13-EFT.WP.Methods.SimStack v1.0 | Chapter 8: Error Budget, Validation & Conservation Checks

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Chapter 8: Error Budget, Validation & Conservation Checks

I. Scope & Objectives

• Establish a complete chain from each Quantity of Interest (QoI) to its error sources, with quotas and combination rules. Normalize quality gates including eps_norm, eps_mass, delta_form, and eps_time_map, and provide executable guidance on statistical power and sample size.
• Objective: in joint scenarios that span the continuous kernel, coupled advancement, and the thread network, provide manifest-backed evidence for unit and dimensional conservation, two-form consistency of arrival time, time–frequency consistency for spectral quantities, and stable, reproducible regression thresholds.

II. Terms & Symbols

1. Quantities of Interest (QoI)
   • Q ∈ { T_arr , M(t) , S_xx(f) , TS.latency.p99 , ... }.
   • Reference and tolerance: Q_ref, gate(Q).
2. Error components & budget
   • e = { e_phys , e_disc , e_time , e_cpl , e_sched , e_data , e_spec , e_form }.
   • Budget vector b = { b_phys , b_disc , b_time , b_cpl , b_sched , b_data , b_spec , b_form }.
3. Conservation & normalization
   • Mass: M(t) = ( ∫_V rho dV ) with residual eps_mass.
   • Normalization: eps_norm = | ( ∫ p dX ) - 1 | when p is a density.
4. Two formulations of arrival time
   • T_arr.general = ( ∫ ( n_eff / c_ref ) d ell ).
   • T_arr.factorized = ( 1 / c_ref ) * ( ∫ n_eff d ell ).
   • delta_form = | T_arr.general - T_arr.factorized |.
5. Time-base mapping & uncertainty
   ts = alpha * tau_mono + beta + epsilon(t); r_rms = std(epsilon(t)).
6. Spectral consistency
   S_xx(f), window constant U_w, and ENBW.

III. Postulates & Minimal Equations (P61-/S62-)

1. P61-16 (QoI-first with consistent gates)
   Each Q must bind to a unique gate(Q) with an evidence manifest. Conservation gates (eps_mass, eps_norm) are decided before performance gates (e.g., TS.latency.p99).
2. P61-17 (Parallel evidence for two forms)
   Any publication involving T_arr must report T_arr.general, T_arr.factorized, and delta_form in parallel, and persist gamma(ell) with d ell.
3. S62-50 (First-order linear uncertainty propagation)
   With parameter vector θ and covariance C_θ, and Jacobian J_Q,
   Var[Q] ≈ J_Q C_θ J_Q^T, σ_Q = sqrt( Var[Q] ).
   For T_arr.general on a discretized path {Δell_i}, the sensitivity is J_i = Δell_i / c_ref.
4. S62-51 (Path decomposition of two-form discrepancy)
   If c_ref = c_ref(ell), then
   delta_form = | ( ∫ n_eff * ( 1 / c_ref - 1 / c_ref_ref ) d ell ) |,
   where c_ref_ref is the constant reference speed adopted in the report.
5. S62-52 (Projecting time-base mapping error into a QoI)
   • σ_ts^2 ≈ ( tau * σ_alpha )^2 + σ_beta^2 + r_rms^2.
   • If Q = Q(ts) is differentiable, then σ_Q_time ≈ | dQ/dts | * σ_ts.
6. S62-53 (Parseval-style check for spectral consistency)
   Let time-domain variance be var_t(x), and spectral integral
   var_f(x) = ( Σ_k S_xx(f_k) * Δf ) / U_w_adj,
   then
   eps_psd = | var_t(x) - var_f(x) | / max( ε , var_t(x) ),
   where U_w_adj is determined by window.type and ENBW.
7. S62-54 (Error combination and total budget)
   • Near-independent case: e_rms(Q) = sqrt( Σ_i e_i(Q)^2 ).
   • General case: e_rms(Q) = sqrt( e(Q)^T C_e e(Q) ), with correlation matrix C_e.
8. S62-55 (Sample size & statistical power)
   For target effect δ, variance σ^2, significance α, and power 1-β,
   N_min ≈ ( z_{1-α} + z_{1-β} )^2 * σ^2 / δ^2, with z_q = inverse_normal_cdf(q).

IV. Data & Manifest Conventions

• QoIs & budgets
  qoi[].name, qoi[].ref, qoi[].gate, qoi[].sigma_est, qoi[].components = { e_phys , ... }, qoi[].combine ∈ [rms , corr].
• Conservation & two-form
  mass.M(t), eps_mass, norm.eps_norm; T_arr.general, T_arr.factorized, delta_form, gamma.param, measure.d_ell.
• Time-base propagation
  time.alpha, time.beta, time.r_rms, sigma_ts, sigma_Q_time.
• Spectral consistency
  window.type, U_w, ENBW, eps_psd.
• Statistical power
  power.alpha, power.target, power.N_min, evidence.samples.
• Regression gates & baseline
  regression.baseline.id/hash, gates[], status ∈ [pass, fail], delta.metric.

