24-EFT.WP.Particle.TopologyAtlas v1.0 | Foreword

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Foreword

One-Sentence Goal
Define the scope, objects, and deliverables of the Topological Atlas, and establish a unified, engineering-ready route from raw data to topological invariants and atlas construction—complete with auditability and reproducibility.

I. Purpose and Scope

• Clarify the objects of interest: structures describable by topological invariants—particles/defects/braids/knots—embedded in a host space that is a manifold M or its discrete approximation.
• Establish a unified workflow: data acquisition → complex/filtration construction → homology/persistent homology → invariant measurement → atlas (Atlas) stitching → comparison & search → publication & audit.
• Standardize outputs into manifest.topo.*, including Atlas.hash, PD_k, invariants, contracts.*, signature, and other key fields.
• Out of scope: no derivation or solution of concrete dynamical field equations; this volume provides metrology and engineering interfaces from fields/trajectories to topological invariants and atlases. Rendering and UI are not covered.

II. Intended Readers and a Fast Path

1. Who should read:
   • Algorithms: topological data analysis and graph computation.
   • Engineering: pipelines and interfaces.
   • Operations: dashboards and contract guardianship.
2. Hands-on path: Chapter 2 (Mathematical Baseline) → Chapter 4 (Field → Topological Density) → Chapter 7 (Complexes & Filtrations) → Chapter 8 (Persistent Homology) → Chapter 9 (Atlas Construction) → Chapter 13 (Comparison & Retrieval) → Chapter 15 (Use Cases).

III. Terminology and Notation (aligned with global settings)

• All inline symbols are enclosed in backticks, e.g., phi(x,t), Xi(x,t), K, Q, β_k, PD_k, Gamma_i(t).
• Any expression with a division, integral, or composite operator must be parenthesized and declare its path/measure, e.g., ( ∫_{gamma(ell)} • d ell ), ( ∫_M • dA ).
• Conflict-name enforcement: T_fil (tension) and T_trans (transmittance) must not be mixed; strictly distinguish n (number density) from n_eff (effective refractive index).
• Unified numbering: axioms P90x-*, minimal equations S90x-*, metrology flows M90-*, implementation bindings I90-*, contracts C90-*.
• Language and units: formulas/symbols/definitions must not use Chinese; every field entering an equation must declare unit(field) and dim(field) and pass check_dim( y - f(x) ).

IV. Deliverables and Reproducibility

1. Engineering outputs
   • Atlas: coverage { (U_i, chi_i) }, transition maps, and local invariant stitching.
   • Persistent homology: { PD_k, barcodes } with stability metrics (bottleneck distance d_B, p-Wasserstein distance d_W).
   • Invariant bundle: Q (topological charge), w (winding number), Lk/Sl (linking/self-linking), β_k (Betti numbers), each with uncertainty.
2. Compliance templates
   • Contract set C90-*: gates for sample size/coverage/latency/stability, plus actions.
   • manifest.topo: hashes, windows, algorithm versions, random seeds, diagnostics, and signatures to guarantee replayable audits.
3. Quality metrics (see Appendix D): d_B_p95, β_k_drift, Q_bias, chart_overlap, SLI_latency, etc.

V. Architecture and Workflow Overview (chapter map)

• Data ingress (Ch. 1): point clouds / meshes / fields / worldlines → metadata checks and unit normalization.
• Mathematical baseline (Ch. 2): π_n, H_k, β_k, and boundary operator ∂_k.
• Object hierarchy (Ch. 3): point/line/surface defects; knot/braid typology; invariant definitions S903-*.
• Field → topological density (Ch. 4): phase unwrapping, gradient circulation, Jacobian zero-indexing.
• Worldlines and braids (Ch. 5): track Gamma_i(t) and braid-word classification.
• Event graphs (Ch. 6): creation/annihilation/reconnection and causal topological conservation.
• Complexes & filtrations (Ch. 7): K ∈ { Rips, Čech, cubical, Delaunay } with F(τ).
• Persistent homology (Ch. 8): barcodes/diagrams, stability, and uncertainty.
• Atlas stitching (Ch. 9): chart coverage, transition consistency, and orientation.
• Metrology & calibration (Ch. 10): discrete–continuous error bounds and bias correction.
• Incremental maintenance (Ch. 11): online updates, conflict merges, and versioning.
• Noise & robustness (Ch. 12): cleaning and robust estimation.
• Comparison & retrieval (Ch. 13): d_B / d_W aggregation and indexing.
• Runtime (Ch. 14): windows/caching/dashboards and fallback.
• Use cases (Ch. 15): 2D vortex braids, 3D link/knot chains, time-series point-cloud anomalies.

VI. Design Principles and Axiom Preview

1. Computability first: every invariant must expose an implementable I90-* interface with an explicit complexity bound.
2. Stability first: use stability theorems to guide filtrations and denoising; every publication includes stability metrics and U.
3. Parallel dual-form convention (cross-volume exemplar):
   • Constant factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ).
   • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
   • By analogy, when computing/publishing persistent homology and atlases, we record, in parallel, numerical-route discrepancies and stability residuals (the topological analogue of delta_form).
4. Traceability: each release must record algo.ver, seed, windows, digest.hash, and contracts.* outcomes.

VII. Data & Dependencies; Cross-Volume References

1. Inputs: gridded fields, phase/amplitude data, point-cloud trajectories, event logs, geometric meshes, and boundary conditions.
2. External anchors:
   • Physical semantics of energy/fields, cleaning strategies, and energy conservation: see companion white paper Energy Filaments, Chapters 2, 3, and 6.
   • Runtime constraints—freshness/coverage/latency: see Energy Filaments, Chapter 14.
3. Evidence chain: reuse the cross-volume manifest and signature standards for provenance and attestation.

VIII. Compliance, Audit, and Safety

• Any field entering equations must declare units/dimensions and pass check_dim; statistical statements must carry u(x), U = k * u_c, nu_eff.
• Prior to publication, contracts C90-* must pass: coverage, sample size, stability, and latency gates. Violations trigger degrade/fallback and uncertainty inflation.
• Audit artifacts: input hashes, parameter snapshots, random seeds, diagnostics, and signatures to ensure offline replay.

IX. Reading Guide and Contribution Policy

• Suggested order: Foreword → Ch. 2 → Chs. 7–9 → Ch. 8 → Chs. 13–15. Ops teams should prioritize Ch. 14 and Appendix D.
• Change process: record breaking interface changes and migration guidance in Appendix G; every new I90-* / C90-* requires a minimal use case and a complexity note.

Summary
This volume provides an executable framework from data to Topological Atlases, spanning the mathematical baseline, engineering interfaces, and quality & audit controls. Subsequent chapters proceed along the P/S/M/I/C rails to ensure invariants are computable, atlases maintainable, publications auditable, and runtime stable.