24-EFT.WP.Particle.TopologyAtlas v1.0 | Appendix E — Error and Confidence Propagation (Topology Edition)

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Appendix E — Error and Confidence Propagation (Topology Edition)

One-Sentence Goal
Provide an end-to-end blueprint for decomposing error sources and propagating uncertainty—from raw fields and algorithmic steps to topological invariants and atlas metrics—using GUM-style linearization and MC/Bootstrap confidence intervals, with outputs written into manifests and governed by contracts.

I. Scope and Objects

• Inputs: observed field x (or tangent/orientation field n(x), phase field θ(x)), noise & priors { V_x, model }, discretization & filtration τ_policy, atlas Atlas = { U_a, φ_a, ψ_a, T_ab }, persistence D, invariant vector Inv = [ Q, Lk, Tw, Wr, ... ], dual forms formA/B, and runtime stream/offline.
• Outputs: u(Inv), U = k * u_c, coverage γ, nu_eff; stability bounds for u(D) (e.g., d_B sensitivity), u(Δ_cyc), and u(delta_form); persisted under manifest.topo.*.uncertainty and contracts.*.
• Boundaries & constraints: complete RefCond, explicit unit()/dim(); if local non-differentiability or non-negligible topology-flip probability exists, switch to MC/Bootstrap.

II. Terms and Variables

• Error budget: ε_sensor, ε_clean, ε_grid, ε_alg, ε_atlas.
• Covariance & Jacobian: V_x = cov(vec(x)), J = ∂g/∂vec(x) for mapping g: x → Inv.
• Persistence objects: D (diagram), λ_D (landscape), distances d_B, W_p.
• Stats: u_c (combined standard uncertainty), U = k * u_c, coverage = γ, nu_eff (Satterthwaite).
• Dual-form gaps: delta_form = | Inv(formA) − Inv(formB) |, delta_form_rt for stream vs offline.

III. Axioms P90E-*

• P90E-1 (Layered additivity) — u_tot^2 ≈ u_stat^2 + u_sys^2.
• P90E-2 (Local differentiability) — use MC/Bootstrap if g is not Gateaux-differentiable near x.
• P90E-3 (Topological stability) — if x ↦ f(x) is 1-Lipschitz, d_B(D(f), D(f+δ)) ≤ ||δ||_∞.
• P90E-4 (Parallel forms) — always co-publish delta_form with percentiles.
• P90E-5 (Explicit units) — all fields declare unit/dim and pass check_dim.
• P90E-6 (Sampling coherence) — bootstrap on i.i.d./exchangeable blocks or modeled correlation; persist block length in RefCond.

IV. Minimal Equations S90E-*

• S90E-1 (Delta/GUM):
  u_c^2(Inv) = ( J V_x J^T ) + Σ_{s∈{ grid, clean, alg, atlas }} u_s^2, with unit(u_c(Inv)) = unit(Inv).
• S90E-2 (Satterthwaite):
  nu_eff = ( ( Σ w_i u_i^2 )^2 ) / ( Σ ( w_i^2 u_i^4 / nu_i ) ), then k = k(γ, nu_eff).
• S90E-3 (MC intervals):
  u_mc^2 = ( 1 / (M−1) ) * Σ ( Inv^{(m)} − mean(Inv) )^2,
  CI_γ = [ quantile(Inv, (1−γ)/2), quantile(Inv, (1+γ)/2) ].
• S90E-4 (Integer invariants): publish Q = round(Q_raw) only if | Q_raw − round(Q_raw) | ≤ tol_int; always store Q_raw ± U.
• S90E-5 (PD bounds): u(d_B) ≤ u( ||δ||_∞ ); u(W_p) ≤ C_p * u( ||δ||_∞ ).
• S90E-6 (Transition propagation): with transition covariance V_T,
  u^2(Δ_cyc) ≈ G V_T G^T, G = ∂Δ_cyc / ∂vec(T).
• S90E-7 (Dual-form compliance): compliant if delta_form ≤ tol_form; publish p95/p99.
• S90E-8 (Hybrid): u_mix^2 = u_lin^2 + u_mc^2.

V. Workflow M90-E1

• Ready: lock RefCond / τ_policy / units / dim; decompose { ε_sensor, ε_clean, ε_grid, ε_alg, ε_atlas }.
• Route: Delta/GUM vs MC/Bootstrap vs Hybrid.
• Delta/GUM: compute J, V_x → u_c, nu_eff, U.
• MC/Bootstrap: sample { x^{(m)} } → { Inv^{(m)}, D^{(m)} } → u_mc, CI_γ, analyze d_B distributions.
• Dual forms & runtime: compute delta_form, delta_form_rt, publish percentiles.
• Check: apply contracts (Sec. VI); fallback via policy cards when violated.
• Persist: write { method, u/U, nu_eff, coverage, samples, delta_form } to manifest.topo.*.uncertainty; sign.

VI. Contracts & Assertions (C90-16x)

• C90-161 coverage ≥ γ_min (default 0.95).
• C90-162 nu_eff ≥ ν_min (default 20); else publish quantiles only.
• C90-163 u(ε_grid) ≤ 0.2 * u_c.
• C90-164 slope of d_B vs ||δ||_∞ ≤ s_max.
• C90-165 delta_form_p95 ≤ tol_form.
• C90-166 rounding only if | Q_raw − round(Q_raw) | ≤ tol_int.
• C90-167 MC/Bootstrap M ≥ M_min (default 500) with convergence (≤ 5% variance change).

VII. Implementation Bindings I90-*

• propagate_uncertainty_delta(g, x, Vx, extras) -> { u_c, nu_eff, U, J }
• propagate_uncertainty_mc(sampler, g, M, seeds) -> { u_mc, CI_γ, samples_hash }
• estimate_pd_stability(D_ref, sampler) -> { sens_dB, sens_Wp }
• compose_error_budget(parts) -> { u_stat, u_sys, u_tot }
• guardband_decision(value, U, rule) -> { pass, gb }
• emit_uncertainty_manifest(manifest, results) -> manifest'

Invariants: non_decreasing(τ); Σ_a ψ_a ≡ 1; check_dim(all); traceable method/RefCond; delta_form persisted.

VIII. Cross-References
Ch. 2 (math/homology); Chs. 3/5/6 (objects/events); Ch. 7 (complexes); Ch. 8 (persistence variance); Chs. 9/10/11 (atlas transition uncertainty); Ch. 12 (cleaning errors); Ch. 13 (interval publication for metrics); Appendix D (metrics/drift); Appendix B (contracts); Appendix C (manifests).

IX. Quality & Risk Control

• Suggested SLOs: coverage ≥ 0.95; nu_eff ≥ 30; delta_form_p95 ≤ 0.02 * scale_τ; u(Δ_cyc) ≤ 5e-4.
• Recalibration triggers: u_mc > 1.5× baseline or sens_dB > 2× baseline → recalibrate and write thresholds to RefCond.
• Fallback ladder: Delta → Hybrid → MC; optionally freeze integer rounding and publish intervals only.
• Audit: persist J / V_x / samples_hash and seeds to enable full replay.

Summary
This appendix codifies topology-specific error budgets, Delta/GUM propagation, MC/Bootstrap intervals, and dual-form consistency. With computable S90E-*, workflow M90-E1, contracts C90-16x, and I90-* bindings, outputs are published under manifest.topo.*.uncertainty, completing an auditable loop uncertainty → contracts → release.