twistedCostSpectrumValue_one_char
The theorem establishes that the twisted cost spectrum value for the constant character equal to 1 coincides exactly with the ordinary cost spectrum value at any natural number n. Number theorists working inside the Recognition Science framework cite this reduction when specializing twisted arithmetic functions back to the untwisted prime cost spectrum. The proof is a short term-mode calculation that unfolds the three relevant definitions and equates the Finsupp sums by congruence followed by ring simplification.
claimFor the trivial character $chi(n) = 1$ for all $n$, the twisted cost spectrum value $c_{chi}(n) = sum_p v_p(n) J(p) chi(p)$ equals the standard cost spectrum value $c(n)$ for every natural number $n$.
background
The CostTwistedLSeries module extends the prime cost spectrum by inserting a completely multiplicative character chi into the sum that defines the cost function. Explicitly, the twisted cost spectrum value is the sum over primes p dividing n of the valuation v_p(n) times the J-cost J(p) times chi(p). The untwisted costSpectrumValue is recovered by dropping the chi factor. This construction rests on the cost definitions in MultiplicativeRecognizerL4 and ObserverForcing, both of which derive costs from the J-cost on positive ratios or recognition events, and on the J-cost structure supplied by PhiForcingDerived.
proof idea
The proof unfolds twistedCostSpectrumValue, costSpectrumValue and twistedPrimeCost to expose the underlying Finsupp sums. It then applies Finsupp.sum_congr to align the summands for each prime p, after which the ring tactic finishes the algebraic identity.
why it matters in Recognition Science
This specialization lemma is invoked inside the master certificate cost_twisted_certificate that records the elementary properties (vanishing at 1, complete additivity) of the twisted cost function. It therefore supports the passage from the ordinary prime cost spectrum to character-twisted L-series inside the Recognition framework, consistent with J-uniqueness (T5) and the Recognition Composition Law. No open scaffolding questions are closed by this result.
scope and limits
- Does not treat non-principal or complex Dirichlet characters.
- Does not establish analytic continuation or functional equations for the twisted series.
- Does not compute numerical values or bounds for specific n.
- Does not address convergence questions for the associated Dirichlet series.
Lean usage
exact twistedCostSpectrumValue_one_char nformal statement (Lean)
93theorem twistedCostSpectrumValue_one_char (n : ℕ) :
94 twistedCostSpectrumValue (fun _ => 1) n = costSpectrumValue n := by
proof body
Term-mode proof.
95 unfold twistedCostSpectrumValue costSpectrumValue twistedPrimeCost
96 refine Finsupp.sum_congr ?_
97 intro p _
98 ring
99
100/-- Negating the character flips the sign of the twisted cost. -/