shiftInvOp
The inverse prime-shift operator V_p^{-1} maps each basis vector e_v in the free real module on multiplicative indices to e_{v - delta_p}. Researchers constructing algebraic models of the Riemann hypothesis cite it when verifying intertwining relations between the reciprocal involution and the shift operators. The definition is a one-line wrapper around Finsupp.lmapDomain that subtracts the singleton basis vector for the chosen prime.
claimThe inverse prime-shift operator $V_p^{-1} : S → S$ is the linear map on the free real vector space $S$ over the multiplicative indices satisfying $V_p^{-1} e_v = e_{v - δ_p}$, where $δ_p$ denotes the standard basis vector for prime $p$.
background
The module builds an algebraic skeleton for a candidate Hilbert-Pólya operator on the free real module StateSpace over the multiplicative indices MultIndex ≔ Nat.Primes →₀ ℤ. This space is isomorphic to the positive rationals under multiplication via the map toRat. The Recognition Science cost function J supplies the diagonal operator, while the shift operators implement multiplication and division by primes. Upstream, Primes is defined as the set of prime numbers, and StateSpace is the Finsupp module MultIndex →₀ ℝ. The local setting is the construction of operators V_p (forward shift), V_p^{-1} (inverse shift), and the involution U that intertwines them to mirror the functional equation of the zeta function.
proof idea
This definition is a one-line wrapper that applies Finsupp.lmapDomain ℝ ℝ to the map sending each index v to v minus the singleton basis vector for the given prime p. It directly encodes the subtraction of the prime exponent in the multiplicative index.
why it matters in Recognition Science
This definition supplies the inverse shift operator required by the intertwining theorems involutionOp_shiftOp and involutionOp_shiftInvOp, which establish that the reciprocal involution U satisfies U V_p = V_p^{-1} U. These relations feed into the master certificate hilbert_polya_candidate_certificate and the definition of primeHop = V_p + V_p^{-1}. Within the Recognition Science framework it realizes the operator-level counterpart of the zeta functional equation ξ(s) = ξ(1-s), using the J-cost symmetry J(x) = J(1/x). The open spectral question remains whether the resulting operator has eigenvalues at the imaginary parts of the zeta zeros.
scope and limits
- Does not establish that the spectrum coincides with the imaginary parts of the non-trivial zeta zeros.
- Does not address the completion of StateSpace to a Hilbert space or self-adjointness.
- Does not prove unitarity beyond the formal algebraic inverse relation.
- Does not incorporate the diagonal cost operator into the shift dynamics.
formal statement (Lean)
145def shiftInvOp (p : Nat.Primes) : StateSpace →ₗ[ℝ] StateSpace :=
proof body
Definition body.
146 Finsupp.lmapDomain ℝ ℝ (fun v => v - Finsupp.single p 1)
147