11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 11 Error Budget and Verification

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Chapter 11 Error Budget and Verification

I. Scope and Objectives

• Establish error classification, allocation, and composition across the full filament drawing or stretching chain, and provide executable budgets and gates that support benchmarking, regression, and publication decisions.
• Define verification gauges for conservation and normalization, producing unified pass standards for gate.mass and gate.norm, consistent with Chapter 3 S12-1, Chapter 7 spectral gauges, Chapter 8 time-base alignment, and Chapter 10 parallel execution.
• Provide minimal implementation bindings and a runnable workflow (Mx-13 conservation-check) so that quality gates are audited and recorded before data ingestion and reporting.

II. Terminology and Symbols

1. Errors and statistics
   • bias[x] (systematic bias), var[x] (variance), std[x] = sqrt( var[x] ), cov[x,y] (covariance).
   • CI_p[x] (confidence interval at confidence level p), power (test power), alpha (significance).
2. Residuals and gates
   • eps_mass (mass-conservation residual), eps_norm (normalization error), eps_T (tension estimation error), eps_v (velocity estimation error).
   • gate.mass.abs, gate.mass.rel, gate.norm.abs, gate.norm.rel, gate.spectrum.leak, gate.threads.hb, gate.delta_form.
3. Frequency domain and windows (as in Chapter 7)
   S_xx(f), U_w, ENBW, window.id.
4. Key physical quantities (recap)
   • lambda(x,t), s(x,t) = ( d/dt ) ( ln( lambda ) ), v(x,t), A(x,t), rho_L(x,t) = rho(x,t) * A(x,t), J(x,t) = rho_L(x,t) * v(x,t), T_fil(x,t).
   • Path and measure: gamma(ell), d ell. Time bases: tau_mono, ts = alpha + beta * tau_mono.

III. Postulates and Minimal Equations

1. P11-20 (dimension and unit gate)
   Any publishable expression expr must satisfy check_dim( expr ) and unit consistency. Invalid expressions are disallowed from ingestion.
2. P11-21 (observability consistency)
   Estimate bias and variance only on samples that already satisfy ts alignment and hb ordering. Any cross-thread splice must retain timebase.alpha, beta and hb evidence.
3. P11-22 (budget additivity)
   To first order, variances of independent error components add. For dependent components, provide explicit correlation coefficients or a conservative envelope.
4. S12-30 (first-order error propagation)
   Let y = f( x ), x ∈ R^n. Linearized at x0:
   y ≈ f( x0 ) + J_x( x0 ) * ( x - x0 ), where J_x = ( ∂f / ∂x ).
   Then var[y] ≈ J_x * cov[x] * J_x^T, and bias[y] ≈ f( E[x] ) - f( x0 ).
5. S12-31 (mass-conservation residual on path interval [ell_a, ell_b] and time window [t1, t2])
   • eps_mass = ( ∫_{gamma(ell_a→ell_b)} rho_L(ell,t2) d ell ) - ( ∫_{gamma(ell_a→ell_b)} rho_L(ell,t1) d ell ) - ( ∫_{t1}^{t2} ( J(ell_a,t) - J(ell_b,t) ) dt ).
   • Pass conditions: | eps_mass | <= gate.mass.abs and ( | eps_mass | / M_ref ) <= gate.mass.rel, where M_ref = max( ∫ rho_L d ell ).
6. S12-32 (normalization checks)
   • For probability or spectral densities:
     eps_norm.time = | 1 - ( ∫ p(t) dt ) |
     eps_norm.spectrum = | var_time - ( ∫ S_xx(f) df ) |, where var_time is the time-domain variance.
   • Pass conditions: each eps_norm.* must be below its corresponding gate.norm.*.
7. S12-33 (sample size and power approximations)
   • For a mean test, sample size: N >= ( ( z_alpha + z_beta )^2 * sigma^2 / delta^2 ), where sigma^2 is the variance estimate and delta the minimum detectable difference.
   • Relative error of the variance estimate: std[ var_hat ] / var ≈ sqrt( 2 / ( N - 1 ) ).
8. S12-34 (publishing the two arrival-time conventions)
   If T_arr is used, compute in parallel
   T_arr.1 = ( 1 / c_ref ) * ( ∫ n_eff d ell ),
   T_arr.2 = ( ∫ ( n_eff / c_ref ) d ell ),
   and report delta_form = | T_arr.1 - T_arr.2 |, with gate delta_form <= gate.delta_form.
9. S12-35 (thread causality gate)
   TS.hb.violations == 0 is required. Otherwise the error budget is void, and all conclusions are downgraded to fail.

IV. Data and Manifest Gauges

1. qc.* fields (to be included under schema.core.drawing/v1)
   • qc.timebase.alpha, qc.timebase.beta, qc.hb.ok.
   • qc.mass.eps_abs, qc.mass.eps_rel, qc.mass.window = [ t1 , t2 ], qc.mass.path = [ ell_a , ell_b ].
   • qc.norm.time, qc.norm.spectrum, qc.spectrum.ENBW, qc.spectrum.U_w, qc.spectrum.leak_ratio.
   • qc.params.CI_p (intervals for K_el, K_vis, etc.), qc.tension.rmse, qc.velocity.rmse.
   • qc.delta_form (if T_arr is enabled), qc.gates.pass (boolean), qc.gates.detail[] (per-gate notes).
2. Traceability and units
   • qc.unit.check = pass|fail. On fail, attach expr and the inconsistent dimensions.
   • Every qc.* entry must record method.id, window.id, runtime.G.hash, and the ts reference.

