22-EFT.WP.Metrology.Instrument v1.0 | Chapter 2 — Postulates & Minimal Equations (Instrument Modeling Baseline)

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Chapter 2 — Postulates & Minimal Equations (Instrument Modeling Baseline)

One-Sentence Objective
Establish a unified family of modeling and metrological equations for electrical/optical/acoustic/RF/multi-modal instruments, serving as the computational baseline for implementation and compliance across this volume.

I. Scope and Objects

1. Scope
   • Measurement mapping from the applied stimulus x(t) through the instrument chain to the readout y, together with uncertainty publication.
   • Coverage of static and dynamic characteristics, broadband noise and discrete artifacts, and time-/path-dependent terms (arrival time).
2. Objects
   • A general measurement function y = f( x; theta, RefCond, tau_mono ), with linear and nonlinear regimes.
   • Frequency response H(f), noise power spectral density S_n(f), quantization and sampling processes, and environmental correction corr_env.
3. Outputs
   Calibration coefficients theta, corrected values y_corr, combined standard uncertainty u_c and coverage uncertainty U = k * u_c, the two arrival-time forms, and delta_form.

II. Terms and Variables

• Signals & responses: x(t), y_raw, y_corr, H(f), phi(f) (phase), tau_g(f) (group delay).
• Sampling & quantization: FS, N, LSB, ENOB, f_s, RBW, VBW, ENBW.
• Noise & jitter: S_n(f), Vn_rms, sigma_t (time jitter), L(f) (phase noise).
• Uncertainty: u(x), u_c, U = k * u_c, J (Jacobian), Sigma (covariance).
• Environment & arrival time: RefCond, corr_env(x; RefCond), T_arr, gamma(ell), delta_form.
• Dimensions & units: unit(z), dim(z), check_dim(expr).

III. Postulates P702-*

• P702-1 (Explicit measurement function): all published quantities satisfy y = f( x; theta, RefCond, tau_mono ), with unit(y) and dim(y) declared.
• P702-2 (Integrated timebase): compute and window internally on tau_mono; publish on ts, recording offset/skew/J.
• P702-3 (Parallel arrival-time forms): whenever propagation/path matters, compute in parallel
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ), and persist delta_form.
• P702-4 (Dimensional conservation): before publication, enforce check_dim( y - f(x) ) = 0.
• P702-5 (Small-signal first): near the operating point, prefer y ≈ (h * x)(t) + n(t); outside the linear region, supply nonlinear corrections or lookup tables.
• P702-6 (Explicit dependence/correlation): encode independence/correlation of error terms explicitly in variance propagation.
• P702-7 (ENBW invariance): data reduction and windowing must preserve the target measurement’s ENBW definition.
• P702-8 (Traceable & reproducible): TraceID spans data and certificates; {theta, RefCond, seed} enables statistical reproduction.

IV. Minimal Equations S702-*

• S702-1 (LTI representation):
  y(t) = (h * x)(t) + n(t), frequency domain Y(f) = H(f) * X(f) + N(f).
• S702-2 (Quantization & LSB):
  LSB = FS / 2^N, quantization variance sigma_q^2 = ( LSB^2 ) / 12, u_q = sqrt( sigma_q^2 ).
• S702-3 (Noise integration):
  Vn_rms^2 = ( ∫_B |H(f)|^2 S_n(f) df ), where B is the effective bandwidth;
  ENBW = ( ∫ |W(f)|^2 df ) / ( |W(0)|^2 ) (window W(f)).
• S702-4 (SNR bound from jitter): for sinusoidal measurement, SNR_j = - 20 * log10( 2 * pi * f_in * sigma_t ).
• S702-5 (Group delay & phase):
  tau_g(f) = - ( d phi(f) ) / ( d ( 2 * pi * f ) ), phi(f) = arg( H(f) ).
• S702-6 (Linearity & scale correction):
  y_corr = a1 * y_raw + a0 or piecewise y_corr = LUT( y_raw ), recording the valid range under RefCond.
• S702-7 (Variance propagation):
  u_c^2 = J * Sigma * J^T, coverage uncertainty U = k * u_c.
• S702-8 (Weighted mean & effective sample size):
  hat{y} = ( ∑ w_i y_i ) / ( ∑ w_i ), n_eff = ( ( ∑ w_i )^2 ) / ( ∑ w_i^2 ).
• S702-9 (Dynamic range & noise floor):
  DR_dB = 20 * log10( V_fullscale_rms / Vn_rms ).
• S702-10 (Difference between arrival-time forms):
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
• S702-11 (Dimensional checks):
  check_dim( y - f(x) ) = 0, unit( Vn_rms ) = unit(y), dim( Vn_rms ) = dim(y).

