33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 12 — Error Budget & System Safeguards

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Chapter 12 — Error Budget & System Safeguards

I. One-Sentence Goal

Establish an end-to-end error model and protection framework for early objects from causation (seed/trigger) → growth (state) → radiation (L_nu/LC) → propagation (n_eff, segmentation, two forms) → observation (T_arr/Delta_T_arr/F_nu/LC). Provide both GUM and MC uncertainty propagation, two-form–specific errors, thin/thick switching tau_switch, interface energy closure and sidedness, differential out-of-band leakage, c_ref drift, coordinate/unit consistency, and auditable criteria for spectral and causation parameter ill-posedness and clamping/saturation—together with falsification lines.

II. Scope & Non-Goals

Covered: error stratification and sources; two-form GUM/MC propagation formulae; spectrum–arrival joint propagation; thin/thick consistency; safeguard checklist; energy closure and sidedness; differential OOB leakage; path discretization and quadrature error; c_ref calibration/drift; K-correction and luminosity distance; guardband and falsification; logging and interface anchors.
Not covered: re-derivation of Chs. 3–9; detailed instrument response; any propagation construct violating n_eff ≥ 1.

III. Minimal Terms & Symbols

• Observables: T_arr_obs(f, gamma), Delta_T_arr_obs(f1,f2, gamma), F_nu_obs(f), LC_obs(t).
• Models: T_arr_mod/Delta_T_arr_mod/F_nu_mod/LC_mod; Residual: Residual = Obs − Mod.
• Consistency indices:
  eta_T = | T_arr^{const} − T_arr^{gen} |,
  tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) |.
• Lower bound & energy: LB = L_path / c_ref; R_env + T_trans + A_sigma = 1; sidedness n_eff^± ≥ 1.
• Guards & isolation: n_eff ≥ 1; T_fil ≠ T_trans; n ≠ n_eff; f_em = f_obs • (1+z_obs) (for K-correction).

IV. Error Stratification & Sources

• Metrology layer: u(T_arr_obs), timebase & sync; F_nu/LC sampling & calibration; u(c_ref); transforms among coords_spec/units_spec/metric_spec.
• Causation/growth layer: u(seed), trigger‐rate u(Λ_event), growth-law params u(θ_growth); prior uncertainty and model misspecification.
• Spectral layer: u(params_sed), K-correction and luminosity distance D_L; spectral resolution and band-edge truncation.
• Propagation/interface layer: u(n_eff) (from F and H_sea params), segmentation { ell_i } & endpoint tolerances, Delta_T_sigma triggers, triplet {R_env,T_trans,A_sigma} estimation error.
• Numerics: path discretization and quadrature errors; interpolation error (no cross-interface interpolation); convergence gate eps_T.
• Band/orientation: differential OOB leakage; frequency mis-registration; directional terms b1/b1_n regression uncertainty; multi-path weights w_m.
• Environment: injected and drifting TBN(x,t).

V. GUM Uncertainty Propagation (Two-Form Arrival Time)

Constant pull-out (discrete approximation)

T_arr ≈ (1/c_ref) * ∑_{k=0}^{N-1} n_eff[k] • Δell[k]

Sensitivities:

∂T_arr/∂c_ref = − T_arr / c_ref

∂T_arr/∂n_eff[k] = Δell[k] / c_ref

∂T_arr/∂Δell[k] = n_eff[k] / c_ref

Combination:

u_c^2(T_arr) ≈ (∂T/∂c_ref)^2 u^2(c_ref)

+ ∑ ( (Δell[k]/c_ref)^2 u^2(n_eff[k]) )

+ ∑ ( (n_eff[k]/c_ref)^2 u^2(Δell[k]) )

+ 2∑∑ ρ_ij (∂T/∂q_i)(∂T/∂q_j) u(q_i) u(q_j)

with q_i ∈ { c_ref, n_eff[*], Δell[*] }. Report the correlation model ρ_ij.

