33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 7 — Metrology & Observable Design

Chapter 7 — Metrology & Observable Design

I. One-Sentence Goal

Design a traceable, auditable, and replayable execution convention centered on the observables T_arr / Delta_T_arr / F_nu / LC. The plan combines multi-band on the same path with multi-path across incidence angles, defines two-form consistency and energy-closure audits, and specifies GUM/MC uncertainty propagation and reproducibility so that the chain object → environment → propagation → observation is end-to-end accountable.

II. Scope & Non-Goals

• Covered: measurable targets and definitions; experiment-design principles; a stepwise workflow M70-*; audits for two-form consistency and energy closure; thin/thick dual-chain consistency; GUM/MC propagation; minimal data & logging set; interface mapping and publication conventions.
• Not covered: re-deriving the modeling and equations of Chs. 3–6; instrument-response and pipeline specifics; any construct that violates n_eff ≥ 1 or circumvents R_env + T_trans + A_sigma = 1.

III. Minimal Terms & Symbols

1. Object & state: O_i, state(t); type ∈ { PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud }.
2. Propagation & path: n_eff(x,t,f) (dimensionless, ≥ 1), c_ref, path gamma(ell), measure d ell, segments gamma_i, interface set Sigma_env.
3. Observables & differentials: T_arr(f, gamma), Delta_T_arr(f1,f2, gamma), F_nu(f), LC(t).
4. Energy closure: R_env + T_trans + A_sigma = 1.
5. Two-form arrival-time dialects:
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

IV. Measurement Targets & Measurability

• Single-frequency, single-path arrival time: T_arr_obs(f, gamma) (unit: s).
• Same-path band differential:
  Delta_T_arr_obs(f1,f2, gamma) = T_arr_obs(f1, gamma) − T_arr_obs(f2, gamma).
• Spectrum & light curve: F_nu_obs(f) (observer frame), LC_obs(t) (observer time base).
• Layer/interface measurables: crossing count N_cross; explicit endpoints { ell_i } with numerical tolerances and interface type tags (continuous / potential-jump / flux-jump / directional).
• Energy-triplet estimators: in-band curves with brackets for R_env(f), T_trans(f), A_sigma(f).

V. Experiment-Design Principles

• Multi-band on the same path. Use Delta_T_arr_obs to cancel n_common and isolate the path term; the two frequency points must share { gamma[k], Δell[k] }, segmentation, and the Delta_T_sigma configuration.
• Cross-angle, multi-path. Vary incidence and geometry to identify structural differences and directional terms (if the model enables dot( grad_Phi_T , t_hat / n_vec )).
• Explicit segmentation. Include endpoints { ell_i } explicitly in integrals; forbid cross-interface interpolation; apply symmetric step-shrinking near { ell_i }.
• Two-form dual solve. Produce T_arr^{const}, T_arr^{gen}, and eta_T in parallel; exceeding the gate first triggers back-tracing of c_ref calibration and the n_eff decomposition.
• Thin/thick dual chain. Run both links within the eta_w neighborhood and report tau_switch; if exceeded, fix to thick.
• Energy-closure audit. For each interface/frequency, output residual curves for R_env / T_trans / A_sigma.

VI. Stepwise Workflow (M70-4 … M70-12)

• M70-4 Path capture & segmentation
  Acquire { gamma[k], Δell[k] }; perform interface detection to obtain { ell_i } and type tags; fix endpoint tolerances tol_x and persist interface_marks.
• M70-5 Band & path orchestration
  Design f ∈ { f_m } and select path sets { gamma_a }; same-path multi-band for differential isolation, cross-angle for directionality and geometry discrimination.
• M70-6 Reference-speed calibration
  Calibrate c_ref using gamma_ref, T_arr_ref; record u_stat(c_ref), u_sys(c_ref) and associated environment blocks.
• M70-7 Joint observation of spectrum & arrival
  Acquire F_nu_obs(f), T_arr_obs(f, gamma), LC_obs(t) synchronously; persist coords_spec / units_spec / metric_spec to the Contract.
• M70-8 Two-form consistency audit
  Compute eta_T = | T_arr^{const} − T_arr^{gen} |; if over the gate, back-trace c_ref, n_eff decomposition, and segmentation implementation.
• M70-9 Differential isolation & OOB assessment
  Evaluate the correlation and slope of Delta_T_arr_obs / Delta_T_arr_mod; fold out-of-band residuals into u_sys and log the leakage ratio.
• M70-10 Energy closure & interface audit
  Estimate in-band curves and brackets of R_env / T_trans / A_sigma; verify R_env + T_trans + A_sigma = 1 and n_eff^± ≥ 1.
• M70-11 Thin/thick consistency
  In the eta_w neighborhood, dual-solve T_arr^{thin} + Delta_T_sigma vs T_arr^{thick}, output tau_switch; if not met, lock to thick and back-trace SeaProfile and endpoint tolerances.
• M70-12 Archival & replay
  Compile a metrology report and a reproducibility bundle: Contract/Log, hash(*), SolverCfg, RNG seed, falsification samples, and replay handles.

