67. A declared template can be recognized without being rediscovered
On this page · Permalink · Source line 2723
General curved-domain learning remains difficult. There is a useful finite case between supplying a finished coordinate certificate and claiming unrestricted domain discovery.
Suppose a structural library supplies a certified template with polynomial coefficients \(C_{a,k}\), mass row \(\lambda\) and embedding \(\mathcal J\). Suppose observed population tables produce coefficients \(\widehat C_{a,k}\) and mass row \(\widehat\lambda\) of the same dimension.
Proposition 60: Exact recognition of a fixed certified template
The equations
are linear in \(T\). Every invertible solution transports the template certificate to the fitted coordinates by \(\widehat{\mathcal J}=\mathcal JT^{-1}\). If the fitted coefficient family is observable, meaning the rows \(\widehat\lambda\widehat C_w\) span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.
Read proof · Recorded Lean scope · Permalink · Source line 2729
Proof
All coefficient matrices are fixed, so the displayed constraints are linear. An invertible solution yields \(\widehat B_a(u)=TB_a(u)T^{-1}\) coefficient by coefficient. Substitution in the template intertwining identity proves the transported certificate. To prove uniqueness, let \(D\) be the difference of two solutions. It satisfies \(\widehat C_{a,k}D=DC_{a,k}\) and \(\widehat\lambda D=0\). Therefore \(\widehat\lambda\widehat C_wD=0\) for every coefficient word. Observability makes every column of \(D\) zero. An initial state outside the transported cone is not repaired by this operator identity, which explains the separate check. \(\square\)
For the worked example, the linear system has the unique solution \(T=Q\) from Section 66. The learner receives the population tables and a fixed structural template. It reconstructs the operators, solves the linear match, transports the embedding and exports a certificate. Independent replay then needs only that certificate. It does not refit, search, or import the template or generator.
This is template recognition, not unrestricted geometry discovery. The supplied template includes the specific operator family. An incompatible model is rejected rather than changed to fit the template. The reference routine also withholds a result for an underdetermined or singular match. Noise would require a declared approximate match and a separate validity or correction argument. This edition's recognition example is exact.
For fixed embeddings, polynomial Gram optimization offers a broader construction route than fixed-template matching. That convex proposal problem is stated in Section 59 but is not implemented as a general solver here. The independent checker remains useful regardless of whether a proposal came from a template, symbolic derivation or an optimization routine.