cost_twisted_certificate
plain-language theorem explainer
The declaration certifies six elementary properties of the twisted cost spectrum c_chi(n) for completely multiplicative chi: vanishing at n=1, complete additivity over positive products, explicit values J(p) chi(p) on primes and k J(p) chi(p) on prime powers, recovery of the untwisted spectrum under the trivial character, and sign flip under negation of chi. Number theorists formalizing arithmetic functions in Recognition Science cite it as the master certificate for the twisted prime cost spectrum. The proof is a term-mode assembly of six preë
Claim. Let $c_chi(n)$ denote the twisted cost spectrum value. Then $c_chi(1)=0$ for every chi; $c_chi(mn)=c_chi(m)+c_chi(n)$ whenever $m,n>0$; $c_chi(p)=J(p)chi(p)$ for every prime $p$; $c_chi(p^k)=k J(p)chi(p)$ for every prime $p$ and $k>0$; the trivial character recovers the untwisted cost spectrum value; and negation of chi negates the spectrum value.
background
The module CostTwistedLSeries generalizes the prime cost spectrum to the character-twisted setting. For each completely multiplicative chi : N to R the twisted cost is defined by $c_chi(n) := sum_p v_p(n) J(p) chi(p)$, where J is the Recognition Science cost function obeying the Recognition Composition Law J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y). This builds directly on the untwisted costSpectrumValue from PrimeCostSpectrum and on the J-cost definitions appearing in ObserverForcing and MultiplicativeRecognizerL4. The local theoretical setting is the number-theoretic layer of Recognition Science, where costs are assigned to positive integers via their prime factorizations in preparation for Dirichlet series and L-functions.
proof idea
The proof is a term-mode construction that packages six prior results into a single conjunction: vanishing at 1 from twistedCostSpectrumValue_one, complete multiplicativity from twistedCostSpectrumValue_mul, the prime case from twistedCostSpectrumValue_prime, the power case from twistedCostSpectrumValue_pow, trivial-character recovery from twistedCostSpectrumValue_one_char, and the negation property from twistedCostSpectrumValue_eq_neg.
why it matters
This master certificate consolidates the basic arithmetic properties of the twisted cost function, enabling later work on twisted prime cost sums and their Dirichlet series. It fills the elementary-properties step in the number-theory development of Recognition Science and links to J-uniqueness (T5) and the phi-ladder via the underlying cost definitions. No open questions are directly touched; the result is fully proved with zero sorry.
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