IndisputableMonolith.NumberTheory.CostTwistedLSeries
This module defines the cost of a prime p twisted by an arithmetic function chi and extends the prime cost spectrum to twisted multiplicative functions on naturals. Researchers extending Recognition Science arithmetic to characters and L-series constructions would cite these definitions when incorporating twisting into cost calculations. The module consists of a collection of definitions for the twisted prime cost together with its values at units, primes, powers, and products, plus basic sum lemmas.
claimFor an arithmetic function $\chi$ the twisted prime cost is $\tilde{c}(p,\chi):=\chi(p)J(p)$. The associated twisted spectrum value on $n$ is the multiplicative extension $\sum_{p^k\Vert n}k\,\tilde{c}(p,\chi)$, satisfying the usual rules at 1, at primes, under powers, and under coprime multiplication.
background
The module imports the cost function J from IndisputableMonolith.Cost and the prime cost spectrum from IndisputableMonolith.NumberTheory.PrimeCostSpectrum. The upstream module states that for each $n\ge1$ one defines $c(n):=\sum_{p^k\Vert n}k,J(p)=\sum_p v_p(n)J(p)$. The present module twists this construction by an arithmetic function chi, replacing each local factor J(p) by a chi-twisted version while preserving multiplicativity.
No new theoretical setting is introduced beyond the prime-factorization definition of cost already given in PrimeCostSpectrum; the module simply equips that definition with a character twist to support later L-series work.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The definitions supply the arithmetic-function layer needed to build twisted L-series from the Recognition Science cost function. They feed any future parent results on twisted prime-cost sums or character-twisted spectra that appear in the NumberTheory section. The construction directly continues the prime-factorization approach quoted in the upstream PrimeCostSpectrum doc-comment.
scope and limits
- Does not prove convergence or analytic continuation of any series.
- Does not relate the twisted cost to physical constants or the T0-T8 forcing chain.
- Does not supply numerical evaluations or explicit examples.
- Does not establish uniqueness or extremal properties of the twisted function.
depends on (2)
declarations in this module (13)
-
def
twistedPrimeCost -
def
twistedCostSpectrumValue -
theorem
twistedCostSpectrumValue_one -
theorem
twistedCostSpectrumValue_zero -
theorem
twistedCostSpectrumValue_prime -
theorem
twistedCostSpectrumValue_pow -
theorem
twistedCostSpectrumValue_mul -
theorem
twistedCostSpectrumValue_one_char -
theorem
twistedCostSpectrumValue_eq_neg -
def
twistedPrimeCostSum -
theorem
twistedPrimeCostSum_zero -
theorem
twistedPrimeCostSum_one_char -
theorem
cost_twisted_certificate