LocalSplittingHypothesis
plain-language theorem explainer
LocalSplittingHypothesis(k,s) encodes the factorization of the genus-one Hadamard product into the explicit E1 factor at the k-th zero multiplied by the product with that term replaced by 1. Analysts deriving local contributions to the completed zeta function cite it to isolate one zero before handling the regular tail. The declaration directly transcribes the identity that follows from Mathlib's multipliable product splitting once the skipped product is shown multipliable.
Claim. Let $H$ be the shifted cost function and $E_1(z)=(1-z)e^z$ the genus-one elementary factor. For natural number $k$ and complex $s$, the local splitting hypothesis asserts that the infinite product satisfies $$∏'_n E_1(s/(H·zeros_n)) = E_1(s/(H·zeros_k)) · ∏'_n (1 if n=k else E_1(s/(H·zeros_n))).$$
background
In the Recognition Science treatment of the completed Riemann zeta, the Hadamard product uses the genus-one factor $E_1(z)=(1-z)exp(z)$. The shifted cost $H(x)=J(x)+1=½(x+x^{-1})$ reparametrizes the recognition composition law into d'Alembert form. The module isolates the contribution of any chosen zero index $k$ by defining the regular tail as the product over remaining indices times the exponential prefactor $exp(A+Bs)$ from the Hadamard identification.
proof idea
The declaration is a direct definition of the Prop as the displayed product identity. No proof is attached because the statement is introduced as a named hypothesis; it will be discharged later by invoking Mathlib's Multipliable.tprod_eq_mul_tprod_ite' after verifying multipliability of the full product and of the version with the k-th term replaced by 1.
why it matters
This hypothesis_interface supplies the local factorization step required by completedRiemannZeta0_local_split and completedRiemannZeta0_local_split_regularTail, which populate the HadamardRegularTailStatus structure. It advances the Hadamard product representation of the completed zeta within the Recognition framework, where genus-one factors trace to J-uniqueness (T5). The declaration leaves open the unconditional verification of multipliability on ℂ and global cost domination on the critical strip, as documented in the module.
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