involutionOp_sum_primeHop
The theorem shows that the reciprocal involution commutes with arbitrary finite linear combinations of prime-shift operators. Researchers constructing Hilbert-Pólya candidates for the Riemann hypothesis in the Recognition Science setting would cite this auxiliary result. The argument proceeds by induction on the finite set of primes, reducing each step to linearity of composition and the single-prime commutation case.
claimLet $U$ be the reciprocal involution operator and $V_p$ the prime-shift operator for each prime $p$. For any finite set $S$ of primes and real coefficients $λ_p$, $U ∘ ∑_{p∈S} λ_p V_p = ∑_{p∈S} λ_p V_p ∘ U$.
background
The module assembles 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 function J. The involution operator implements the map q ↦ 1/q, which preserves J-cost by the reciprocal property J(x) = J(1/x). The prime-shift operators implement multiplication by a fixed prime in this group. The theorem concerns commutation of the involution with finite weighted sums of these shifts. It rests on the single-prime commutation property and the algebraic structure of linear maps over the reals.
proof idea
The proof applies classical logic and Finset induction on S. The empty-set base case simplifies to the zero operator, which commutes with everything. In the inductive step the sum is split by insert, additivity of composition and scalar multiplication isolates the new term, and the single-prime commutation lemma finishes the step.
why it matters in Recognition Science
This auxiliary result is invoked directly in the proof of the main commutation theorem for the full candidate operator (diagonal cost plus sum of shifts). That theorem supplies the structural symmetry mirroring the functional equation ξ(s) = ξ(1-s). In the Recognition Science framework the commutation follows from J-uniqueness and reciprocal invariance (T5), supporting the Hilbert-Pólya analogy while leaving open whether the spectrum matches the imaginary parts of the zeta zeros.
scope and limits
- Does not establish self-adjointness or domain issues on a completed Hilbert space.
- Does not identify eigenvalues with imaginary parts of zeta zeros.
- Applies only to finite sums; convergence questions for infinite sums are untouched.
- Remains purely algebraic and does not address spectral theory or closure.
Lean usage
rw [involutionOp_sum_primeHop]formal statement (Lean)
249private theorem involutionOp_sum_primeHop
250 (S : Finset Nat.Primes) (lam : Nat.Primes → ℝ) :
251 involutionOp ∘ₗ S.sum (fun p => lam p • primeHop p)
252 = S.sum (fun p => lam p • primeHop p) ∘ₗ involutionOp := by
proof body
Term-mode proof.
253 classical
254 refine Finset.induction_on S ?_ ?_
255 · simp
256 · intro p S hp ih
257 rw [Finset.sum_insert hp]
258 rw [LinearMap.comp_add, LinearMap.add_comp, ih]
259 congr 1
260 rw [LinearMap.comp_smul, LinearMap.smul_comp]
261 congr 1
262 exact involutionOp_primeHop p
263
264/-- The candidate operator commutes with the reciprocal involution.
265 This is the Hilbert--Pólya-style structural symmetry: any spectrum
266 of the (closure of the) operator decomposes into eigenspaces of
267 the involution, mirroring `s ↔ 1-s`. -/