Reciprocal Cost and an Eight-Tick Register: A Conditional Recognition-Operator Construction

We study a conditional finite-dimensional operator construction based on two imported Recognition-Science inputs: a continuous comparison cost satisfying a multiplicative coherence axiom and equilibrium calibration, and a three-coordinate, eight-tick ledger schedule. The first input uniquely determines the reciprocal cost J(x)=12(x+x−1)−1. A cyclic Gray traversal of the configuration graph Q3 fixes the cyclic ordering of an assumed eight-sample register with shift P. The additional half-cycle antiperiodicity criterion selects the odd Fourier sector V=ker(P4+I); unitarity alone does not select this sector because P is unitary on the full register. On the realification of V, P2 defines an additional complex structure, and, after assigning an independent beat duration, the principal logarithm defines one self-adjoint stroboscopic interpolation, unique only after imposing the principal-zone convention. We distinguish the exact reciprocal action from a local adapter hypothesis needed to relate its mismatch coordinates to Hilbert-space defect coordinates. For the regular shift, the rational cyclotomic decomposition identifies V as the minimal faithful Φ8-block. Under the stated relative normalization of the local bridge, leading cost–defect agreement is equivalent to an isometric adapter tangent, while a positive self-adjoint single-filter commit that commutes with the beat has at most three non-negative mode factors on the minimal register; defect descent restricts them to [0,1]. The relaxed commit is a contraction and, in a normalized quantum interpretation, represents a trace-nonincreasing accepted branch whose complementary trace weight is recorded by a measurement outcome or environment; the in-sector component nevertheless evolves exactly and unitarily. For coupled ququart cores, adjoining the standard conjugate Weyl shift to the clock inherited from P spans the full finite-dimensional operator algebra. This proves algebraic representability of arbitrary finite-dimensional Hermitian dynamics, not a physical rule selecting their coefficients.