10-EFT.WP.Core.Tension v1.0 | Chapter 9 — Identification, Uncertainty, and Information Bounds

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Chapter 9 — Identification, Uncertainty, and Information Bounds

I. Objectives and Scope

• Objective. Recover the tension field T_fil(ell,t) and its derived quantities from observations while delivering statistical uncertainty and information-theoretic lower bounds, so that estimates are comparable, auditable, and reproducible across experiment design, simulation, and deployment. Observation types include: arrival time T_arr, modal frequencies f_n, spectra S_xx(f), node reflection R_ref, and transmission T_trans (dimensionless).
• Scope. We work with a parameter vector theta (e.g., theta = { T0 , rho_l , ... } or piecewise parameters { T_i , rho_{l,i} }), covering likelihood construction, estimators, Fisher information, the Cramér–Rao lower bound (CRLB), multi-source fusion, and experiment design.
• Conventions. All variables and measures are explicit; along the path gamma(ell) we write ( ∫ • d ell ). T_fil (N) must not be conflated with T_trans (dimensionless). The cross-calibration between c = sqrt( T / rho_l ) and c = L_gamma / T_arr follows Chapter 4 and Chapter 8.

II. Observation Channels and Forward Models

• Arrival-time (mean model):
  h_toa(theta) = ( ∫ ( 1 / c(ell;theta) ) d ell ), with c(ell;theta) = sqrt( T_fil(ell;theta) / rho_l(ell;theta) ).
  Under dual-form calibration, T_arr can also be obtained from n_eff, c_ref (see Chapter 8).
• Modal frequencies (uniform-string approximation):
  f_n(theta) = ( n / ( 2 * L_gamma ) ) * sqrt( T0 / rho_l ), n = 1,2,....
  For non-uniform strings, solve the eigenproblem ∂_ell( T_fil ∂_ell u ) + rho_l (2*pi*f)^2 u = 0 with boundary conditions and normalization to obtain the discrete spectrum.
• Reflection/transmission at a junction:
  r_amp = ( Z2 - Z1 ) / ( Z2 + Z1 ), R_ref = | r_amp |^2, T_trans = 1 - R_ref (lossless), Z = rho_l * c.
  Measured R_ref constrains the ratio of impedances, hence indirectly T_fil or rho_l.
• Spectral channel (aligned with Core.Sea):
  S_vv(f) = ( 2 * pi * f )^2 * S_uu(f), S_PP(f) = Z * S_vv(f) (lossless segment). With window correction U_w and equivalent noise bandwidth ENBW_Hz applied, use the spectrum to characterize amplitudes and noise.

III. Noise Models and Likelihood

• Joint Gaussian across channels. Stack observations as y = [ y_toa , y_f , y_R , y_S ]^T, with y = h(theta) + w, w ~ N(0, Σ).
  The negative log-likelihood is -log L(theta) = (1/2) * ( y - h(theta) )^T Σ^{-1} ( y - h(theta) ) + const.
• Non-Gaussian and robust alternatives. Use Huber loss rho_H(•) or Student–t likelihood for impulsive noise/outliers; use circular/Rice likelihoods for phase-/envelope-based TOA.
• Spectral leakage and windows. In the S_xx(f) channel, embed U_w and ENBW_Hz explicitly into the diagonal of Σ for frequency-domain weighting.

IV. Estimators (MLE / MAP / WLS)

• Maximum likelihood (MLE):
  theta_hat = argmin_theta ( y - h(theta) )^T Σ^{-1} ( y - h(theta) ).
  Gauss–Newton iteration:
  theta_{k+1} = theta_k + ( J^T Σ^{-1} J )^{-1} J^T Σ^{-1} ( y - h(theta_k) ), with J = ∂h/∂theta.
• MAP with priors:
  -log p(theta|y) = -log L(theta) - log p(theta). For an engineering prior T0 ~ N(μ_T, σ_T^2), add the corresponding quadratic penalty on T0.
• WLS closures (by channel). Under the uniform-string model, let g_n(theta) = f_n - ( n / ( 2 * L_gamma ) ) * sqrt( T0 / rho_l ). Linearizing yields a closed-form update for ΔT0. A similar first-order solution exists for the T_arr channel via Δc, then back-projected to ΔT0.

V. Fisher Information and CRLB (Core Relations)

• Information (general):
  S72-15 : I_F(theta) = E[ ( ∂_theta log L )^T ( ∂_theta log L ) ] = - E[ ∂^2_{theta,theta} log L ].
  For Gaussian residuals, I_F(theta) = J^T Σ^{-1} J.
• Lower bound:
  S72-16 : cov( theta_hat ) ≥ I_F^{-1}(theta) (Cramér–Rao bound; Löwner partial order).
• Uniform string, frequency set { f_n }, Σ_f = diag(σ_{f,n}^2) (closed forms):
  ∂ f_n / ∂ T0 = ( n / ( 4 * L_gamma ) ) * ( 1 / sqrt( T0 * rho_l ) ),
  ∂ f_n / ∂ rho_l = - ( n / ( 4 * L_gamma ) ) * sqrt( T0 ) / ( rho_l^{3/2} ).
  Hence I_F = J_f^T Σ_f^{-1} J_f, var( T0_hat ) ≥ [ I_F^{-1} ]_{T0,T0}.
• TOA single-parameter bound (c = L_gamma / T_arr):
  ∂ c / ∂ T_arr = - L_gamma / T_arr^2, ∂ T0 / ∂ c = 2 * rho_l * c,
  u^2( T0_hat ) ≥ ( 2 * rho_l * c )^2 * ( L_gamma^2 / T_arr^4 ) * u^2( T_arr ) (first-order propagation).

