24-EFT.WP.Particle.TopologyAtlas v1.0 | Chapter 11 — Incremental Maintenance and Merging of the Atlas

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Chapter 11 — Incremental Maintenance and Merging of the Atlas

One-Sentence Goal
Maintain the Topological Atlas incrementally with auditable patches and commutative merges, so coverage, transitions, and invariants remain consistent, stable, and roll-backable under new data, new regions, and parallel team edits.

I. Scope and Objects

1. Inputs
   • Existing atlas: Atlas = { (U_a, phi_a, K_a(τ), D_a, Γ_a, G_{event,a}), T_ab, ψ_a } (see Chapter 9).
   • New data or peer atlas: data', Atlas', and candidate diffs { ΔK, ΔD, ΔΓ, ΔT, Δψ }.
   • Policies & thresholds: policy = { tol_match, tol_cyc, tol_glue, tol_cover, tol_fresh }, version and timestamp ver, ts.
2. Outputs
   • Incremental patch: patch = { ops, scope, evidence, signatures }, with operation semantics below.
   • Merged result: Atlas* (after applying patch or merge(Atlas, Atlas')) and rollback snapshot.
   • Reports & manifests: manifest.topo.atlas.patch, manifest.topo.atlas.merge.
3. Boundaries & constraints
   • Preserve Σ_a ψ_a(x) ≡ 1 (within numerical tolerance) and non_decreasing(τ).
   • Update T_ab / ψ_a only in affected regions; any global invariant change must have causal evidence.

II. Terms and Variables

• Atomic ops op ∈ { ADD_CHART, DROP_CHART, UPDATE_CHART, UPDATE_TRANS, REWEIGHT_PU, SPLIT_CHART, MERGE_CHART, STITCH_TRACK, MERGE_EVENT, REINDEX, REPARAM_TAU }.
• Scope & influence cone: scope(op) ⊂ M, neighborhood N(scope) (overlapping charts & transition edges).
• Atlas distance: d_Atlas(Atlas_1, Atlas_2) (see S911-5).
• Two-form delta (pre/post change):
  delta_form_glue^Δ = dist_intervals( D_global^{recomp,pre}, D_global^{glue,post} ).
• Freshness: fresh = ts_now − ts(data'); coverage redundancy κ_overlap.

III. Axioms P911-*

• P911-1 (Locality) — Patches affect only scope(op) ∪ N(scope); other charts and transitions remain fixed.
• P911-2 (Conservation & monotonicity) — After a patch, enforce Σ_a ψ_a ≡ 1 and non_decreasing(τ); all check_dim must pass.
• P911-3 (Transition coherence) — Updated T_ab satisfy d_B(D_a, T_ab# D_b) ≤ tol_match and cocycle Δ_cyc ≤ tol_cyc.
• P911-4 (Parallel two forms) — Compute global-by-glue and global-by-recompute before/after a patch and record delta_form_glue^Δ.
• P911-5 (Commutative / idempotent) — Replaying the same patch with the same TraceID is a no-op; patches with disjoint scopes commute.
• P911-6 (Rollbackable) — Every commit includes a snapshot and inverse operations op^{-1}.
• P911-7 (Freshness) — If fresh > tol_fresh, tag stale=true and use down-weighted merge strategies.

IV. Minimal Equations S911-*

1. Patch semantics (examples)

• S911-1 (ADD_CHART): add (U_c, phi_c, K_c, D_c, Γ_c), compute T_ca, T_ac on U_c∩U_a, and adjust ψ_{a/c} so
  Σ_a ψ_a + ψ_c ≡ 1 with 0 ≤ ψ_* ≤ 1.
• S911-2 (UPDATE_TRANS): on U_a∩U_b, regress r_ab: τ_a = s_ab τ_b + b_ab and geometry g_ab to minimize
  d_B^{ab} = d_B( D_a , T_ab# D_b ) under Δ_cyc ≤ tol_cyc.
• S911-3 (REWEIGHT_PU): ψ'_a = ψ_a + δψ_a with Σ_a δψ_a = 0 and supp(δψ_a) ⊂ N(scope).

2. Merge operations

• S911-4 (Three-way merge): with base, ours, theirs, build conflict set
  C = { x | x(ours) ≠ x(theirs) ∧ x(base) ≠ x(ours) ∨ x(theirs) }.
  Resolve T_ab / ψ / Γ / events on C per policy to produce Atlas*.

3. Atlas distance & rebuild triggers

• S911-5 (Atlas distance):
  d_Atlas = max( max_{a,b} d_B( D_a , T_ab# D_b ), sup_x | Σ_a ψ_a(x) − 1 | ).
• S911-6 (Recompute trigger): if d_Atlas > tol_merge ∨ delta_form_glue^Δ > tol_glue, run recompute_subset.

4. Tracks & events

• S911-7 (STITCH_TRACK): on overlaps, minimize
  E = Σ_t ( || g_ab(Γ_b(t)) − Γ_a(t) ||_2^2 + λ | κ_a − κ_b | ) to obtain Π_ab and merge trajectories.
• S911-8 (MERGE_EVENT): if d_t ≤ tol_t ∧ d_B^{ab} ≤ tol_match, merge event nodes and keep multi-source evidence.

