Period three is the first unary sign-only failure

Exact statement

For unary residual period dividing two, $P\le0$ is sufficient for an unprimed formula; therefore period three is the smallest unary coordinate period at which the sign condition alone fails.

StatusProved internally and finitely corroborated

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The result places the counterexample at the first possible unary period rather than merely giving an isolated failure.

Hypotheses and scope

Unary, zero-at-zero ZILP functions.
Unprimed acyclic gate syntax.

Proof or evidence

For period two, with $A=f(2)$ and $a=f(1)$ , the sign condition is $A\ge2a$ , and $f(n)=an+(A-2a)\lfloor n/2\rfloor$ is an unprimed formula. Period one is affine; UPT-02 supplies the period-three failure.

Period-two formula and minimality →

Verification notes

The exact period-two construction and minimality step were checked. The companion enumeration reports 15,616 period-two formula values as corroboration.

Limitations

Minimality is only among unary coordinate periods.
No statement is made about multi-input period lattices.

Open work

Retain period three as the exact first obstruction when studying larger unary periods.