twistedPrimeCost
The definition sets the twisted prime cost of a prime p under an arithmetic function chi to the product of the untwisted prime cost and chi(p). Researchers extending prime cost spectra to twisted Dirichlet series in Recognition Science cite it as the local factor inside the twisted cost spectrum sum. It is realized as a direct one-line multiplication with no lemmas or tactics required.
claimLet $chi : ℕ → ℝ$ be an arithmetic function and let $p ∈ ℕ$. The twisted prime cost is defined by $c_chi(p) := c(p) · chi(p)$, where $c(p)$ is the prime cost of $p$ (equal to the recognition cost $J(p)$).
background
The module generalizes the prime cost spectrum to the character-twisted setting. For each completely multiplicative arithmetic function chi : ℕ → ℝ (typically the real part of a Dirichlet character), the twisted cost spectrum value is the sum over the prime factorization: $c_chi(n) := Σ_p v_p(n) · J(p) · chi(p)$. This rests on the untwisted prime cost spectrum imported from PrimeCostSpectrum and on the J-cost defined in upstream modules. The cost induced by a multiplicative recognizer is the derived cost of its comparator on positive ratios; the cost of any recognition event is its J-cost.
proof idea
Direct definition. The body is the product of the prime cost at p and the twisting function evaluated at p, supplying the atomic term that downstream sums use to build the full twisted spectrum value.
why it matters in Recognition Science
This supplies the local factor for twistedCostSpectrumValue, which in turn supports the elementary identities for negation of chi, reduction to the trivial character, and the prime-power cases. It advances the Recognition framework by preparing the twisted L-series whose Dirichlet series factorizes through the corresponding L-function (paper-level result). The construction aligns with the multiplicative structure enforced by the Recognition Composition Law and the J-uniqueness property.
scope and limits
- Does not assume or enforce that chi is completely multiplicative.
- Does not compute the twisted cost for composite arguments directly.
- Does not address analytic continuation or convergence of the associated L-series.
- Does not derive explicit closed forms for the twisted L-function.
formal statement (Lean)
40def twistedPrimeCost (chi : ℕ → ℝ) (p : ℕ) : ℝ :=
proof body
Definition body.
41 primeCost p * chi p
42
43/-- The cost spectrum value of `n` twisted by `chi`, defined via the
44 prime factorization. -/