33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 13 — Application Scenarios & Worked Cases

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Chapter 13 — Application Scenarios & Worked Cases

I. One-Sentence Goal

Deliver six core, ready-to-run scenarios (A…F) plus one end-to-end reference case that span causation → growth → radiation → propagation → observation for early objects. For each: state inputs/outputs, stepwise execution, mapping to the template interface family and I70-*, acceptance criteria and falsification lines, and a minimal logging set—so implementations are auditable, reproducible, and comparable.

II. Scope & Non-Goals

Covered: scenario objectives, prerequisites, required objects/environment/paths/bands, execution steps, interface mapping, audit and publication stance, common risks and mitigations.
Not covered: re-derivations from Chs. 3–12; instrument/pipeline specifics; any construct that violates n_eff ≥ 1 or circumvents R_env + T_trans + A_sigma = 1.

III. Minimal Terms & Symbols

• Objects & state: Catalog, Seeds, Trajectory (state(t)).
• Fields & environment: Phi_T, grad_Phi_T, SeaProfile, Sigma_env.
• Propagation & path: n_eff(x,t,f) (dimensionless, ≥ 1), c_ref, gamma(ell), endpoints { ell_i }, thin-layer correction Delta_T_sigma.
• Observables & differentials: L_nu(f), F_nu(f), LC(t), T_arr(f, gamma), Delta_T_arr(f1,f2, gamma).
• Consistency & thresholds: two-form eta_T, thin/thick tau_switch; energy closure R_env + T_trans + A_sigma = 1; lower bound T_arr ≥ L_path / c_ref.
• Naming isolation: T_fil ≠ T_trans; n ≠ n_eff.

Two-form arrival-time exemplars (unified across the volume)

T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) # constant pull-out

T_arr = ( ∫ ( n_eff / c_ref ) d ell ) # general form

IV. Scenario A — Parameter Identification (Joint Object/Environment Inversion)

Goal: Jointly invert θ_state, θ_sed, θ_path and (optionally) key SeaProfile parameters using T_arr / Delta_T_arr / F_nu / LC.

Inputs: Catalog/Seeds; Observations:{ T_arr, Delta_T_arr, F_nu, LC }; f_grid; gamma; optional SeaProfile; c_ref/CalibCref.
Outputs: theta_hat, Cov; consistency indices eta_T, tau_switch; energy-closure and lower-bound residuals.

Flow (Template → suggested I70-*):

• I.Path.Capture|Segment → { gamma[k], Δell[k] }, { ell_i }.
• I.Build.Phi|Neff → assemble n_eff.
• I.Fit.Profile / custom → fit_object_params or estimate_coupling_params.
• I.Consistency.DualMode|ThinThick → eta_T, tau_switch.
• I.Report.Emit → report + logs.

Accept: |Residual| ≤ GB; eta_T, tau_switch within gates; energy closure and lower bound pass.
Falsify: persistent n_eff < 1; long-term two-form/thin-thick inconsistency with no resolvable back-trace.

V. Scenario B — Cross-Layer Propagation (Region_in / layer / out Segmentation)

Goal: With explicit layering, compute T_arr_total and audit interface energy closure and sidedness.

Inputs: SeaProfile/Sigma_env; gamma and { ell_i }; Phi_T/grad_Phi_T or T_fil+G(•); mode; c_ref.
Outputs: per-segment T_arr_i, composite T_arr_total; {R_env,T_trans,A_sigma} residual curves and sidedness report.

Flow: apply_sea_matching → estimate_neff_* → segment_integrals (+ interface_correction_sea when thin) → estimate_energy_triplet.
Accept/Reject: energy closure & n_eff^± ≥ 1 pass/fail; lower bound & two-form consistency pass/fail.

VI. Scenario C — Band-Differential Isolation of the Path Term

Goal: Identify n_path using same-path, multi-frequency Delta_T_arr, and quantify out-of-band (OOB) leakage.

Inputs: Observations:{ T_arr }; f_grid; same gamma (share { gamma[k], Δell[k] } and segmentation/correction).
Outputs: polynomial coefficients c_m of n_path, differential correlation and slope, OOB leakage ratio.

Flow: I.Arrival.Delta → delta_arrival_in_sea / predict_arrival_signature; I.Report.Log to capture consistency & leakage.
Accept/Reject: differential linear-region / designated order passes; OOB folded into u_sys and still passes / fails.