V. Algorithms & Implementation Bindings (I60-*)

• I60-17 estimate_error_budget(qoi:list, sources:any, cov:any) -> BudgetReport
  Compute σ_Q and component shares for each Q, supporting rms and correlation-aware combination; surface hotspots.
• I60-18 check_conservation(state:any) -> ConsReport
  Compute eps_mass, eps_norm, eps_psd, and delta_form; write them into the manifest and return pass/fail.
• I60-19 plan_power(q:dict, sigma:float, alpha:float, power:float, delta:float) -> NPlan
  Estimate N_min per S62-55 and suggest windowing and sampling configurations.
• I60-20 build_regression_suite(spec:any, baseline:any) -> RegrReport
  Generate regression cases, thresholds, and controls; output pass/fail with a deviation table.

VI. Conservation & Normalization Tests (Mx-62, detailed)

1. Inputs & preconditions
   Provide manifest, a snapshot of the continuous kernel, and the path manifest. Preconditions: pass check_dim(expr) and unit audits.
2. Steps
   • Mass conservation: compute M(t) and the discrete continuity residual from fluxes and sources to obtain eps_mass.
   • Normalization error: for probability or intensity density p, compute eps_norm.
   • Arrival time (two forms): produce T_arr.* and delta_form.
   • Spectral consistency: compute eps_psd.
   • Gate aggregation: status = all( eps_* ≤ gate(*) ).
   • Failure handling: shrink step size or raise method order, rebuild path measures or switch windowing, record compensations and alerts.
3. Artifact
   ConsReport = { eps_mass , eps_norm , eps_psd , delta_form , status , remediation }, persisted into audit.trail.

VII. Error Sources & Budget Allocation (Execution Conventions)

• Layered quotas
  Continuous kernel e_phys/e_disc, coupled advancement e_cpl, time-base e_time, threads & scheduling e_sched, data & spectra e_data/e_spec, two-form e_form.
• Apportionment strategy
  Allocate the budget b(Q) in proportion to sensitivities |∂Q/∂θ_i|. When strong correlations exist, first reduce the largest contributing covariance pairs.
• Optimization loop
  Map hotspots to actions: raise spatial order, improve advance_dt control, switch synchronization mode, adjust window and ENBW, refine gamma(ell) sampling density.

VIII. Statistical Power & Sample Size (Design Conventions)

• One-sided gates (e.g., eps_mass ≤ gate)
  With allowable deviation δ = gate - μ0 (μ0 is target), estimate N_min using S62-55. For correlated data, replace N with effective N_eff = N / ( 1 + 2 Σ_{k≥1} ρ_k ).
• Difference detection (baseline comparison)
  With Δ = | Q_new - Q_base | and variance σ^2, estimate N_min similarly after sample alignment. Record near_iid evidence or apply correlation correction in the manifest.

IX. Regression Testing & Gates

1. Regression case families
   Micro-kernels (analytically comparable), system-level (end-to-end TS.*), and cross-scale (parallel path and spectrum checks).
2. Gate policy
   • Conservation gates (mandatory): eps_mass, eps_norm, delta_form, eps_psd.
   • Performance gates (contextual): TS.latency.p99, TS.throughput.rps.
   • Confidence gates: σ_Q ≤ gate_sigma(Q).
3. Reporting
   RegrReport returns pass/fail, principal deviations, and recommended actions (step-size, sync, placement, or windowing).

X. Cross-References & Dependencies

• With the Continuous Kernel (Chapter 2)
  Units and measures for n_eff(x,t), rho(x,t), and c_ref determine comparability of T_arr and eps_mass.
• With Coupled Advancement (Chapter 4)
  advance_dt and synchronization strategies impact e_cpl and σ_ts; higher-order or adaptive methods reduce error hotspots.
• With Time Calibration (Chapter 5)
  alpha/beta/r_rms feed σ_ts and σ_Q_time; outputs of Mx-61 are prerequisites here.
• With Data Persistence (Chapter 6)
  The manifest must include qoi[], budgets, gates, and evidence. eps_* and delta_form are mandatory fields.
• With Parallelization (Chapter 7)
  Scheduling and batching affect e_sched and performance gates; migrations and retries must preserve hb and be logged in audit.trail.

XI. Risks, Limitations & Open Questions

• Risks
  Ignoring inter-component correlations yields false budget passes; poor window and ENBW choices introduce systematic bias; sparse path measures understate delta_form.
• Limitations
  This chapter provides first-order propagation and empirical gates. Strongly nonlinear or non-Gaussian regimes require specialized uncertainty propagation.
• Open questions
  Joint optimization of step-size × synchronization × placement; equivalence-class merging and confidence evaluation for multi-path gamma(ell); a unified calibration procedure for spectral consistency.

XII. Deliverables & Versioning

• Deliverables
  budget.policy, Mx-62 scripts, reference implementations for I60-17/18/19/20, regression.suite, dashboard templates, and gate configurations.
• Versioning
  From v1.0, freeze key fields qoi[].*, eps_*, delta_form, sigma_ts. Add new statistics in a backward-compatible manner with migration guidance.

XIII. New Terms & Symbols (to memorize)

• Budgets & components: e_phys, e_disc, e_time, e_cpl, e_sched, e_data, e_spec, e_form, b(*).
• Conservation & normalization: M(t), eps_mass, eps_norm.
• Two forms & paths: T_arr.general, T_arr.factorized, delta_form, gamma(ell), d ell.
• Time-base propagation: sigma_ts, σ_Q_time, alpha, beta, r_rms.
• Spectral consistency: S_xx(f), U_w, ENBW, eps_psd.
• Statistical power: N_min, α, β, z_q, near_iid, N_eff.
• Reports & manifests: BudgetReport, ConsReport, RegrReport, audit.trail, gate(Q).