Exact unary period-three classification

Exact statement

For a zero-at-zero unary ZILP function with residual period dividing three, unprimed-DAG computability, superadditivity, $2f(1)\le f(2)$ and $2f(2)\le f(3)+f(1)$ , and an explicit one-layer unprimed formula are equivalent.

StatusProved internally and finitely corroborated

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

Context

In this smallest nontrivial unary subclass, arbitrary unprimed depth collapses to one layer.

Definitions

Put $A=f(3),\ a=f(1),\ b=f(2)$ .

Hypotheses and scope

Unary ZILP function, zero at zero.
Periodic residual has period dividing three.

Proof or evidence

Superadditivity at $1+1$ and $2+2$ gives $2a\le b$ and $2b\le A+a$ . These make the coefficients in $f(n)=an+(A+a-2b)\lfloor n/3\rfloor+(b-2a)\lfloor2n/3\rfloor$ nonnegative; substitution on the three residue classes proves sufficiency.

Unary period-three theorem →

Verification notes

The implication cycle and residue-class substitution were checked from the full Run 8 artifact. The companion enumeration reports 1,881 sign-admissible and 1,041 superadditive parameter tuples plus 63,501 formula-value checks; the analytic proof carries the theorem.

Limitations

The classification is unary and period at most three.
No arbitrary-period or multi-input characterization follows.

Open work

Determine whether superadditivity remains sufficient for every zero-at-zero unary ZILP function, or find the smallest later failing period.