24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 5 — Worldlines, Braids, and Knots (Spacetime Topology)

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Chapter 5 — Worldlines, Braids, and Knots (Spacetime Topology)

One-Sentence Goal
Elevate particle/defect worldlines Gamma_i(t) into computable braid and knot objects, defining an engineering route from time-ordered observations to topological invariants—braid word, Lk, Sl, Tw/Wr—that remains auditable and persistable in discrete spacetime.

I. Scope and Objects

1. Inputs
   • Worldlines / trajectories: Γ = { Gamma_i: [t0, t1] → R^3 } (or punctured-plane trajectories in R^2 × [t0,t1]).
   • Fields & densities (optional): phi(x,t), q_* (from Chapter 4) for assisting localization/tracking and integrality checks.
   • Projections & views: π_view: R^3 → R^2 or π_axis: R^3 → R to establish braid ordering.
   • Metadata: coeffs, ts / tau_mono, sampling interval Δt, noise and filtering policies.
2. Outputs
   • Braids & knots: braid word w ∈ B_n, its permutation perm(w), and Garside normal form; after closure, a summary of link/knot types.
   • Invariants: Lk(Γ_i, Γ_j), Sl(Γ_i) = Tw + Wr, crossing counts, and an event log.
   • Manifest: manifest.topo.braid (parameters, two-form discrepancies, diagnostics, signature).
3. Boundaries & constraints
   • Assume generic position for the projection by default: no instantaneous triple crossings or tangencies; when degeneracies occur, break them with perturbed π_view' or temporal jitter and record it.
   • Link/knot analysis applies to closed worldlines or open trajectories closed by a declared policy; closure rules must be persisted.

II. Terms and Variables

• Worldlines & parameterization: Gamma_i: [t0,t1] → R^3, unit tangent t_hat = d Gamma_i / dt / |•|.
• Braid group & generators: B_n = ⟨ σ_1, …, σ_{n-1} | σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, σ_i σ_j = σ_j σ_i (|i-j|>1) ⟩.
• Braid word: w = σ_{k1}^{ε1} σ_{k2}^{ε2} …, ε ∈ {+1, -1}; permutation map perm: B_n → S_n.
• Projection & crossing sign: π_view(•); crossing c has sgn(c) ∈ {+1, -1}.
• Link invariants: Lk, self-linking Sl, twist/writhe Tw/Wr.
• Events: E = {birth, death, swap, reconnection, boundary-hit}, with ts(e).
• Metrology: u(x), U = k * u_c, nu_eff; dual-form discrepancy delta_form_braid/link.

III. Axioms P905-*

• P905-1 (Temporal monotonicity) — Any worldline parameter uses event time t ∈ [t0,t1] monotonically; publication time ts is recorded separately.
• P905-2 (Generic projection) — The projection π_view used for braid computation must be in general position; otherwise perturb π_view or time to break degeneracies and log the action.
• P905-3 (Parallel two-form routes) — Compute Lk both via Gauss integral and crossing sum, recording delta_form_link.
• P905-4 (Closure policy) — For open trajectories, knot/link analysis requires closure_policy ∈ { Alexander, plat, periodic, geodesic }.
• P905-5 (Boundary & topological events) — Except for explicit reconnection / birth / death events, Lk and Sl are invariant under continuous deformations; such events must be detected and persisted.
• P905-6 (Reproducibility) — Persist π_view, ordering rules, filtering scale, thresholds, closure policy, and algo.ver / seed.

IV. Minimal Equations S905-*

1. Equivalence of crossing sum and Gauss integral

• S905-1 (crossing route): for two oriented components Γ_i, Γ_j under a regular projection π_view,
  Lk(Γ_i, Γ_j) = ( 1 / 2 ) * ( Σ_{c ∈ C_ij} sgn(c) ),
  where C_ij includes only inter-component crossings, stabilized within discrete time windows.
• S905-2 (integral route):
  Lk(Γ_i, Γ_j) = ( 1 / ( 4*pi ) ) * ( ∬_{Γ_i×Γ_j} ( ( (r_i - r_j) / |r_i - r_j|^3 ) ⋅ ( d r_i × d r_j ) ) ).
• Two-form discrepancy: delta_form_link = | Lk_cross − Lk_gauss |.

2. Braid-word generation

• S905-3: let x_i(t) = π_axis( Γ_i(t) ) define the instantaneous ordering; whenever t* satisfies x_k(t*) = x_{k+1}(t*) and the projected crossing has sgn(c) = ±1, append σ_k^{sgn(c)} to w in temporal order.
• S905-4: the permutation perm(w) equals the product of adjacent swaps accumulated in time.

3. Self-linking decomposition & projection coupling

• S905-5: with a chosen framing f_hat, Sl(Γ_i) = Tw(Γ_i, f_hat) + Wr(Γ_i), where writhe may be computed by the principal-value integral
  Wr(Γ) = ( 1 / ( 4*pi ) ) * ( ∬_{Γ×Γ} ( ( (r − r') / |r − r'|^3 ) ⋅ ( d r × d r' ) ) ).

