24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 6 — Spacetime Event Graphs and Topological Transitions

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Chapter 6 — Spacetime Event Graphs and Topological Transitions

One-Sentence Goal
Using worldlines and defect evolution as primary objects, construct a spacetime event graph G_event and a computable taxonomy of topological transitions together with conservation/jump laws, so that invariants are conserved on nonsingular segments and auditable at events.

I. Scope and Objects

1. Inputs
   • Worldlines & defect tracks: Γ = { Gamma_i(t) } (see Chapter 5), defect sets and invariants (Chs. 3–4).
   • Density layers & evidence: q_*, j_topo (Ch. 4), geometric context, and boundary ∂M.
   • Time axes & bases: event time t ∈ [t0,t1], publication time ts, monotone base tau_mono.
2. Outputs
   • Event graph G_event = (V_event, E_event): nodes are events or steady segments; edges encode temporal/associative links.
   • Event types & jumps: type ∈ { birth, death, merge, split, reconnection, phase-slip, boundary-hit, Reidemeister-II/III, … }; Δinv = { ΔQ, ΔLk, ΔSl, … }.
   • Compliance report & manifest: manifest.topo.events (conservation checks, form discrepancies, diagnostics).
3. Boundaries & constraints
   • Generic position & finite energy: away from events, curves are regular and θ/n_hat are differentiable; events are isolated instants or short windows.
   • Relative homology / boundary terms must be enabled when ∂M ≠ ∅.

II. Terms and Variables

• Event node: v ∈ V_event, annotated with type(v), ts(v), Δinv(v).
• Steady segment: seg = (t_a, t_b, id_set) on which invariants remain constant.
• Conserved quantities & jumps: inv = { Q, w, Lk, Sl, Tw, Wr }; Δinv = inv^+ − inv^-.
• Event graph: a DAG G_event topologically ordered by t; E_event(u→v) expresses causality and tracking association.
• Quality & uncertainty: u(Δinv), U = k * u_c, nu_eff; dual-form discrepancy delta_form_evt.

III. Axioms P906-*

• P906-1 (Off-event conservation) — On nonsingular intervals, d inv / dt = 0 (including Q, Lk, Sl).
• P906-2 (Locality) — Events are local: a radius ρ_evt exists such that Δinv depends only on geometry/densities within B(ρ_evt).
• P906-3 (Paired creation/annihilation) — For births/deaths away from boundaries, net charge is conserved: Σ ΔQ = 0.
• P906-4 (Reconnection allowance) — reconnection can produce ΔLk, ΔSl ≠ 0, determined by crossing changes and local geometry.
• P906-5 (Parallel two-form routes) — Every ΔLk is evaluated both by crossing-difference and Gauss-difference, and delta_form_evt is recorded.
• P906-6 (Causal acyclicity) — G_event is topologically ordered by t and acyclic; cross-segment tracking obeys monotone timing and matching thresholds.
• P906-7 (Boundary consistency) — At ∂M, events are represented in relative homology / boundary flux; Δinv includes boundary terms.

IV. Minimal Equations S906-*

1. Net-charge conservation (local pairs)

• S906-1: for a birth/death window W,
  Σ_{v∈W} ΔQ(v) + Φ_boundary(W) = 0,
  where Φ_boundary is the relative flux through the boundary during W.

2. Linking-number jumps (two routes)

• S906-2 (crossing route): within an event window,
  ΔLk(Γ_i, Γ_j) = ( 1 / 2 ) * ( Σ_{c∈C_ij^+} sgn(c) − Σ_{c∈C_ij^-} sgn(c) ),
  where + / − denote post-/pre-event crossing sets.
• S906-3 (Gauss-difference):
  ΔLk = ( 1 / ( 4*pi ) ) * ( ∬_{Γ_i×Γ_j}^{t^+} … − ∬_{Γ_i×Γ_j}^{t^-} … ).
• S906-4: delta_form_evt = | ΔLk_cross − ΔLk_gauss |.

3. Self-linking jumps

• S906-5: ΔSl = ΔTw + ΔWr, where ΔWr uses the principal-value difference of the self-double integral and ΔTw comes from framing twist changes.

4. Phase slips

• S906-6: in 2-D phase fields, if a 2π phase-slip occurs then
  Δw(∂A) = ( 1 / ( 2*pi ) ) * ( ∮_{∂A}^{t^+} dθ − ∮_{∂A}^{t^-} dθ ) = k ∈ Z,
  paired with Q birth/death.

5. Reidemeister-type transitions (in projection)

• S906-7:
  • Type II: pair creation/annihilation of crossings, ΔΣ sgn(c) = 0, ΔLk = 0.
  • Type III: reorder crossings only, ΔLk = 0.
  • Type I: affects self-crossing, may change Wr and projection-based metrics, but not Lk.

