involutionOp_shiftOp
The declaration shows that the reciprocal involution operator U intertwines each prime-shift operator V_p with its inverse: U composed with V_p equals the inverse shift composed with U. Number theorists exploring operator models for the Riemann hypothesis would reference this when checking symmetries of a candidate Hilbert-Pólya operator built from the Recognition cost function. The short proof reduces the operator equality to an identity on basis vectors by direct computation of the action and cancellation in the exponent group.
claimLet $U$ be the reciprocal-involution operator and $V_p$ the shift operator for prime $p$ on the free real module over the multiplicative index space. Then $U$ composed with $V_p$ equals the inverse of $V_p$ composed with $U$.
background
The module constructs an algebraic candidate for the Hilbert-Pólya operator on the free real module over the multiplicative group of positive rationals, realized as the free abelian group on the primes. Key objects include the multiplicative index space, the map sending an index to its rational value, the cost function given by the J-cost, the state space, the diagonal cost operator, the prime-shift operators, and the involution operator realizing the map q to 1/q. This setup draws on the Recognition Science cost algebra and the symmetry J(x) = J(1/x) from the ledger factorization. The local setting is the search for a self-adjoint operator whose eigenvalues would be the imaginary parts of the zeta zeros, with the present theorem supplying one of the required intertwining relations.
proof idea
The proof applies extensionality to an arbitrary vector v in the state space. It then simplifies the compositions of the linear maps using the explicit single-action formulas for the shift, involution, and inverse-shift operators together with the Finsupp single-application rule. The resulting equality of coefficients reduces to an abelian cancellation in the exponents.
why it matters in Recognition Science
This theorem supplies one of the structural properties collected in the master certificate that assembles all symmetries of the candidate operator. It is also invoked to show that the involution preserves the prime-hop operator. Within the Recognition Science framework it realizes at the operator level the reciprocal symmetry of the J-cost, which is the algebraic counterpart of the zeta functional equation s to 1-s. The open spectral question whether the eigenvalues match the zeta ordinates remains untouched.
scope and limits
- Does not prove that the operator is self-adjoint on a completed Hilbert space.
- Does not identify the spectrum with the imaginary parts of zeta zeros.
- Does not address the continuous spectrum or scattering states.
- Does not derive the explicit form of the eigenvalues.
Lean usage
unfold primeHop; rw [involutionOp_shiftOp, involutionOp_shiftInvOp]formal statement (Lean)
181theorem involutionOp_shiftOp (p : Nat.Primes) :
182 involutionOp ∘ₗ shiftOp p = shiftInvOp p ∘ₗ involutionOp := by
proof body
Term-mode proof.
183 ext v
184 simp only [LinearMap.coe_comp, Function.comp_apply,
185 shiftOp_single, involutionOp_single, shiftInvOp_single,
186 Finsupp.lsingle_apply]
187 congr 1
188 abel
189
190/-- Symmetric form of the previous: `U ∘ V_p^{-1} = V_p ∘ U`. -/