V. Metrology Procedure M70-2 (Modeling Baseline)

• Readiness: complete time_align and anchor RefCond; acquire calibration data and noise baselines.
• Modeling: estimate H(f) and theta; delineate linear regions and nonlinear correction rules.
• Noise & jitter: measure S_n(f), compute ENBW and Vn_rms, evaluate sigma_t and SNR_j.
• Environmental correction: apply y_corr = corr_env( f(x; theta); RefCond ) and log the applicability window.
• Uncertainty: with J, Sigma, compute u_c and U = k * u_c, stating the coverage factor k.
• Arrival time: compute the two T_arr forms in parallel and record delta_form.
• Verification & persistence: enforce contracts C70-2*, generate manifest.instrument.*, and sign artifacts.

VI. Contracts & Assertions (Selected)

• C70-21 check_dim: check_dim(all) = true.
• C70-22 enbw.consistency: processing-chain ENBW matches configuration (tolerance tol_enbw).
• C70-23 noise.floor: Vn_rms ≤ Vn_spec_max.
• C70-24 linear.region: at the operating point, |INL| ≤ inl_max and |DNL| ≤ dnl_max, with no missing codes.
• C70-25 jitter.bound: SNR_j ≥ snr_min_jitter.
• C70-26 arrival.forms: delta_form ≤ tol_Tarr.
• C70-27 coverage.k: coverage factor k matches the stated confidence; U lies within guard bands.
• C70-28 weights.norm: ( ∑ w_i ) / N ≈ 1 and n_eff / N ≥ neff_min_ratio.

VII. Implementation Bindings I70-2* (Interface Prototypes)

• fit_transfer_function(ds, method) -> H(f), phi(f), tau_g(f)
• estimate_noise_psd(ds, window, avg) -> S_n(f), ENBW
• infer_jitter_from_phase_noise(Lf, f_in) -> sigma_t, SNR_j
• linearize_readout(ds, scheme) -> y_corr, theta
• propagate_uncertainty(y_corr, J, Sigma, k) -> u_c, U
• apply_env_correction(y, RefCond) -> y_corr
• compute_arrival_two_forms(path, n_eff, c_ref) -> T_arr_1, T_arr_2, delta_form
• assert_instrument_baseline(ds, rules) -> report
  Invariants: delta_form ≤ tol_Tarr; sum(w)/N ≈ 1; unit/dimension assertions always hold; interfaces are idempotent; TraceID is complete.

VIII. Cross-References

• Timebase and jitter metrics: see EFT.WP.Metrology.TimeBase v1.0, Chapters 6–7.
• Sync servo and link asymmetry: see EFT.WP.Metrology.Sync v1.0, Chapters 6–7.
• Data cleaning and release freeze: see EFT.WP.Methods.Cleaning v1.0, Chapter 10.
• Optical imaging response mapping: see EFT.WP.Methods.Imaging v1.0, Chapter 5.

IX. Quality & Risk Control

• SLI/SLO (recommended): latency_ms_p99, contract_violation_rate, recalibration_age_days, drift_level, uptime.
• Drift strategy: when drift_level > threshold, trigger re-calibration or switch to redundant channels, preserving the lower bounds on U and ENOB.
• Audit & rollback: retain computational traces and hash_sha256(blob); upon violation of C70-2*, roll back to the last valid certificate.

Summary
The chapter establishes a common baseline for instrument modeling and metrology via postulates P702-* and equations S702-*, complemented by the M70-2 workflow, contracts C70-2*, and interfaces I70-2*. Together, they support subsequent chapters on calibration, operation, and compliant publication.