finitePrimeLedgerPartition_insert_prime
plain-language theorem explainer
Inserting a new prime p outside a finite set S multiplies the finite Euler ledger partition by the local factor at p. Number theorists building finite Euler products would cite this when extending partial products toward the zeta function. The proof unfolds the product definition and simplifies using the non-membership and primality conditions.
Claim. Let $P(s,S)$ denote the finite product over primes in the set $S$ of the local partition factor at each prime. For a complex number $s$, a finite set $S$ of naturals, and a prime $p$ not in $S$, $P(s,S cup {p}) = frac{1}{1-w(s,p)} cdot P(s,S)$, where $w(s,p)$ is the prime posting weight.
background
The module packages the Euler product as the partition function of independent prime-ledger postings. The finite prime-ledger partition is the product over a finite set S of the local factor attached to each prime, where the local factor is the reciprocal of one minus the posting weight. Non-prime entries contribute a factor of one. This finite statement is proved directly, while the identification with the Riemann zeta function is deferred to a separate certification to isolate the analytic import. Upstream results include the definition of the finite partition as that product and the local partition as the inverse factor. The primality condition uses the standard predicate for primes.
proof idea
Unfolding the definition of the finite partition exposes the product over the inserted set. Simplification with the hypotheses that p is absent from S and is prime then multiplies in the new local factor exactly once.
why it matters
This establishes the multiplicative structure needed to construct the Euler product inductively from individual prime factors. It underpins the finite-product statements in the Euler ledger partition module and supports the formal equality to the zeta function in the companion certification. Within Recognition Science it realizes the independent composition of ledger atoms, consistent with the recognition composition law for multiplicative structures.
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