candidateOp
plain-language theorem explainer
candidateOp assembles a finite truncation of the proposed Hilbert-Pólya operator on the free real module over multiplicative indices as the diagonal J-cost operator plus a weighted sum of prime-hop terms. Number theorists modeling the Riemann zeta spectrum algebraically would cite this construction as the explicit skeleton before any Hilbert-space completion. The definition is a direct one-line assembly from the pre-defined diagOp and scaled primeHop components.
Claim. Let $D$ be the diagonal J-cost operator and let $V_p$ denote the prime-shift operator on the free real module over multiplicative indices. For finite $S$ of primes and weights $λ$, the candidate operator is $T_S(λ) = D + ∑_{p∈S} λ(p) (V_p + V_p^{-1})$.
background
The module builds an algebraic model for the Hilbert-Pólya conjecture inside Recognition Science by using the J-cost function on the multiplicative group of positive rationals. StateSpace is the free real module on MultIndex, the free abelian group generated by the primes. diagOp is the multiplication operator that sends each basis vector e_v to J(toRat v) · e_v. primeHop p implements the symmetric shift V_p + V_p^{-1} that encodes the reciprocal symmetry J(x) = J(1/x). Upstream results supply the J-cost convexity from PhiForcingDerived and the prime set from EulerInstantiation.
proof idea
One-line wrapper that adds diagOp to the Finsupp sum over S of λ(p) scaled by primeHop p.
why it matters
This supplies the explicit operator whose commutation with the reciprocal involution is recorded in involutionOp_candidateOp and whose full structural certificate appears in hilbert_polya_candidate_certificate. It translates the Recognition Science J-cost into an operator that respects the zeta functional equation at the algebraic level, using the eight-tick octave and phi-ladder only indirectly through the cost definition. The open spectral question, whether the eigenvalues match the imaginary parts of the zeta zeros, remains untouched.
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