from_curl(θ, grid) -> q_wind_2D (curl route)
• I90-45 skyrmion_density_2d(n_hat, mesh) -> q_sk_2D (Berg–Lüscher triangulation)
• I90-46 monopole_density_3d(n_hat, tetmesh) -> q_mono_3D (solid-angle / triple-Jacobian)
• I90-47 assemble_topo_current(θ, q_wind_2D, ρ) -> j_topo (sparse–smooth fusion)
• I90-48 compare_forms(q_a, q_b, domain) -> delta_form_topo
• I90-49 emit_density_layers(layers, policy) -> manifest.topo.density

Invariants: domain & measure are explicit; orientation is consistent; delta_form_topo ≤ tol_topo; integrality passes—or degrade per policy and inflate U.

VIII. Cross-References

• Defect typology & invariant reads: Chapter 3 (I90-33 / I90-35 integrate q_*).
• Persistence stability & noise robustness: Chapters 8 and 12 (guiding ℓ_reg and filtration scales).
• Atlas stitching & transition coherence: Chapter 9 (cross-chart transitions/averages of density layers).
• Error & uncertainty propagation: Appendix E (derivative/triangulation errors and spectral truncation).
• Runtime dashboards & cache/fallback: Chapter 14.

IX. Quality & Risk Control

• SLOs: density-layer success rate ≥ 99%; delta_form_topo_p95 ≤ tol_topo; integrality failures ≤ 1%.
• Fallback path: full-res → coarser-grid → larger ℓ_reg → loop-only summary; at each step increase U and tag fallback.stage.
• Audit: persist unwrapping events, derivative scales, orientation-conflict rate, two-form discrepancy distributions, integrality/boundary checks, and signature.

Summary
This chapter fixes computable routes from fields to topological densities/currents as P904 / S904 / M90-4 / C90-41x / I90-4*. Parallel two-form evaluation and integrality gates ensure results are auditable and integrable; subsequent chapters can directly integrate/aggregate q_* and j_topo to drive defect tracking and atlas stitching.