twistedPrimeCostSum_one_char
plain-language theorem explainer
The declaration shows that the twisted prime cost sum for the constant character equals the ordinary sum of prime costs over primes up to N. Researchers formalizing Recognition Science number theory would invoke this reduction when specializing twisted L-series to the untwisted prime cost spectrum. The proof is a direct unfolding of the sum definition followed by case distinction on primality.
Claim. Let $N$ be a natural number and let $chi$ be the constant function $chi(n)=1$. Then the twisted prime cost sum equals the sum over primes $p leq N$ of the prime cost $J(p)$, that is, $sum_{p leq N} J(p) chi(p) = sum_{p leq N, p text{ prime}} J(p)$.
background
The Cost-Twisted Arithmetic Functions module generalizes the prime cost spectrum to the character-twisted setting. For a completely multiplicative chi, the twisted prime cost spectrum value is defined as $c_chi(n) := sum_p v_p(n) J(p) chi(p)$, and the prime-restricted partial sum is $Pi_chi(x) := sum_{p leq x} J(p) chi(p)$. The J-cost itself originates from the Recognition Science forcing chain, specifically J-uniqueness where $J(x) = (x + x^{-1})/2 - 1$, and from upstream cost definitions in MultiplicativeRecognizerL4 and ObserverForcing that derive cost as the J-cost of recognition events. The local setting establishes elementary properties of these twisted functions before any analytic L-function results.
proof idea
The proof unfolds the definition of twistedPrimeCostSum, applies congruence to equate the two sums, extends the equality over the finite range, and performs case analysis on whether each p is prime. In the prime case it simplifies directly to primeCost p; otherwise it yields zero, with both branches discharged by simp.
why it matters
This result forms part of the master certificate for the elementary properties of the cost-twisted arithmetic function. It permits immediate reduction of the twisted prime cost sum to the untwisted version precisely when the character is trivial. Within the Recognition framework it supports the passage from the phi-ladder mass formulas and eight-tick octave to the spectral layer in number theory, without touching open questions since the statement is fully proved.
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