24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 2 — Mathematical Baseline (Manifolds / Homotopy / Homology / Bundles)

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Chapter 2 — Mathematical Baseline (Manifolds / Homotopy / Homology / Bundles)

One-Sentence Goal
Establish the minimal mathematical substrate—manifold → atlas → complex → homotopy/homology → fiber bundles—on which the Topological Atlas depends, and ground the engineering notion of “computable invariants” within the P/S/M/I/C (Axioms/Equations/Metrology/Interfaces/Contracts) framework.

I. Scope and Objects

• Objects: oriented or non-oriented manifolds M, possibly with boundary; local charts chi_i and an atlas Atlas; chain complexes C_k with boundary maps ∂_k; homotopy groups π_n and homology groups H_k; principal/vector bundles with connection/curvature.
• Inputs: manifold_spec = { M, ∂M?, orientable?, metric? g }, candidate charts { (U_i, chi_i) }, coefficient field coeffs, a discrete model K (simplicial/cubical, etc.), and—when needed—fields phi(x,t) or worldlines Gamma_i(t).
• Outputs: mathematical consistency checks, domains for computable invariants, engineering constraints (complexity, tolerances), and anchors for persistence into manifest.topo.*.
• Boundaries: algorithmic optimizations and runtime dashboards are deferred to Chapter 14; physical dynamics are covered in the companion Energy Filaments white paper.

II. Terms and Variables

• Manifold & atlas: M, U_i ⊂ M, chi_i: U_i → R^d, Atlas = { (U_i, chi_i) }.
• Complexes & filtrations: K (K ∈ { simplicial, cubical, rips, cech, delaunay }), F(τ) (a filtration with F(τ1) ⊆ F(τ2)).
• Chains & boundaries: C_k(K; coeffs), ∂_k: C_k → C_{k-1}, H_k(K; coeffs) = ker(∂_k) / im(∂_{k+1}), β_k = rank H_k.
• Homotopy: π_n(M, x0); when appropriate, approximate generators of π_1 via graph-based shortest cycle bases.
• Fiber bundles: principal/vector bundles P → M, connection A, curvature F = dA + A∧A.
• Worldlines & measures: Gamma(t), path integrals ( ∫_{gamma(ell)} • d ell ), explicit domains/measures such as ( ∫_S • dA ), ( ∫_V • dV ).
• Metrology & units: unit(field), dim(field); topological invariants such as Q, β_k, Lk are, by default, dimensionless "[1]".

III. Axioms P902-*

• P902-1 (Atlas Regularity) — Atlas provides coverage ⋃_i U_i = M; the regularity (continuous / differentiable / homeomorphic) and orientation of chi_j ∘ chi_i^{-1} are specified at the level required by downstream computation.
• P902-2 (Coefficient Consistency) — The homology coefficient field coeffs ∈ { Z, Z_p, R } remains fixed across the entire pipeline.
• P902-3 (Algebraic Boundary Consistency) — ∂_k ∘ ∂_{k+1} = 0 must pass numerical verification; otherwise publication is blocked.
• P902-4 (Discrete–Continuous Alignment) — Discretization from M to K preserves bounds on connectivity and fundamental-group generators; provide error bounds where necessary.
• P902-5 (Explicit Measure) — Every integral declares its domain and measure; probabilistic densities are handled separately from geometric/physical densities.
• P902-6 (Computability) — Each invariant exposes an executable I90-* interface with an explicit asymptotic complexity bound.

IV. Minimal Equations S902-*

1. Atlas and chart transitions

• S902-1 Coverage: ⋃_i U_i = M.
• S902-2 Transition chain rule:
  D(chi_j ∘ chi_i^{-1})(p) = D chi_j (chi_i^{-1}(p)) • D(chi_i^{-1})(p)；
  if a metric g is provided, pull back across charts: g_i = (chi_i^{-1})^* g.

2. Chain complex and homology

• S902-3 Chain complex: … → C_{k+1} \xrightarrow{∂_{k+1}} C_k \xrightarrow{∂_k} C_{k-1} → … with ∂_k ∘ ∂_{k+1} = 0.
• S902-4 Homology & Betti: H_k(K; coeffs) = ker(∂_k)/im(∂_{k+1}), β_k = rank H_k.
• S902-5 Euler characteristic: χ(K) = ∑_{k=0}^d (-1)^k β_k = ∑_{k=0}^d (-1)^k f_k (where f_k counts k-simplices).

3. Homotopy and degree (examples)

• S902-6 Degree (2-D phase field example):
  deg(φ; S^1) = ( 1 / 2π ) * ( ∮_{γ} dθ ), where γ encircles a singularity and dθ is the phase increment.
• S902-7 Gauss linking integral (knots/links):
  Lk(Γ1, Γ2) = ( 1 / 4π ) * ( ∬_{Γ1×Γ2} ( ( (r1 - r2) / |r1 - r2|^3 ) ⋅ ( dr1 × dr2 ) ) ), with r1, r2 the parameterized positions.

