# Verification of the scoped result

The explicit four-level Lagrange formula proves interpolation for all
compatible data on the constructed family. The sum of the first two
vanishing ideals is (x^m,y,z), with exact quotient nilpotence order m.
The construction is irredundant and uses proper reduced pure curves.
Both directions of the hereditary replacement and the finite-intersection
extension argument are written in full.

Normal and optimized standard-library Python runs each pass 2,119 explicit
checks. These include 613 Buchberger S-pairs, two-sided ideal membership
certificates, 80 polynomial restriction fixtures and examples m=2,...,7.
A deliberately corrupted Lagrange coefficient is rejected. The tests are
diagnostic support, not a substitute for the all-data, all-m proof.

The originating researcher has self-audited the stated scope. No known
mathematical gap remains in that scope. Independent review, formal
verification and novelty certification have not occurred. The general
frozen interpolation problem and the source's PDE questions are not
marked solved by this paper.

The desktop LaTeX compiler succeeds after correcting a nested optional
citation argument. The portable PDF was also compiled with Tectonic.
Final page and source-archive checks are recorded in the package metadata.