5. Two-form delta (pre/post)

• S911-9: delta_form_glue^Δ = dist_intervals( D_global^{recomp,pre}, D_global^{glue,post} ) (bottleneck or landscape norm).

6. Units & dimensions

• S911-10: check_dim(s_ab) = "[1]", check_dim(b_ab) = unit(τ),
  check_dim(d_Atlas) = unit(τ), check_dim(ψ) = "[1]".

V. Metrology Workflow M90-11

• Ready: lock policy & thresholds; parse data' / Atlas'; generate candidate diffs { ΔK, ΔD, ΔΓ } and determine scope.
• Local rebuild: within scope, update K(τ), D, Γ / events; compute neighbor T_ab and d_B^{ab}.
• Transitions & partition-of-unity updates: run UPDATE_TRANS / REWEIGHT_PU; verify Δ_cyc and Σ ψ = 1.
• Two-form comparison: compute D_global^{glue} and D_global^{recomp}; obtain delta_form_glue^Δ.
• Merge (optional): three-way merge Atlas with Atlas'; resolve conflicts C; locally rebuild or fuse by weights.
• Quality gates: if d_Atlas ≤ tol_merge and delta_form_glue^Δ ≤ tol_glue and κ_overlap ≥ tol_cover, commit; else backtrack or enlarge-scope rebuild.
• Commit & persist:
  manifest.topo.atlas.patch = { ops, scope, d_B^{ab}, Δ_cyc, Σψ deviation, delta_form_glue^Δ, κ_overlap, evidence, ver, ts };
  for merges, also publish manifest.topo.atlas.merge. Save snapshot and op^{-1}.

VI. Contracts & Assertions C90-111x (suggested gates)

• C90-11101 (Coverage): κ_overlap_p99 ≥ 2 (critical regions double-covered).
• C90-11102 (Registration): d_B^{ab}_p95 ≤ tol_match (suggest 0.05 * scale_τ).
• C90-11103 (Cocycle): Δ_cyc_p95 ≤ tol_cyc (suggest 1e-3).
• C90-11104 (Two-form): delta_form_glue^Δ ≤ tol_glue (suggest 0.02 * scale_τ).
• C90-11105 (Partition-of-unity): sup_x | Σ_a ψ_a(x) − 1 | ≤ 1e-6.
• C90-11106 (Freshness): fresh ≤ tol_fresh; otherwise tag stale=true and down-weight.
• C90-11107 (Idempotent/commutative): reapplying a patch with the same TraceID leaves Atlas* unchanged; disjoint-scope patches commute.
• C90-11108 (Rollback-ready): snapshot and op^{-1} are complete; otherwise reject.

VII. Implementation Bindings I90-11*

• I90-111 diff_atlas(Atlas, data'?) -> { ΔK, ΔD, ΔΓ, scope }
• I90-112 build_local_update(scope, policy) -> { K(τ), D, Γ }
• I90-113 repair_transitions(U_a∩U_b, D_a, D_b) -> { T_ab, d_B^{ab}, Δ_cyc }
• I90-114 reweight_partition_of_unity({ ψ_a }, scope) -> { ψ'_a }, sup|Σψ'−1|
• I90-115 recompute_subset(Atlas, scope) -> Atlas_subset
• I90-116 merge_atlases(base, ours, theirs, policy) -> { Atlas*, conflicts, evidence }
• I90-117 stitch_tracks_events({ Γ, G_event }, { T_ab }) -> { Γ*, G_event* }
• I90-118 atlas_distance(Atlas1, Atlas2) -> d_Atlas
• I90-119 assert_patch_contracts(results, rules) -> report
• I90-11A emit_atlas_patch_manifest(results, policy) -> manifest.topo.atlas.patch
• I90-11B emit_atlas_merge_manifest(results, policy) -> manifest.topo.atlas.merge

Invariants: non_decreasing(τ); Σ ψ ≡ 1; Δ_cyc ≤ tol_cyc; delta_form_glue^Δ ≤ tol_glue; all check_dim pass; patches are idempotent.

VIII. Cross-References

• Complex/filtration & stability: Chs. 7–8 (local rebuilds & stability gates).
• Atlas & transitions: Ch. 9 (T_ab, ψ_a, Δ_cyc).
• Invariants & calibration: Ch. 10 (pre/post Inv deltas and causal evidence).
• Runtime & dashboards: Ch. 14 (patch pipelines & audit metrics); manifest keys: Appendix C; error propagation: Appendix E.

IX. Quality & Risk Control

• SLOs: patch-commit success ≥ 99%; gate pass-rates for d_Atlas and delta_form_glue^Δ ≥ 99%; mean rollback time ≤ one batch cycle.
• Fallback path: local repair → expanded-scope rebuild → reduced order (H0 + events only) → freeze conflicting regions and bypass—inflating U at each step.
• Audit: persist registration matchings γ*, T_ab parameter trajectories, Σψ error fields, conflict sets and resolutions, resource traces, and seeds; sign and hash each patch.

Summary
This chapter establishes the engineering convention for incremental maintenance and merging of the Atlas as P911 / S911 / M90-11 / C90-111x / I90-11*. With local patches, transition coherence, partition-of-unity conservation, and two-form comparisons, the Atlas remains consistent, reproducible, and rollback-capable as it evolves—uniformly published via manifest.topo.atlas.patch/*merge*.