Bounds to qualify before native prediction

These equations describe an assumed Gaussian error family with explicit deterministic model-error bounds. They do not establish those assumptions for airborne observations. The native adapter and its global remainder treatment remain unqualified.

Shared scale

For one covariance alternative, let a fixed calibration reference satisfy \(z=Xb+e+s\eta\), with \(\eta\sim N(0,I)\) and \(\|e\|\leq B\). Project away the fixed reference design and call the observed residual norm \(R\). If its residual degrees of freedom are \(\nu>0\), then

\[ s\leq s_+=\frac{R+B}{\sqrt{\chi^2_\nu(\alpha_s)}} \]

except on a lower-tail event of probability \(\alpha_s\). The reference coefficient bounds cover any numerically discarded design response. No positive degrees of freedom, unknown remainder, or an adaptively learned reference design means no scale bound is supplied.

This reference is shared across physical candidates. Re-estimating its scale from each candidate's own errors would be a different method. No independence between the shared calibration and later training events is required for the error-budget union bound.

Bounded nuisance region

In whitened candidate training coordinates, retain every parameter column. Write the retained numerical decomposition as \(X_r=U_rS_rV_r^T\). Bound \((X-X_r)b\) over the entire declared parameter box and add that bound to the training remainder. Then

\[ \|S_rV_r^T(b-\hat b)\|\leq s_+\sqrt{\chi^2_k(1-\alpha_b)}+B_{\rm total}=\rho. \]

For a fixed design, \(k=r\). For a Jacobian learned from the same observations, use \(k=n\), the full training-noise dimension: its projection depends on the noise and cannot claim a fixed-rank chi-square distribution. Zero-rank designs retain the whole parameter box rather than zero uncertainty.

For a prediction direction \(l\), separate its row-space and null components. The row-space support is bounded by \(l^T\hat b\pm\rho\|S_r^{-1}V_r^Tl\|\). Add the null component's support over the original box, then intersect with the full-box support. This is an outer bound; it does not assert that every point in the resulting rectangular prediction region is jointly feasible.

Conditional prediction and deterministic error

Let \(A=C_{pt}C_{tt}^{-1}\). For approximate training/future means \(X_tb\) and \(X_pb\), the prediction center and nuisance directions are

\[ c=Ay_t,\qquad D=X_p-AX_t. \]

Known intercept terms are included consistently before whitening. Conditional future noise is Gaussian with covariance \(C_{pp}-C_{pt}C_{tt}^{-1}C_{tp}\). For \(m\) prediction coordinates, the simultaneous marginal noise allowance uses \(s_+\Phi^{-1}(1-\alpha_p/(2m))\sqrt{v_i}\). Bonferroni requires no coordinate independence.

If \(\|L_t^{-1}e_t\|\leq B_t\), the additional conditional-mean error is bounded coordinatewise by \(B_{p,i}+\|(A L_t)_{i,:}\|B_t\). The structured forecast calculates these response norms without a dense cross-covariance. Omitting the training-remainder term would understate uncertainty.

Family and selection limits

Allocate \(\alpha_s+\alpha_b+\alpha_p\) before evaluation. Under the stated assumptions this bounds false interval exclusions without relying on independence of the three events. Union the envelopes across the predeclared covariance family; if the true member belongs to that family, its valid envelope is retained. Unsupported alternatives are not silently dropped.

The bound is not a coverage claim conditional on passing a convergence, selection or adequacy gate. An unsupported calculation abstains; abstention frequency must be reported. Software coverage controls count unsupported outputs as failures rather than allowing an always-abstaining method to qualify.

Finite-family sensitivity does not cover arbitrary unmodeled dependence. Nonlinear remainder limits must hold over the declared parameter region; saved axis probes cannot prove that. These equations supply no production decision threshold or calibrated significance for the existing 53 preferences.