Cyclic deletion rigidity and Macaulay shadows in the Tu--Deng problem

Abstract

We determine all equality cases in the Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$. If the $k$-bit cyclic word of $t$ has $R$ ones, $Z$ zeros, and cyclic one-gap lengths $g_1,\ldots,g_Z$, then equality holds if and only if $g_i\ge Z-1$ for every $i$. This resolves Conjecture~3.20 of Flori, Randriambololona, Cohen and Mesnager, and we also enumerate all equality parameters. For $R\ge Z$ we determine the sharp first stability gap and all extremal words, while for $R<Z$ we obtain an exact quantization of the deficit and an explicit run-sensitive lower bound. The proofs are structural: an explicit matrix conjugation identifies the auxiliary enumerators in the two recent complete proofs of the Tu--Deng conjecture. We then develop a rooted coarsening model for all coefficients, prove one-sided deletion rigidity and an exact Macaulay-flux identity, and derive a Macaulay--M\"obius formula from the bounded simplex at the highest cyclic level.