MMP.12:4.2 - Locate the part that needs regularization

Use C.16.IR’s compatible cases or sufficient bounds to establish what the current records and premises resolve. Reuse that result. This pattern adds the construction of a recovery rule and analysis of the rule’s sensitivity.

For a linear relation y=Ax, a change h with Ah=0 is invisible to the records. If both x and x+h are admitted and T(x+h) differs from T(x), full recovery of that target needs another premise. If T is unchanged, the invisible direction may be irrelevant to the current use. For unknown influences, vary x and z jointly.

Next examine changes that are visible but weak. In finite dimensions, after meaningful scaling, singular values of A describe the response to orthogonal input directions. A small nonzero singular value sigma means that direct inversion multiplies the corresponding record error by 1/sigma. Estimate the effect on T, not merely on an unnecessarily detailed reconstruction. For nonlinear models, a derivative can reveal local weak directions; it does not establish global uniqueness or exclude another branch.

Distinguish poor conditioning from a discontinuous inverse. An invertible finite matrix has a continuous inverse, although its error amplification may be unacceptable. In a function-space problem, data changes tending to zero can produce target changes that do not tend to zero. The spaces and norms determine this claim; :5.3 gives an explicit example. Refining a finite discretization can expose progressively larger amplification.

If the supported target bound is already sufficient, stop. If the premises are inconsistent, return to their diagnosis through C.16.IR. A penalty cannot make an incompatible observation account true.