4. Fiber bundles and topological charge (examples)

• S902-8 First Chern number: c1 = ( 1 / 2π ) * ( ∫_{S} F ), where F is the curvature 2-form and S ⊂ M is a closed surface.
• S902-9 Pontryagin / Chern–Simons (volume integral):
  CS(A) = ( 1 / 4π ) * ( ∫_{V} tr( A ∧ dA + (2/3) A ∧ A ∧ A ) )，with domain V and differential-form calculus declared.

5. Persistence placeholders

• S902-10 Filtration coherence: F(τ1) ⊆ F(τ2), PD_k = { (b_i, d_i) }; stability theorems are developed in Chapter 8—here we fix symbols and domains.

V. Metrology Workflow M90-2 (Ready → Model → Verify → Persist)

1. Ready: load manifold_spec and candidate atlas; lock coeffs; declare whether a metric g is used.
2. Model:
   • Register charts and transitions; for discrete settings, generate K and F(τ).
   • Select an invariant set { β_k, χ, deg, Lk, c1, … } with their computational domains/paths.
3. Verify:
   • Check ∂_k ∘ ∂_{k+1} = 0; cross-check χ via both β_k and f_k.
   • Validate atlas coverage/orientation; perform randomized pullback tests on overlaps.
4. Persist:
   • manifest.topo.math = { coeffs, atlas.hash, trans.check, chi, beta, euler, invariants:set, metric? }.
   • Record algo.ver, seed, numerical tolerances, and diagnostics.

VI. Contracts & Assertions C90-21x (suggested gates)

• C90-2101 Coverage: measure(⋃_i U_i)/measure(M) ≥ 0.98.
• C90-2102 Transition consistency: max_{p∈U_i∩U_j} angle( det D(chi_j ∘ chi_i^{-1})(p) ) ≤ tol_orient.
• C90-2103 Zero boundary composite: ||∂_k ∘ ∂_{k+1}|| = 0 (numerically ≤ tol_boundary).
• C90-2104 Euler double-count coherence: | χ_{β} - χ_{f} | ≤ 0 (exact equality for discrete integers); with down-sampling, allow ≤ 1 and tag degrade.
• C90-2105 Integrality of degree/linking: dist_to_Z(deg) ≤ tol_int, dist_to_Z(Lk) ≤ tol_int.
• C90-2106 Integrality of Chern numbers: dist_to_Z(c1) ≤ tol_int; for non-closed domains, include boundary compensation or block.

VII. Implementation Bindings I90-* (chapter interfaces)

• I90-21 register_charts(charts) -> Atlas — returns coverage, overlap statistics, and transition diagnostics.
• I90-22 align_transitions(Atlas) -> transitions — evaluates regularity/orientation of chi_j ∘ chi_i^{-1}.
• I90-23 build_chain_complex(K, coeffs) -> { C_k, ∂_k }
• I90-24 compute_homology(C_k, ∂_k) -> { H_k, β_k, χ }
• I90-25 degree_curve(phi, γ) -> deg — phase unwrapping + closed-loop integration.
• I90-26 gauss_linking(Γ1, Γ2) -> Lk
• I90-27 chern_number(A, S) -> c1 — discrete exterior calculus / Stokes assembly.
• I90-28 assert_math_contracts(results, rules) -> report
• I90-29 emit_math_manifest(results, policy) -> manifest.topo.math

Invariants: ∂_k ∘ ∂_{k+1} = 0; atlas coverage and orientation consistency; integrality dist_to_Z(•) ≤ tol_int.

VIII. Cross-References

• Data → complex & filtrations: Chapter 7; persistence & stability: Chapter 8.
• Defect/braid objects & instantiated invariants: Chapters 3–5.
• Atlas construction & transition engineering: Chapter 9.
• Topological uncertainty and error propagation: Appendix E.
• Runtime and publication manifests: Chapter 14 & Appendix C.

IX. Quality & Risk Control

• SLOs (suggested): register_charts success ≥ 99%; compute_homology completes within T_budget at ~10⁶ simplices; p95 integrality deviation ≤ tol_int.
• Fallback: if ∂∘∂ ≠ 0 or integrality fails, downgrade to coarser filtrations / lower-order complexes or reduce the domain; mark fallback in manifest.topo.math and inflate uncertainty.
• Audit: persist input hashes, complex summary (f_k), coeffs, atlas coverage/transition diagnostics, integrality reports, and signatures.

Summary
This chapter fixes the core definitions and computable conventions of atlases, complexes, homotopy/homology, and bundles as P902 / S902 / M90-2 / C90-21x / I90-2*. Consequently, any downstream use case can compute and publish topological invariants under explicit domain/measure/coefficients/integrality constraints, stably persisted to manifest.topo.math.