General form (discrete approximation)

T_arr ≈ ∑ ( n_eff[k] / c_ref[k] ) • Δell[k]

Sensitivities:

∂T_arr/∂n_eff[k] = Δell[k] / c_ref[k]

∂T_arr/∂c_ref[k] = − n_eff[k] • Δell[k] / c_ref[k]^2

∂T_arr/∂Δell[k] = n_eff[k] / c_ref[k]

Model the spatio-temporal correlation of c_ref[k] (and its correlation with n_eff[k]) explicitly in V_ξ.

VI. Spectrum↔Arrival Joint Propagation (GUM View)

Observation model:

F_nu(f_obs) = L_nu(f_em)/(4π D_L^2) • K(z_obs), f_em = f_obs • (1+z_obs)

Illustrative first-order sensitivities:

∂F_nu/∂L_nu = [K / (4π D_L^2)], ∂F_nu/∂D_L = − 2 F_nu / D_L,

∂F_nu/∂z ≈ (∂K/∂z) • L_nu / (4π D_L^2)

Build a joint block covariance over {T_arr, F_nu, LC}, carrying cross-terms from θ_growth / params_sed / n_eff / c_ref for joint fitting and GUM composition of u_c.

VII. MC Propagation (Nonlinearity, Clamping & Events)

Use when: n_eff ∈ [1,n_max] clamping, discrete Delta_T_sigma triggers, interface type switching, significant c_ref(x,t,f) variation, or strongly nonlinear multi-path weights.
Flow:

• Sample the joint { θ_growth, params_sed, c_ref, n_eff[k], Δell[k] } (preserve shared correlations).
• Per sample compute { T_arr, Delta_T_arr, F_nu, LC }.
• Report median/quantiles, tail risk, clamping rate; compare against GUM for consistency.

VIII. Two-Form–Specific Error & Consistency

• Form selection: if max_ell |δc_ref/c_ref| ≤ eta_c, use constant pull-out; else use general and record the c_ref(x,t,f) model and u(c_ref[k]) in the contract.
• Index: ensure eta_T = | T_arr^{const} − T_arr^{gen} | ≤ threshold. Exceedance back-trace order: c_ref calibration → n_eff decomposition & assembly → segmentation & endpoints → Delta_T_sigma consistency.

IX. Thin/Thick Switching Error & Safeguards

• Definition: tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) |.
• Rule: if Delta_k/L_char ≤ eta_w, pick thin; in the gate neighborhood, dual-run and record tau_switch; if out-of-gate, lock thick chain.
• Defense: symmetric endpoint root-finding with strict tolerances; reduce step in high Xi_k(chi) regions; log trigger statistics and impact.

X. Differential & Out-of-Band Leakage

Same-path differential:

Delta_T_arr = (1/c_ref) ∫ ( n_path(f1) − n_path(f2) ) d ell (constant pull-out)

Delta_T_arr = ∫ ( ( n_path(f1) − n_path(f2) ) / c_ref ) d ell (general form)

Consistency: share the same { gamma[k], Δell[k] }, segmentation, and Delta_T_sigma settings across the two bands; fold OOB residuals into u_sys and record leakage ratio & alignment policy.

XI. Path Discretization, Quadrature & Interpolation

• Three-gate adaptivity: curvature ‖ d^2γ/dℓ^2 ‖, medium change | d n_eff / dℓ |, layer strength Xi_k(chi); explicitly include endpoints in the integral.
• Error estimates: intra-segment dual-order quadrature difference; global error as RSS; target | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Interpolation: fixed order on grids; symmetric stencils near interfaces; no cross-interface interpolation—must segment.

XII. c_ref Calibration & Drift Guard

• Calibration:
  c_ref = ( ∫ n_eff_ref d ell ) / T_arr_ref (constant pull-out) or solve numerically for general form; record environment block and uncertainty.
• Drift: track c_ref(t); add cross-environment reuse via drift_budget into u_sys(c_ref); eta_T must still pass after cross-application.