4. Events and conservation

• S905-6: without reconnection/boundary events, d Lk / dt = 0, d Sl / dt = 0.
• S905-7: upon reconnection(Γ_a, Γ_b), allow ΔLk ≠ 0; record ΔLk, Δ|components|, and local geometric evidence (curvature / nearest-distance).

5. Closure and knot invariants (placeholders)

• S905-8: applying closure_policy to endpoints yields an oriented link K = closure(Γ); compute projection-based invariants (e.g., crossing number, an interface to the Alexander polynomial), and persist method/parameters in the manifest.

V. Metrology Workflow M90-5

1. Ready
   • Choose π_view and π_axis; set sampling Δt, filtering scales, denoising; declare closure_policy if applicable.
   • Validate continuity and monotonic time for Γ: non_decreasing(t), with gap handling.
2. Worldline construction & ordering
   Track piecewise-linear Γ_i(t) from defect centers or density peaks (Chs. 3/4); build instantaneous order via π_axis and stabilize within Δt.
3. Crossing detection & sign
   Detect crossings c in π_view; assign sgn(c) by right-hand rule and oriented tangents; remove spurious crossings (depth occlusions / distinct layers).
4. Braid word & permutation accumulation
   Append σ_k^{±1} in temporal order; normalize using the braid relations (σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} and far commutativity) via Garside / left normal form.
5. Invariant estimation
   Compute Lk in both routes and Sl/Tw/Wr; report u_c, U. If closed, compute link/knot projection invariants.
6. Event cataloging
   Identify { birth, death, swap, reconnection, boundary-hit }; record ts, local geometry, and ΔLk / ΔSl.
7. Checks & persistence
   Enforce C90-51x; generate
   manifest.topo.braid = { π_view, π_axis, Δt, w, perm, Lk, Sl, Tw, Wr, delta_form_link, events, closure_policy?, algo.ver, seed }, and sign it.

VI. Contracts & Assertions C90-51x (suggested)

• C90-5101 Temporal monotonicity: timestamps on each Gamma_i strictly increase; violations block publication.
• C90-5102 Projection regularity: share of triple crossings/tangencies ≤ 0.1%; exceedances require perturbation/degrade and tag projection.degenerate=true.
• C90-5103 Two-form coherence: delta_form_link_p95 ≤ tol_link (suggest tol_link = 0.05).
• C90-5104 Event compliance: if ΔLk ≠ 0 then a reconnection/boundary-hit record with geometric evidence must exist.
• C90-5105 Permutation consistency: perm(w) equals the permutation of the final strand order; failure blocks.
• C90-5106 Closure compliance: when closure is used, closure_policy is mandatory and compatible with the invariant calculators.
• C90-5107 Units/hashes: check_dim(Lk) = check_dim(Sl) = "[1]"; Γ, w, events, params all carry hashes and signatures.

VII. Implementation Bindings I90-5*

• I90-51 track_worldlines(cands, Δt, policy) -> Γ, diag (data association / interpolation / gap repair)
• I90-52 braid_word_from_crossings(Γ, π_view, π_axis, Δt) -> { w, perm, crossings }
• I90-53 linking_matrix(Γ) -> { Lk_ij } (crossing & Gauss routes in parallel; returns delta_form_link)
• I90-54 self_linking_decompose(Γ, frame_policy) -> { Sl, Tw, Wr }
• I90-55 detect_topo_events(Γ, crossings, boundary, tol) -> events
• I90-56 normalize_braid_word(w, method) -> w_norm (Garside / left normal form)
• I90-57 close_and_summarize(Γ, closure_policy) -> { K, invariants } (placeholder; returns projection-based metrics)
• I90-58 assert_braid_contracts(results, rules) -> report
• I90-59 emit_braid_manifest(results, policy) -> manifest.topo.braid

Invariants: non_decreasing(t); delta_form_link ≤ tol_link; perm(w) matches the final state; event logs are complete.

VIII. Cross-References

• Defect localization and invariant sources: Chs. 3 & 4 (q_*, j_topo seed worldlines and supply integrality checks).
• Complex/persistence stability: Chs. 7–8 (robust events and window choices).
• Atlas stitching & transition coherence: Ch. 9 (harmonization of w / Lk / Sl across overlapping charts).
• Uncertainty propagation: Ch. 10 & Appendix E (crossing jitter, projection perturbations, Gauss-integral discretization).
• Runtime and publication: Ch. 14; manifest keys: Appendix C.

IX. Quality & Risk Control

• SLOs: crossing-detection F1 ≥ 0.99; delta_form_link_p95 ≤ tol_link; braid normalization success ≥ 0.999; event miss-rate ≤ 1%.
• Fallback path: full 3D Gauss → sparse Gauss + crossing sum → crossing-only (no Gauss) → order-only (perm); inflate U and tag fallback.stage at each step.
• Audit: store keyframes, crossing slices, π_view / π_axis, local geometry near events, and nearest-distance curve segments; attach signatures and verification hashes.

Summary
This chapter turns time-ordered worldlines into computable braid/knot objects, formalized as P905 / S905 / M90-5 / C90-51x / I90-5*. With two-form coherence, event compliance, and normalized closure, invariants such as w / Lk / Sl remain auditable and reproducible in discrete spacetime, with manifest.topo.braid.* as the publication anchor.