The full zero-sum prefix-count characterization

For every positive integer sequence satisfying a constant integer recurrence, let $P$ be the characteristic polynomial of its minimal recurrence. For every finite $D\subset\mathbb Z$, count the ordered original-coordinate assignments $S_n(D)=#{d\in D^n:\sum d_i a_i=0}$. The assertion is that all these count sequences are C-finite if and only if $P$ has no unit-circle roots, all its expanding roots are simple, and all ratios of two expanding roots are roots of unity. The target is the entire displayed iff of MathOverflow 516232 revision 2.

Cayley–Hamilton makes observations of powers of one finite matrix satisfy a fixed recurrence. The issue is whether exact prefix counts have such a representation under the proposed root restrictions.

The source's sequence class includes alternating growth scales. Positive terms need not increase, and repeated equal weights retain distinct digit coordinates. A fixed-base word count alone does not identify these prefix counts.

For $a_{2k-1}=4^k$, $a_{2k}=2^k$, regroup the actual prefix coordinates by binary exponent and count their ordered local assignments. Bounded carries give finite integer transfer matrices. Adding four to the prefix length adds the lower pair of transfer factors on the left and the upper pair on the right. Vectorization and a four-phase state turn that evolution into a single integer matrix recurrence.

One positive integer recurrence sequence whose exact counts are C-finite for every finite integer digit set, and whose actual minimal polynomial violates one asserted root clause, refutes the whole characterization.

The versioned source is John C's question 516232 revision 2, revision GUID 583B6E92-78BD-4BC4-89F2-21EB26E02150, under CC BY-SA 4.0. The corresponding library note states the source and literature limits. The polynomial for the displayed alternating sequence is $(X^2-4)(X^2-2)$; the ratio $2/\sqrt2$ has modulus greater than one.

theorem. The proposed resolution is a refutation of the entire iff by falsifying its only-if direction. A replacement classification is a separate question.

What the settlement shows

Proved by D5/S1/Recurrence/MoZeroSumCountRefutation.result: the full characterization is false. The decisive bridge retains every ordered original coordinate while grouping weights by binary exponent. Its multiplicity word grows by fixed factors on both ends, giving a finite integer matrix evolution after vectorization and phase synchronization. The proof includes every finite integer digit set, empty sets, and all positive prefix lengths. A replacement classification, extensions to other alternating sequences, and consequences for any other source conjectures remain open; no such extension is asserted here.

Historical priority and complete prior coverage by Frougny's original 1992 article remain UNKNOWN. No global openness, literature absence, historical originality, or new-solution credit follows from the pinned question or from bounded source retrieval.