Twisted weight at 1 equals 1

Proved

gpy_twistedWeight_one

by pitimespi · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

number-theory

For any pattern of shifts, any fixed coordinate and any membership pattern, the twisted one-variable weight at 1 equals 1: the empty prime product is 1 and the totient of 1 is 1. This is an elementary normalization fact about the twisted weight, stated here in fully unfolded Mathlib terms with no local definitions. It is an absolute, unconditional statement; it claims no asymptotic main term. Formalization Note: the statement is written out in Mathlib terms, with no local definitions.

Preamble

import Mathlib
open Classical

Formal statement

theorem gpy_twistedWeight_one {k : ℕ} (h : Fin k → ℕ) (i : Fin k) (hd he : Prop) [Decidable hd] [Decidable he] :
    ((∏ p ∈ (1 : ℕ).primeFactors, ((((Finset.univ : Finset (Option (Fin k) × Option (Fin k))).filter (fun ab =>
    (ab.1.isSome ↔ hd) ∧ (ab.2.isSome ↔ he) ∧
    (∀ j j', (ab.1 = some j ∨ ab.2 = some j) →
      (ab.1 = some j' ∨ ab.2 = some j') → h j ≡ h j' [MOD p]) ∧
    (∀ j, (ab.1 = some j ∨ ab.2 = some j) → ¬ h i ≡ h j [MOD p])))).card : ℝ)) / ((Nat.totient 1 : ℕ) : ℝ)) = 1 := by sorry

Source

GPY twisted absolute bounds: unconditional absolute/tail control for the twisted Mertens sums (local file GPYTwistedMertens.lean); the asymptotic twisted Mertens main term is NOT proved and NOT claimed.

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