24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 1 — Definition and Scope of the Topological Atlas Domain

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Chapter 1 — Definition and Scope of the Topological Atlas Domain

One-Sentence Goal
Specify the inputs (point clouds / meshes / fields / worldlines), outputs (Atlas, PD_k, invariants, manifest.topo), and boundaries (static/dynamic, single/multi-domain, with/without metric), establishing a single coherent convention for computation and publication that downstream chapters can rely on.

I. Scope and Objects

1. Input objects
   • Spacetime fields: phi(x,t), Xi(x,t) (phase / vector / order-parameter grids); units and dimensions follow the source data.
   • Geometric data: point clouds / triangle meshes / voxel grids points|mesh|voxel; trajectories / worldlines Gamma_i(t).
   • Metadata: coeffs (homology coefficient field), manifold_spec = { M, boundary, orientable, metric? g }, and RefCond (sampling cadence / windows).
2. Outputs
   • Atlas: Atlas = { (U_i, chi_i, inv_i) }, transition maps chi_j ∘ chi_i^{-1} with consistency diagnostics.
   • Persistent homology: PD_k = { (b_i, d_i) }, barcodes, and stability metrics.
   • Invariant bundle: Q, w, Lk, Sl, β_k with associated uncertainties.
   • Publication manifest: manifest.topo.
3. Boundaries & assumptions
   • Host space M is a 2D/3D manifold or a discrete approximation; may include boundary ∂M and/or a metric g.
   • Time semantics are dual: event time vs processing time; both are persisted as tau_mono/ts (shared cross-volume convention).
   • Out of scope: solving dynamical field equations and visualization (see the companion Energy Filaments white paper, Chs. 2/6).

II. Terms and Variables

• Carrier & atlas: M, U_i ⊂ M, chi_i: U_i → R^d, Atlas.
• Complexes & homology: K, F(τ), C_k, ∂_k, H_k(K; coeffs), β_k = rank H_k.
• Persistence objects: PD_k, d_B, d_W.
• Defects & invariants: Q (topological charge, dim(Q) = "[1]"), w, Lk, Sl.
• Metrology & time: ts, tau_mono, u(x), U = k * u_c, nu_eff.
• Units & dimensions: every field entering an equation declares unit(field), dim(field) and passes check_dim().

III. Axioms P901-*

• P901-1 Atlas coverage: ⋃_i U_i = M; near ∂M, charts extend into the outward normal neighborhood.
• P901-2 Transition regularity: each chi_j ∘ chi_i^{-1} meets the regularity required by downstream computation (continuous / differentiable / homeomorphic) and is orientation-consistent.
• P901-3 Explicit coefficient field: coeffs ∈ { Z, Z_p, R } is mandatory and remains fixed end-to-end.
• P901-4 Discrete–continuous coherence: construction of the discrete complex K must preserve upper/lower bounds on connectivity and fundamental-group features.
• P901-5 Separate probability from topology: statistical weights affect uncertainty propagation only; they do not alter the domains of topological operators.
• P901-6 Reproducibility: persist algo.ver, seed, windows, numerical tolerances, and cache policy.

IV. Minimal Equations S901-*

• S901-1 Atlas and transitions: Atlas = { (U_i, chi_i) }, ⋃_i U_i = M.
• S901-2 Chain complex and boundary: ∂_k ∘ ∂_{k+1} = 0, H_k = ker(∂_k) / im(∂_{k+1}).
• S901-3 Betti numbers: β_k = rank H_k(K; coeffs).
• S901-4 Persistent homology: for a filtration F(τ), PD_k = { (b_i, d_i) }, with stability d_B(PD_k(X), PD_k(Y)) ≤ ||f_X - f_Y||_∞.
• S901-5 Topological charge (example): Q = ( ∫_{S} j_topo ⋅ dA ), with declared integration domain and measure dA.
• S901-6 Data → complex: K = build_complex(data, method, params), method ∈ { rips, cech, cubical, delaunay }.

V. Metrology Workflow M90-1

• Ready: validate metadata and manifold_spec; declare coeffs; set windows / sampling and tolerances.
• Complex construction: K ← build_complex(points|mesh|voxel, method, params) and derive filtration F(τ).
• Homology / persistence: compute H_k, β_k, PD_k, and stability metrics d_B/d_W.
• Invariant estimation: from fields/trajectories, compute Q, w, Lk, Sl and produce u_c, U.
• Atlas assembly: register the Atlas and transition maps, and merge local invariants inv_i.
• Checks & contracts: execute C90-11x (coverage / consistency / stability / latency).
• Publish: generate manifest.topo, including hashes, parameters, diagnostics, and signature.

VI. Contracts & Assertions C90-11x (suggested)

• C90-1101 Coverage: measure(⋃_i U_i) / measure(M) ≥ 0.98; otherwise degrade/merge_charts.
• C90-1102 Transition consistency: max_i || D(chi_j ∘ chi_i^{-1}) - I || ≤ tol_trans (for locally Euclidean approximations).
• C90-1103 Boundary-operator check: ||∂_k ∘ ∂_{k+1}|| = 0; nonzero implies block.
• C90-1104 Persistence stability: d_B(PD_k(data), PD_k(boot)) ≤ tol_PD.
• C90-1105 Time freshness: ts_now - ts_data ≤ T_max; otherwise use cache/fallback and inflate uncertainty.
• C90-1106 Units & dimensions: all fields pass check_dim; failures block publication.

VII. Implementation Bindings I90-* (chapter-local)

• I90-11 validate_topo_inputs(meta) -> report
• I90-12 declare_domain(manifold_spec) -> M
• I90-13 build_complex(data, method, params) -> { K, F }
• I90-14 compute_topo_invariants(K, field, tracks) -> { β_k, PD_k, Q, w, Lk, Sl }
• I90-15 assemble_topology_atlas(charts, invariants) -> Atlas
• I90-16 assert_domain_contracts(objects, rules) -> report
• I90-17 emit_topo_manifest(results, policy) -> manifest.topo

Invariants: ⋃U_i = M; ∂_k ∘ ∂_{k+1} = 0; manifest.topo includes algo.ver / seed / windows / contracts.*.

VIII. Cross-References

• Complex / persistence algorithms: Chs. 7–8; atlas construction and transitions: Ch. 9.
• Invariant metrology & uncertainty: Ch. 10 and Appendix E.
• Noise & cleaning: Ch. 12; runtime & dashboards: Ch. 14; use cases: Ch. 15.
• Energy/physical semantics, cleaning, and runtime constraints: companion Energy Filaments white paper, Chs. 2/6/14.

IX. Quality & Risk Control

• SLOs: build_complex success ≥ 99%; PD_k timeout ≤ 1%; d_B_p95 held within baseline ±20%.
• Fallback ladder: live → cached → reduced_K → summary_PD, with stepwise U inflation.
• Audit: persist input hashes, coefficient field, windows, parameters, numerical tolerances, random seeds, and contract outcomes.

Summary
This chapter fixes the objects, I/O, and boundaries of the topological-atlas domain and operationalizes them via P901-* / S901-* / M90-1 / C90-11x / I90-*. Subsequent chapters elaborate computation, stitching, stability, and runtime execution against this domain definition, with all deliverables persisted under manifest.topo.*.