27-EFT.WP.Packets.Light v1.0 | Appendix E — Error & Uncertainty Propagation (Packets.Light Edition)

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Appendix E — Error & Uncertainty Propagation (Packets.Light Edition)

One-sentence goal: Establish a unified framework for uncertainty propagation across the entire Packets.Light chain using GUM linearization (LPU) and Monte Carlo (MC), covering dual-form deltas and guardband design, and persist results into manifests to support release, audit, and replay.

I. Scope & Objects

• Objects: the twelve domains phys / frame / mod / label / comp / fso / tarr / swrt / queue / meas / mpath / sec / rt key quantities and their dual-form deltas delta_form_*.
• Inputs: source-term uncertainties and covariance V_ξ (model / measurement / environment / approximation / runtime drift), dual-form artifacts with RefCond, and unit/dimension mappings & conversions.
• Outputs: combined standard uncertainty u_c(z), coverage interval U = k•u_c, effective dof nu_eff, dual-form uncertainty u(delta_form_*), guardbands, and a publication manifest manifest.packet.u.*.
• Boundary: default to weak nonlinearity and small perturbations; for strong nonlinear sections use MC or piecewise linearization; this appendix does not replace security/governance policies (see Ch. 13).

II. Terms & Variables

• Source vector: ξ = [ x, n, θ, Λ, α, β2, D_pmd, RBW, cal, drift, approx, … ]^T.
• Covariance: V_ξ ⪰ 0; sub-blocks such as R_x (signal/noise), V_θ (kernel params), V_Λ (spectral bounds/structure), V_cal (calibration).
• Target quantity: z = h(ξ) (e.g., T_arr*, OSNR, EVM, BER, lat_total, Avail, …).
• GUM Jacobian: J = ∂h / ∂ξ |_{ ξ̂ }; combined uncertainty u_c(z) = sqrt( J V_ξ J^T ).
• Dual forms: z^{ pred | config }, z^{ meas }, with delta_form = | z^{ pred } − z^{ meas } |.
• Coverage: U = k • u_c (k ≈ 2 ↔ ~95%); nu_eff via Welch–Satterthwaite.
• Guardband: gb = k • u_c(z) or asymmetric gb_± = k_± • u_c(z).

III. Postulates P60E-*

• P60E-1 (Dual forms in parallel): For each core z, evaluate u_c(z^{pred}) / u_c(z^{meas}) and u(delta_form) and persist them.
• P60E-2 (Explicit measures & domains): Any time/frequency/set/path integrals are explicit: ( ∫_W • dt ), ( ∫_B • df ), ( ∑_{pkt∈S} • ), ( ∫_{gamma} • d ell ).
• P60E-3 (Dimensional compliance): All sources and targets declare unit / dim and pass check_dim( y − f(x) ); log↔linear conversions recorded as scale.note.
• P60E-4 (Closed source set): V_ξ must include at least five classes: measurement noise, model parameters, structure/spectral bounds, approximation error, runtime drift.
• P60E-5 (Method choice): Prefer GUM for small perturbations; use MC or piecewise linearization for strong nonlinearity / thresholds / discrete events; record the discrepancy ρ = u_c^{GUM} / u_c^{MC}.
• P60E-6 (Conservation-first): If physical constraints exist (energy / boundary / guard bits), project onto the constraint set before publishing U.

IV. Minimal Equations S60E-*

1. GUM linearization & composition
   • S60E-01: u_c^2(z) = J V_ξ J^T.
   • S60E-02: nu_eff = ( ∑ u_i^4 / ν_i ) / ( ∑ u_i^2 )^2 (Welch–Satterthwaite).
   • S60E-03: U = k • u_c, with k = t_{ nu_eff, 1−α/2 } (use k ≈ 2 as nu_eff → ∞).
2. Sensitivity exemplars
   • S60E-11 T_arr* = g( n_eff, L, ΔT_* ):
     ∂T_arr*/∂n_eff = (1/c_ref) ( ∫ d ell ), ∂T_arr*/∂L = (1/c_ref) n_eff; ΔT_geom / med / inst / proc add linearly.
   • S60E-12 OSNR_dB = 10 log10( P_sig / ( N0 • RBW ) ):
     u(OSNR_dB) ≈ (10/ln10) • sqrt( (u(P_sig)/P_sig)^2 + (u(N0)/N0)^2 + (u(RBW)/RBW)^2 ).
   • S60E-13 EVM^2 ≈ 1 / SNR_lin:
     u(EVM) ≈ (1/2) EVM • u(SNR_lin)/SNR_lin; if only OSNR is known, use the Chapter 4/11 mapping and chain-rule sensitivities.
   • S60E-14 lat_total = ∑ lat_i:
     u(lat_total) = sqrt( ∑ u^2(lat_i) + 2 ∑_{i<j} cov(lat_i, lat_j) ).
   • S60E-15 Avail = 1 − P_outage (ΓΓ/lognormal):
     u(Avail) = | ∂F / ∂θ | • u(θ) via shape/variance sensitivities.
3. Uncertainty of dual-form delta
   • S60E-21 delta_form = | z^{pred} − z^{meas} |:
     u(delta_form) ≈ sqrt( u^2(z^{pred}) + u^2(z^{meas}) − 2 • cov(z^{pred}, z^{meas}) ).
   • S60E-22 Publication gate: delta_form + k • u(delta_form) ≤ tol_* (contractual).
4. Including approximation errors
   • S60E-31 Chebyshev/rational approximations: treat ε_poly / ε_rat as independent terms:
     u_c^2 ← u_c^2 + u^2(ε_poly) + u^2(ε_rat) (add cross terms if correlated).
   • S60E-32 Analytic↔measurement conversions (OSNR↔SNR, dB↔linear) enter via u(scale).
5. MC propagation
   • S60E-41 Sampling: ξ^{(m)} ~ N( ξ̂, V_ξ ) or empirical/bootstrap; use mixtures for discrete/threshold effects.
   • S60E-42 Forward: z^{(m)} = h( ξ^{(m)} ); coverage U = [ q_{α/2}, q_{1−α/2} ], u_c = std( z^{(m)} ).
   • S60E-43 Convergence: stderr ≤ η • u_c (suggest η = 0.05) or KS distance below threshold.
   • S60E-44 GUM vs MC consistency: record ρ = u_c^{GUM} / u_c^{MC}.
6. Constraint projection (conservation)
   • S60E-51 Linear constraints C y = d: project y' = y − C^T ( C C^T )^{−1} ( C y − d ); update
     V_{y'} = ( I − C^T ( C C^T )^{−1} C ) V_y ( I − … )^T.
   • S60E-52 Guard bits: before publishing, verify T_guard ≥ 2σ_J + ΔCD + ΔPMD; if violated, increase gb_guard and reduce throughput (Chs. 3 / 10).
7. Guardband synthesis
   • S60E-61 Baseline: gb(z) = k • u_c(z).
   • S60E-62 Runtime margin: gb'(z) = k • u_c(z) + β • drift_score • range(z) (β and drift_score from Appendix D).