VII. Uncertainty Models & Propagation (GUM / MC)

GUM (constant-factored, discrete form)

T_arr ≈ (1/c_ref) * ∑ n_eff[k] • Δell[k]

∂T/∂c_ref = −T_arr/c_ref

∂T/∂n_eff[k] = Δell[k]/c_ref

∂T/∂Δell[k] = n_eff[k]/c_ref

Correlations are incorporated via the covariance.

GUM (general, discrete form)

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

∂T/∂n_eff[k] = Δell[k]/c_ref[k]

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

MC (primary for nonlinearity / clamping / event triggers)
Jointly sample { c_ref, n_eff, Δell } with shared correlations preserved; compute distributions, quantiles, and tail metrics for T_arr / Delta_T_arr. When clamping n_eff ∈ [1, n_max] or discrete triggers Delta_T_sigma are active, prefer MC for reporting.

VIII. Traceability & Minimal Logging Set

• Physics & geometry: hash(state), hash(SeaProfile), hash(Phi_T), hash(n_eff), hash(gamma), Sigma_env tags and { ell_i } tolerances.
• Conventions & thresholds: mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr_obs − L_path/c_ref.
• Energy & differentials: residuals for R_env / T_trans / A_sigma, counts for Delta_T_sigma triggers, differential linear-regime checks and OOB leakage ratio.
• Uncertainty: u_stat, u_sys, u_c, GUM/MC configurations, k, RNG seed.
• Reproducibility: SolverCfg, coords_spec / units_spec / metric_spec, a hash manifest, and replay commands.

IX. Acceptance Criteria & Falsification Lines

1. Accept if
   • T_arr_obs ≥ L_path / c_ref;
   • eta_T within gate;
   • tau_switch within gate;
   • R_env + T_trans + A_sigma = 1;
   • Delta_T_arr meets linear-regime / specified-order criteria;
   • dimensional checks pass; segment endpoints { ell_i } explicitly integrated with no cross-interface interpolation.
2. Falsify if
   • Stable n_eff < 1;
   • Persistent two-form inconsistency with no resolution;
   • tau_switch over threshold;
   • energy-closure violation;
   • differential linear-regime failure not attributable to OOB leakage;
   • segmentation errors in the path implementation.

X. Interfaces & Implementation Bindings (aligned with template I70-*)

• calibrate_c_ref( gamma_ref, T_arr_ref, n_eff_ref, mode ) -> c_ref
• capture_path( raw_track, coord_spec ) -> { gamma[k], Δell[k] }
• detect_interfaces( gamma, Sigma_env ) -> { ell_i } (or reuse LayeredSea interfaces)
• predict_arrival_signature( n_eff, gamma, mode, c_ref ) -> { T_arr, Delta_T_arr }
• estimate_energy_triplet( data, Sigma_env ) -> { R_env, T_trans, A_sigma }
• propagate_uncertainty_GUM/MC( inputs ) -> { u_c or dist }
• emit_measurement_report( contract, logs, artifacts ) -> Report
  Constraints: consistent units/dimensions at entry; mandatory audits for n_eff ≥ 1, lower bound, and energy closure; naming isolation is required.

XI. Data Packaging & Publication Convention

• Objects: Contract / SeaProfile / Path / Observations / RTParams / CalibCref / Report.
• Formats: JSONL/Parquet (static), Zarr/NetCDF (gridded fields); timestamps in ISO-8601 (UTC).
• Publication: no external links in the main body; references, hashes, and replay commands are appended in the report; include falsification samples and replay handles.

XII. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (minimal equations), Ch. 4 (seed/trigger), Ch. 5 (coupling & growth laws), Ch. 6 (radiation & propagation), Ch. 9 (numerics), Ch. 12 (error budget).
• EFT.WP.Cosmo.LayeredSea v1.0: Chs. 6–8 (segmentation, interface matching, consistency).
• EFT.WP.Propagation.TensionPotential v1.0: two forms, differentials, and path conventions.

XIII. Deliverables

• Metrology workflow manifest: inputs/outputs and threshold fields for M70-4 … M70-12.
• Acceptance & falsification templates: eta_T, tau_switch, lower-bound margins, energy-closure margins, and GB configuration.
• Reproducibility & audit bundle: minimal datasets, code, contracts, logs, and a hash manifest.