VII. Scenario D — Thin/Thick Decision & Switching

Goal: Select and lock execution chain online by Delta_k/L_char and tau_switch.

Flow:

• pre-evaluate consistency_thin_vs_thick_* over representative paths and step regimes;
• online, record eta_w, dual-run in gate and compare tau_switch;
• if over threshold, lock thick, back-trace { ell_i } tolerances & SeaProfile;
• log_artifacts_* the reason and difference curves.

Accept/Reject: after switch, eta_T, lower bound, and energy closure hold / long-term failures recorded.

VIII. Scenario E — Long-Term Drift Monitoring & Guarding (Streaming)

Goal: Track drifts in c_ref(t), n_common(x,t), and slow variables of object/environment, preserving calibration stance.

Flow: periodic calibrate_c_ref; sliding-window fit_object_params/fit_sea_profile; compute GB = k_guard • u_c, eta_T, tau_switch; on breach, back-trace + alert; log time series via log_artifacts_*.
Accept/Reject: in-band drift accepted; unexplainable out-of-band drift → falsify current calibration.

IX. Scenario F — Risk Assessment & Guardband Setting

Goal: Prior to deployment, estimate tail risk, set guardbands and runtime thresholds.

Flow: propagate_uncertainty_MC to obtain { T_arr, Delta_T_arr, F_nu, LC } distributions; evaluate n_eff clamping rate, Delta_T_sigma trigger stats, OOB leakage; set k_guard and GB and write into the Contract.
Accept/Reject: target coverage attained / excessive tails → re-plan path/band layout or raise data-quality gates.

X. End-to-End Case — BHSeed + Layered Sea (Joint Inversion & Consistency)

Goal: For a BHSeed coupled to SeaProfile, complete joint parameter inversion, two-form and thin/thick audits, energy closure, and differential identification.

Steps:

• Prepare: build_early_object_catalog, seed_and_trigger.
• Evolve & spectrum: evolve_object_state, synthesize_spectrum.
• Propagate: estimate_neff_sea, detect_interfaces, segment_integrals (±Delta_T_sigma), predict_arrival_signature.
• Invert jointly: fit_object_params / estimate_coupling_params.
• Consistency & uncertainty: check_dual_arrival_consistency, consistency_thin_vs_thick_*, propagate_uncertainty_GUM/MC.
• Report: emit_measurement_report; persist hash(*)/SolverCfg/metric_spec, eta_T/tau_switch, {R_env,T_trans,A_sigma} residuals, falsification samples.

Accept: residuals within GB; eta_T, tau_switch, energy closure, lower bound all pass; differential linear region confirmed.

XI. Minimal Logging Set (Common to All Scenarios)

• 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 linear region & OOB leakage ratio.
• Uncertainty & replay: u_stat,u_sys,u_c, k, seed, coords_spec/units_spec/metric_spec, SolverCfg, hash manifest.

XII. Interface & Implementation Mapping (Scenario → Template → suggested I70-*)

• Scenario A: I.Path.Capture|Segment → I.Build.Phi|Neff → I.Fit.Profile/fit_object_params → I.Consistency.* → I.Report.Emit.
• Scenario B: I.Interface.ApplyMatching → I.Build.Neff → I.Path.Segment|InterfaceCorrection → I.RT.Estimate.
• Scenario C: I.Arrival.Delta → I.Consistency.DualMode → I.Report.Log.
• Scenario D: I.Consistency.ThinThick → I.Report.Log.
• Scenario E: I.Calibration.Cref → sliding-window I.Fit.Profile → write guard metrics.
• Scenario F: I.Uncertainty.MC → I.Report.Emit.

XIII. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Chs. 3–12 (objects, causation, growth, radiation, propagation, metrology, interfaces, numerics, error).
• EFT.WP.Cosmo.LayeredSea v1.0: segmentation, matching, energy closure, thin/thick consistency.
• EFT.WP.Propagation.TensionPotential v1.0: two forms and differential flows.

XIV. Deliverables

• Scenario-centric workflow checklists (A…F) with parameter templates.
• Audit templates: eta_T / tau_switch, lower-bound & energy-closure margins, differential linear-region panels.
• End-to-end case replay pack layout: data / code / parameters / SolverCfg / random seeds / hash manifest & replay scripts.