V. Metrology Pipeline M60-E* (Ready → Model → Propagate → Check → Persist)

1. Ready: freeze RefCond, unit/dimension maps, B/RBW, frame_spec, path/device inventories; collect source priors and covariance V_ξ; select method (GUM/MC) and k, α.
2. Model: for each domain, construct h(ξ) and Jacobians J (or MC samplers); define dual-form targets and tolerances tol_*.
3. Propagate: run GUM and/or MC to obtain u_c(z), U, and u(delta_form); record ρ.
4. Check:
   • Dual-form gate: delta_form + k • u(delta_form) ≤ tol_*;
   • Conservation gate: energy/boundary/guard-bit contracts satisfied;
   • Units/dimensions: pass check_dim(*); on failure, trigger strategy cards (Appendix B).
5. Persist:
   manifest.packet.u.* = { targets:{ z_list }, two_form:{ delta_form, u(delta_form) }, u_c, U, k, nu_eff, method:{ GUM | MC }, ρ, sources:{ V_ξ, approx, scale.note }, RefCond, signature }.

VI. Contracts & Assertions C60E-* (Suggested thresholds)

• C60E-01 (GUM/MC consistency): 0.8 ≤ ρ ≤ 1.25; otherwise publish MC as the source of truth and annotate.
• C60E-02 (Coverage publication): report U with confidence ≥ 95% (k ≈ 2 or quantile interval).
• C60E-03 (Dual-form gate): delta_form + k • u(delta_form) ≤ tol_* (domain-specific in chapters & Appendix B).
• C60E-04 (Conservation & dimensions): apply constraint projection before publishing U; check_dim(*) == true.
• C60E-05 (Approximation share): u(ε_poly)/u_c ≤ 0.5 and u(ε_rat)/u_c ≤ 0.5; otherwise raise polynomial order or change kernel.
• C60E-06 (Freshness): source covariance and RefCond updated within Δt_max; otherwise reject or degrade.

VII. Implementation Bindings I60-E* (interfaces, invariants)

• I60-E1 build_sensitivity(models, z_spec) -> { J_handles } (domain-specific Jacobians / Jacobian-vector ops)
• I60-E2 propagate_gum(J_handles, V_ξ) -> { u_c, nu_eff, parts }
• I60-E3 propagate_mc(sampler, N, α, η) -> { stats, U, N_eff }
• I60-E4 merge_approx_uncert(u_core, u_eps) -> { u_total }
• I60-E5 design_guardband(metrics_u, drift_score, policy) -> { gb, actions }
• I60-E6 assert_uncert_contracts(u_report, rules) -> { pass, report }
• I60-E7 emit_uncert_manifest(results, policy) -> { uri, status }
  Invariants: V_ξ ⪰ 0; two_forms_present = true; RefCond and units/dimensions consistent; persisted outputs are replayable.

VIII. Cross-References

• Chapters 2–14 (target definitions & gates); Chapter 15 (end-to-end); Appendix A (interfaces); Appendix B (contracts); Appendix C (manifests); Appendix D (drift).
• Companion volumes: PathCorrection / TimeBase / Instrument / Sync for dual forms, timebase, calibration, and replay.

IX. Quality & Risk Control

• Primary monitors: u_c(z)_p95, U/|z|, u(delta_form)_p95, ρ(GUM/MC), coverage, gb / tol_* ratio.
• Rollback ladder: raise approximation order / change kernel → increase MC samples → constraint projection / expand guard bits → lower-order / bandwidth-limit → bypass / rollback.
• Audit: retain V_ξ sources, Jacobian/sampling scripts, MC convergence logs, guardband change records, and signature chains with replay scripts.

Summary

• This appendix unifies link-wide uncertainty as a GUM/MC parallel workflow with dual-form gates, conservation projections, and guardband synthesis.
• With manifest.packet.u.* and C60E-* contracts, Packets.Light deployments become measurable, auditable, and rollback-ready for both release and runtime governance.