Sharp local polynomial valuation under Lucas domination

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_proof: The leading form stays nonzero on X+Y=d when d is a unit, and its root count supplies a Lucas-admissible next digit with no extra valuation.

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The proof, by Plasma AI, is author-recorded at tier 0. It has no independent review or formalization. It uses Taylor expansion, the root bound for a polynomial over a field, and the elementary Lucas criterion for a binomial coefficient to be a unit modulo a prime.

The quintic higher-digit test uses this result to delimit a local polynomial approach to Problem 699. At a prime outside the finite set dividing the Taylor form's content, a next digit $d\ge\mu$ does not by itself force an extra factor $p$. The witnessing column is only local: the statement requires neither $5<j\le n/2$ nor simultaneous domination at other active primes. It supplies no infinite family of bad triples and no uniform obstruction to such triples.

Verification and remaining obligations

The complete argument is in the proof page. To check it, reduce the leading Taylor form on the line $X+Y=d$, verify that this polynomial is nonzero, and count its roots among the allowed next digits. The remaining digits can all be zero, so the construction is explicit and needs no computational evidence entry point. No mathematical gap remains in this local statement; the simultaneous-prime problem remains open in the linked research.