A Singular-Safe Finite-State Proof for the Dakota Number Wall and Shallit's Three-Value Hankel Problem

Shallit asked whether a sequence taking only three real values can have every shifted Hankel determinant of positive order nonzero. The construction is based on Fred Lunnon's Dakota sequence. Lunnon's unpublished 2018 manuscript already proves the relevant zero-free Dakota-wall result over nontrivial ternary extensions and its characteristic-zero consequence for irrational parameters; Claudemir de Souza Cavalcante gave the first located public complete account in July 2026 and published a replayable version 2 in September. This paper gives two further finite proofs of the bilateral finite-field theorem. Over F_9 = F_3[u]/(u^2+1), with y_n = 1 + u d_n, the first proof represents the complete Dakota wall by 1,482 exact diamond tiles and uses a reconstruction procedure that remains valid in the presence of zero windows. In the verified F_9 replay every reconstructed cell takes the regular Desnanot-Jacobi branch. The scalar closures used by the second proof exercise the zero-window and singular-frame branches on genuine zeros. The direct determinant half-plane contains 188,992 valid outputs and no zero. The second proof shows that a Hankel determinant of y vanishes exactly when the corresponding determinants of d and d+1 vanish simultaneously, and excludes that simultaneous zero state by a synchronized finite certificate over F_3. For each order and shift the determinant of t+d_n is a nonzero integer polynomial in t of degree at most one. Hence every irrational real t gives a sequence taking exactly the three values t-1, t, t+1 with all shifted Hankel determinants nonzero. The record contains the manuscript PDF, the manuscript and figure source, and a portable exact proof archive. The proof archive distinguishes the complete 4,134-pattern neighbouring language from the 4,126 lower-diamond reconstruction contexts and explicitly verifies the eight input-boundary cases. After extracting the proof archive, run python -B CHECK_PUBLIC_RELEASE.py from its root. The release checker uses Python 3.9 or later and the standard library only. The uploaded proof archive has SHA-256 c0cfdaeebd3c4aef79d361a8ba466378d72d1ff4e45d8faa596ab63879a9dac1.