A smaller optimal controller for even moduli

Exact statement

For every even modulus $m$ , an $(m+2)$ -state positional controller has Dyck trace $K_m$ and entry cost $\beta_D(K_m)$ . At $m=4$ it is isomorphic to the explicit six-state controller in FPRD-T57.

StatusProved and internally audited

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

Context

Even moduli admit a smaller access-optimal presentation than the uniform 2m-state construction.

Hypotheses and scope

The modulus m is even and at least two.

Proof or evidence

The controller tracks terminal position on an m-cycle when the combined terminal-descent/height parity is odd, and uses two hopeless parity states otherwise. Exact invariants prove the trace, and balanced-distance bounds give cost 2(m-1) for even m at least four, with cost four at m=2.

Even-modulus construction →

Verification notes

The construction was reverse-engineered independently from the modulus-four witness and checked by exact balanced-product equivalence and entry distances.

Limitations

The construction is a state upper bound, not a general minimum-state theorem.
The proposed even/odd minimum-state law remains open.

Open work

Determine whether m+2 is minimal at optimal cost for any even modulus beyond four.

Notes

This is a constructive state upper bound, not a general state-minimality result. The proposed even/odd minimum-state law remains open.