shiftOp
plain-language theorem explainer
shiftOp defines the prime-shift operator V_p that maps each basis vector e_v in the state space to e_{v + δ_p} for a given prime p. Researchers constructing algebraic models of the Riemann zeta function or Recognition Science operator candidates cite this when assembling the Hilbert-Pólya skeleton. The definition is a direct one-line application of Finsupp.lmapDomain to realize the index shift by the prime generator.
Claim. Let $V_p : H → H$ be the linear operator on the free real module $H$ over the multiplicative index space such that $V_p(e_v) = e_{v + δ_p}$, where $δ_p$ is the generator for prime $p$. This realizes multiplication by $p$ on the positive rationals encoded by the indices.
background
The module builds an algebraic candidate for the Hilbert-Pólya operator on the free real module over the multiplicative group of positive rationals, using the Recognition Science cost J. StateSpace is the free ℝ-module on MultIndex, the free abelian group on the primes (isomorphic to ℚ_{>0}^×). The diagonal operator applies J at each index while shiftOp supplies the multiplicative shifts. Upstream results from PhiForcingDerived supply the J-cost J(x) = (x + x^{-1})/2 - 1 and its convexity, while the module setting stresses that reciprocal symmetry J(x) = J(1/x) induces operator intertwining with the involution.
proof idea
The definition is a one-line wrapper that applies Finsupp.lmapDomain ℝ ℝ to the function adding Finsupp.single p 1 to the index v. This directly encodes the required shift on the support of finitely supported functions representing vectors in the state space.
why it matters
shiftOp supplies the V_p building blocks for primeHop and the intertwining relations such as involutionOp_shiftOp. It feeds the hilbert_polya_candidate_certificate that assembles the structural symmetries mirroring the zeta functional equation. In the Recognition Science framework it advances the T5 J-uniqueness construction toward an operator on the J-cost space whose spectrum is conjectured to relate to zeta zeros, leaving the spectral identification as the open Hilbert-Pólya question.
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