6. Graph-level conservation checks

• S906-8: for any connected non-boundary subgraph H ⊂ G_event,
  Σ_{v∈H} ΔQ(v) = 0, Σ ΔLk(v) = 0, Σ ΔSl(v) = 0.

V. Metrology Workflow M90-6

• Ready: choose the time grid and event windows W_evt = [ t_k − τ, t_k + τ ]; set ρ_evt, matching/unassociation thresholds, tol_evt.
• Track association & segmentation: from Γ derive steady segments seg, detect breakpoints and candidate event times.
• Local evidence gathering: within W_evt × B(ρ_evt), sample q_*, j_topo, nearest distances, curvature, crossing sets, and boundary flux.
• Event classification: label { birth, death, merge, split, reconnection, phase-slip, boundary-hit, Reidemeister-* } using geometric/density evidence and output confidence.
• Jump quantification: compute ΔQ, ΔLk, ΔSl (= ΔTw + ΔWr) in parallel routes with u/U, yielding delta_form_evt.
• Build G_event: create nodes/edges with type, ts, Δinv, evidence.hash; topologically sort by time; merge duplicates.
• Conservation & coherence checks: enforce S906-1/8 and boundary consistency; on failure, revisit classification or enlarge W_evt / ρ_evt.
• Persist:
  manifest.topo.events = { G_event, Δinv, delta_form_evt, quality, params, algo.ver, seed }, with signature.

VI. Contracts & Assertions C90-61x (suggested gates)

• C90-6101 Off-event constancy: on any steady segment, | inv(t_b) − inv(t_a) | ≤ tol_inv.
• C90-6102 Net-charge conservation: for any non-boundary component H, | Σ ΔQ | ≤ tol_int.
• C90-6103 Two-form coherence: delta_form_evt_p95 ≤ tol_evt (suggest tol_evt = 0.05).
• C90-6104 Causality: G_event has no directed cycles; topological ordering exists.
• C90-6105 Sufficient evidence: any event with nonzero Δinv must carry density/geometric evidence and boundary annotations.
• C90-6106 Boundary compliance: boundary-hit must include relative flux Φ_boundary; absence blocks publication.
• C90-6107 Quality gate: classification confidence ≥ p_min (suggest p_min = 0.8); below-threshold events are tagged ambiguous and U is inflated.
• C90-6108 Units/dimensions: check_dim(ts) = "[T]", check_dim(Δinv) = "[1]".

VII. Implementation Bindings I90-6*

• I90-61 segment_worldlines(Γ, policy) -> { segs, breakpoints }
• I90-62 gather_local_evidence(data, W_evt, ρ_evt) -> evidence
• I90-63 classify_events(segs, evidence) -> { events, confidence }
• I90-64 delta_invariants_before_after(Γ, events) -> { ΔQ, ΔLk, ΔSl, u/U, delta_form_evt } (crossing & Gauss routes in parallel)
• I90-65 build_event_graph(events, segs) -> G_event
• I90-66 check_conservation(G_event, Δinv, boundary) -> report
• I90-67 assert_event_contracts(results, rules) -> report
• I90-68 emit_event_manifest(G_event, results, policy) -> manifest.topo.events

Invariants: monotone time; parallel two-form evaluation; conservation checks pass; boundary policy explicit.

VIII. Cross-References

• Defects & invariant sources: Chapter 3; densities & topological currents: Chapter 4.
• Worldlines, braids, and linking: Chapter 5 (event ΔLk / ΔSl depends on its gauges).
• Complexes & persistence stability (event robustness and scale selection): Chapters 7–8.
• Atlas stitching & transition coherence: Chapter 9 (align events across chart overlaps).
• Uncertainty propagation & error models: Chapter 10 and Appendix E.
• Runtime & dashboards: Chapter 14; manifest keying: Appendix C.

IX. Quality & Risk Control

• SLOs: event-detection recall ≥ 0.98; delta_form_evt_p95 ≤ tol_evt; net-charge conservation failure ≤ 1%; G_event cycle-free rate 100%.
• Fallback path: full evidence → reduced geometry → crossings-only → time-slice diffs, inflating U and tagging fallback.stage stepwise.
• Audit: persist event-window volumetric slices, crossing sets, boundary fluxes, Gauss-difference logs, and seeds; emit evidence.hash and signatures for replay.

Summary
This chapter provides the engineering convention for spacetime events and topological transitions as P906 / S906 / M90-6 / C90-61x / I90-6*. Through the event graph, two-form differencing, conservation checks, and boundary consistency, it ensures that invariants undergo auditable jumps during evolution, supporting atlas stitching and query at scale.