XIII. K-Correction, Luminosity Distance & Spectral Errors

• K-correction: the correlation of K(z) with L_nu enters the joint covariance (coupling to T_arr/Delta_T_arr governed by band dependence of n_eff(f)).
• D_L model: must be declared in the contract with units; sensitivity ∂F_nu/∂D_L = − 2 F_nu / D_L; propagate cosmology uncertainty into u_sys.

XIV. Clamping, Saturation & Nonlinearity

• Clamping: enforce n_eff ∈ [1, n_max]; triggers reduce local sensitivity and bias estimates.
• MC defense: truncated/reflective sampling; report clamping rate and T_arr/Delta_T_arr bias; persistent triggers indicate modeling or data-quality issues → back-trace.

XV. Bias Detectors, Falsification Lines & Guardband

Detectors:

• Lower bound: T_arr_obs − L_path/c_ref < −k•u_c.
• Form: eta_T > threshold.
• Thin/thick: tau_switch > gate.
• Differential: Delta_T_arr nonlinearity or slope mismatch not due to OOB leakage.
• Energy: R_env + T_trans + A_sigma ≠ 1.
• Sidedness: n_eff^± < 1.

Falsification: meet any detector and rule out implementation/metrology error → register a falsification sample and enter Ch. 11 audit; three independent reproducible falsifications on the same dimension trigger P70-* and parameterization review.
Guardband: GB = k_guard • u_c; edge samples enter a review queue and receive MC re-checks.

XVI. Logging & Audit (Minimal Fields)

• Physics & geometry: hash(Catalog/Seeds/Trajectory/SeaProfile/Phi_T/n_eff/gamma), Sigma_env labels and { ell_i } tolerances.
• Modes & thresholds: mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr − L_path/c_ref.
• Energy & differential: {R_env,T_trans,A_sigma} residuals, Delta_T_sigma trigger stats, Delta_T_arr linearity and OOB leakage ratio.
• Spectral & causation: params_sed and θ_growth hashes/priors/covariances; K and D_L stances and units.
• Uncertainty: u_stat, u_sys, u_c, GUM/MC configs, k, seed.
• Audit bundle: data, code, parameters, SolverCfg/metric_spec, hash manifest, falsification samples, replay entrypoints.

XVII. Interfaces & Implementation Binding (Aligned with Template; I70-* Safety Checks)

• check_dimension( expr ) -> DimReport — dimensional/unit checks.
• propagate_uncertainty_GUM( inputs ) -> u_c / propagate_uncertainty_MC( inputs, Nsamples, seed ) -> dist — twin propagation paths.
• check_dual_arrival_consistency( inputs ) -> eta_T — two-form gate.
• consistency_thin_vs_thick_EO( inputs ) -> tau_report — thin/thick gate.
• estimate_energy_triplet( data, Sigma_env ) -> { R_env, T_trans, A_sigma } — energy closure audit.
• log_artifacts_EO( meta, hashes, metrics ) -> Log — logging & audit hardening.

Constraints: enforce n_eff ≥ 1 and lower-bound auditing at ingress; differential reuse of identical path discretization and correction settings.

XVIII. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (minimal equations), Ch. 4 (causation), Ch. 5 (coupling & growth), Ch. 6 (radiation & propagation), Ch. 7 (metrology), Ch. 8 (interfaces), Ch. 9 (numerics), Ch. 11 (validation).
• EFT.WP.Cosmo.LayeredSea v1.0: interface/layer consistency and tau_switch.
• EFT.WP.Propagation.TensionPotential v1.0: two-form and differential flows.
• EFT.WP.Core.Metrology v1.0 / Errors v1.0: metrology and falsification baselines.

XIX. Deliverables

• Error-budget checklist: GUM/MC inputs, correlation assumptions, output metrics templates (including spectrum–arrival joint items).
• Safeguard playbooks: operational runbooks for forms, segmentation, energy closure, thin/thick switching, differentials, c_ref, K & D_L, directionality & multi-path, clamping/saturation.
• Audit templates: bias-detector dashboard, falsification cards, guardband configuration, tau_switch/eta_T & energy-closure reports, replay guide and hash manifest.