The Dual Cipher, Structurally Induced GF(4) Multiplication, and Real Spectral Structure: Companion-Matrix Realization of the Register Automorphism, Frobenius Reversal, and the Order-One Commutant Condition

Abstract

The two-bit register map $\chi(a,b)=(b,a+b)$ determines an explicit companion-matrix construction of $\mathrm{GF}(4)$. Its matrix satisfies $M_\chi^2+M_\chi+I=0$, and evaluation at a selected cyclic unit transports operator composition to a multiplication on the structural alphabet. Register reversal becomes Frobenius, the simultaneous flip becomes translation by one, and alignment is the field trace. The resulting action on three registers has fixed-point counts $(64,0,8,8)$ and 20 orbits. The RNA dictionary is evaluated consistently through the same map. A perimeter locant correspondence commutes with inversion, while the dimer-to-node identification is formulated as a unitary coordinate isomorphism with a separate molecular-realization hypothesis.

At the spectral level, a symmetrized support filtration yields a self-adjoint compact-resolvent operator with bounded algebraic commutators and a compatible real involution. An explicit local-unitary counterexample shows that both the original and symmetrized filtrations fail the global order-one condition. Separately, the finite matrix bimodule operator $L(H)+\epsilon'R(H)$ satisfies the real sign and order-one relations and retains them under self-adjoint inner fluctuations. Even doubling realizes the KO-sign patterns for dimensions 0 and 6. These results establish the finite field and internal real spectral constructions while identifying the global order-one extension and molecular implementation as distinct targets. Exact verification and finite Lean certificates accompany the proofs.