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Counterexample

UPT-02

The negative-periodic sign condition is not sufficient

Exact statement

The unary increasing ZILP function h(n)=n−1{n≡2(mod3)}h(n)=n-\mathbf1_{\{n\equiv2\pmod3\}} has periodic residual P≤0P\le0 but is not computable by any unprimed acyclic gate network.

StatusProved internally; refutes a precise sufficiency conjecture
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The counterexample distinguishes unprimed computation rather than exploiting a failure of the ZILP conditions.

Proof or evidence

The function is increasing, zero at zero, and has P=−1n≡2(mod3)≤0P=-\mathbf1_{n\equiv2\pmod3}\le0, but h(2)=1<2h(1)=2h(2)=1<2h(1)=2, violating UPT-01. The primed formula ⌊n/3⌋+⌊(2n+1)/3⌋\lfloor n/3\rfloor+\lfloor(2n+1)/3\rfloor computes it.

Sign-condition counterexample →

Verification notes

The analytic witness and primed contrast were checked against the Run 8 proof. The companion computation corroborates 121 values but is not used as the proof.

Limitations

  • This refutes only sufficiency of the sign condition for unprimed acyclic networks.
  • The primed formula is not an unprimed realization.

Open work

Keep superadditivity in every future characterization attempt; do not revive sign-only sufficiency.