V. Algorithms and Implementation Bindings

1. I10-8 compute_error_budget( series:dict , model:dict , window:list ) -> dict
   • Inputs: aligned lambda, s, v, A, T_fil with covariance estimates.
   • Outputs: bias[*], var[*], CI_p[*].
   • Contract: idempotent. If hb is not satisfied, raise E_HB_INVALID.
2. I10-9 check_conservation( rho_L:J:trace , window:list , path:list ) -> dict
   Compute eps_mass and eps_norm, compare against gate.mass and gate.norm, return pass|fail with details.
3. I10-10 run_regression_suite( dataset:list , gates:dict ) -> Report
   For each benchmark case, run I10-8 and I10-9, then aggregate pass rates and tail behavior.
4. Pseudocode (core fragment)

I10-9 check_conservation(trace, window=[t1,t2], path=[ell_a,ell_b]):

rhoL_t1 <- integrate_path(rho_L(.,t1), path)

rhoL_t2 <- integrate_path(rho_L(.,t2), path)

flux <- integrate_time( J(ell_a,.) - J(ell_b,.) , [t1,t2] )

eps_mass <- ( rhoL_t2 - rhoL_t1 ) - flux

eps_rel <- abs(eps_mass) / max(rhoL_t1, rhoL_t2, epsilon)

return {eps_abs: abs(eps_mass), eps_rel: eps_rel}

VI. Metrology Flow and Run Graph

1. Mx-13 conservation-check (aligned with Chapter 8)
   • timebase-attach: collect alpha, beta, map all data to ts.
   • units-guard: batch check_dim( expr ) across all expressions.
   • mass-norm: for each window.id call I10-9, producing qc.mass.* and qc.norm.*.
   • budget-prop: call I10-8 to compute bias/var/CI_p for T_fil, v, and the parameter set.
   • threads-hb: audit TS.hb.violations and the queue gates (see Chapter 10).
   • form-dual: if T_arr is present, compute both conventions and delta_form.
   • gate-eval: aggregate pass|fail and downgrading policies per gates.*, then write the manifest.
2. Fallback and alerts
   Any gate failure triggers ALERT.QC_FAIL and sets publish.blocked = true. Allow re-calibration or downsampled re-test branches.

VII. Verification and Test Matrix

1. Minimum required cases
   • Constant-speed draw: eps_mass near zero, and eps_norm.spectrum matches the time-domain variance.
   • Step draw: bias[T_fil] on the rising edge stays within CI_p. TS.jitter.rms does not exceed its gate.
   • Ramp draw: tracking errors rmse[K_el], rmse[K_vis] remain within budget.
2. Boundary and extreme scenarios
   • Downsampling and aliasing: verify gate.spectrum.leak with ENBW correction maintains eps_norm.spectrum within limits.
   • Injected time-base drift: assess eps_mass sensitivity under beta offset, then re-align to ts and re-test.
   • Thread jitter and packet loss: when TS.backlog.max exceeds its gate, confirm hb is intact and I10-9 either passes or correctly rejects.
3. SLO and conservation gates
   Pre-publication example thresholds:
   1. gate.mass.abs <= M_ref * 1e-3, gate.mass.rel <= 1e-3.
   2. gate.norm.abs <= 1e-3, gate.norm.rel <= 1e-3.
   3. gate.threads.hb: TS.hb.violations == 0.
   4. gate.delta_form <= T_ref * 1e-3 (if T_arr is used).

VIII. Cross-References and Dependencies

• Chapter 2: observations of lambda, s, v, A and geometric approximations directly affect error sources and observability.
• Chapter 3: S12-1 continuity governs eps_mass computation and discrete implementation.
• Chapter 4: constitutive families for T_fil determine the sensitivity matrix J_x and parameter uncertainty in I10-8.
• Chapter 7: spectral gauges S_xx(f), U_w, ENBW are used in eps_norm.spectrum and leakage assessment.
• Chapter 8: ts alignment and calibration confidence feed qc.timebase.*.
• Chapter 9: schema.core.drawing/v1 carries qc.* and gates.*.
• Chapter 10: concurrency effects on hb, back-pressure, and tail latency enter the error budget and rejection criteria.

IX. Risks, Limits, and Open Questions

• Non-Gaussian and heavy-tailed noise weakens the S12-30 linear propagation approximation. Record robust estimators in qc.method.id and publish sensitivity analyses.
• Under non-stationarity, discrepancies between S_xx(f) and time variance cannot be corrected by ENBW alone. Use time-varying spectra or short windows and report deviation bounds.
• Slenderness and incompressibility approximations can fail at extreme draw ratios, propagating to eps_mass and parameter bias. Run parallel body-line hybrid audits.
• OS or VM clocking may introduce cross-node offsets. Record source and confidence under qc.timebase.*.

X. Deliverables and Version Management

1. Artifacts
   • qc.report.json (per window.id qc.* metrics and gates.* decisions).
   • qc.methods.yaml (methods, windows, ENBW, leakage corrections, and sensitivities).
   • qc.trace.hb (causality audits and violation list).
   • qc.changelog.md (gate and algorithm changes).
2. Change policy
   • Gate updates are marked MOD and must backfill re-evaluation on historical windows. New error sources or methods are ADD. Fixes to units or dimensions are FIX.
   • Any change affecting hb or ts alignment must ship a regression matrix and baseline rebuild before publication.