VI. Identifiability and Experimental Design

• Structural identifiability. A necessary condition is rank( I_F ) = dim(theta). With { f_n } alone, T0 and rho_l are coupled proportionally; include T_arr or R_ref/Z to break that scale ambiguity.
• Excitation and sampling. Choose { n } coverage and the geometric baseline L_gamma to maximize trace( I_F ) or minimize det( I_F^{-1} ) (D-optimality).
• Boundary/connectivity. At junctions, concurrent acquisition of R_ref improves sensitivity to Z, reducing var( T0_hat ), subject to the energy constraint of Chapter 6.

VII. Multi-Source Fusion and Joint Likelihood

• Joint likelihood (conditionally independent):
  log L_tot(theta) = log L_toa(theta) + log L_f(theta) + log L_R(theta) + log L_S(theta).
  Equivalently, a block-diagonal WLS; information adds: I_F^{tot} = I_F^{toa} + I_F^{f} + I_F^{R} + I_F^{S}.
• Nuisance elimination. Use profile-likelihood or the Schur complement to eliminate { rho_l , damping , ... } and obtain marginal information and bounds for T0.
• Cross-device weighting. Each channel’s Σ must include contributions from window correction, bandwidth, and time-base registration uncertainty (cf. Chapter 8).

VIII. Interval Estimation and Uncertainty Propagation

• Linearized confidence intervals:
  theta_hat ± z_{alpha/2} * sqrt( diag( I_F^{-1} ) ) (z ≈ 1.96 for ~95%).
• Delta method for derived quantities:
  u^2( g(theta) ) ≈ ( ∂_theta g )^T cov(theta_hat) ( ∂_theta g ), e.g., for c = sqrt( T0 / rho_l ), Z = rho_l * c, T_star = T0 / T_ref.
• Resampling and Bayesian alternatives. For strong nonlinearity or small samples, use bootstrap or posterior sampling (e.g., random-walk MH) to produce bands or credible intervals.

IX. Bias Sources and Robustness

• Modeling bias. Unmodeled bending stiffness, pre-tension gradients, or variability in rho_l(ell) induce systematic errors; expand theta or add suitable penalties in the likelihood.
• Arrival-time gauge bias. If the two T_arr forms diverge beyond tolerance (delta_form), down-weight or exclude the TOA channel and trace the sources for n_eff, c_ref (Chapter 8).
• Robust estimation. Deploy Huber/Tukey weights in the objective; mask phase flips and amplitude saturation in R_ref.

X. Consistency and Validation

• Energy consistency. At a lossless interface, verify R_ref + T_trans = 1 (Chapter 6) and regularize extreme frequencies with this constraint.
• Cross-gauge check. The ratio between c_hat = L_gamma / T_arr and c_hat = ( 2 * L_gamma / n ) * f_n should be close to 1; deviations localize geometric or time-base error.
• Extrapolation and leave-one-out. Fit a subset of modes, predict the rest; examine standardized residuals and spectral consistency error ΔE.

XI. Standard Workflow (Mx-78: Tension → Uncertainty → Information Bounds)

• Load data & metadata. trace, gamma(ell), L_gamma, rho_l(ell), boundary and junction info, window U_w and ENBW_Hz.
• Extract channels. Measure T_arr (both forms, record delta_form), { f_n }, R_ref, and the required S_xx(f).
• Build the likelihood. Specify y, h(theta), Σ; choose robust likelihood where outliers exist.
• Initialization. From Chapter 4 & Chapter 8 derive c_init, T0_init = rho_l_mean * c_init^2; use R_ref to seed Z ratios at interfaces.
• Iterative estimation. Solve theta_hat = argmin ( y - h(theta) )^T Σ^{-1} ( y - h(theta) ) until step/residual convergence.
• Covariance and bounds. Compute I_F = J^T Σ^{-1} J; publish cov(theta_hat) ≈ I_F^{-1} and CRLB; propagate to derived quantities via the Delta method.
• Consistency checks. Energy/gauge consistency, leave-one-out prediction, residual normality/independence diagnostics.
• Report & bind. Publish theta_hat, cov, CRLB, delta_form, and quality metrics; register via I70-7 the uncertainty/bounds reference object.
• Decision/design loop. Use I_F sensitivity maps to re-plan experiments (mode selection, baseline L_gamma, interface placement) to reduce det( I_F^{-1} ).

XII. Interfaces and Implementation Mapping (I70-7 and Related)

• fisher_information_tension(model:any, theta:dict) -> matrix — returns I_F = J^T Σ^{-1} J or expected form (supports multi-channel stacking and block-diagonal Σ).
• crlb_tension(model:any, theta:dict) -> matrix — returns I_F^{-1} and Delta-method uncertainties for derived quantities.
• Related: estimate_tension_from_modes(...), calibrate_tension_by_toa(...) supply initial values/channel pieces; junction_solve(...), transmission_coeff(...) inject the R_ref / T_trans constraints.
• Outputs: theta_hat, cov_theta, crlb_theta, u(T_star), u(c), u(Z), delta_form, and consistency indicators.

XIII. Required Symbols and Cross-References

• Symbols. theta, J, Σ, I_F, CRLB, T0, rho_l, c, Z, f_n, T_arr, R_ref, T_trans, delta_form, L_gamma.
• Cross-references. c = sqrt( T0 / rho_l ) and the wave equation (Chapter 4); c = L_gamma / T_arr and dual-form T_arr (Chapter 8); impedance and transmission (Chapter 6); measures and geometry (Chapter 1); constitutive/static priors (Chapter 2); interface bindings and data conventions (Appendix E, Appendix B).
• Conflict discipline. T_fil (N) vs. T_trans (dimensionless) are strictly distinct. Any integral must specify path and measure explicitly, e.g., ( ∫ n_eff d ell ), ( ∫ ( 1 / c(ell) ) d ell ).