Constants Euler Mascheroni Euler Mascheroni Implies Ne Zero

The bound and its reach

The Euler-Mascheroni constant γ is the limit of the difference between the harmonic series and the natural logarithm: γ = lim_n→∞ (H_n − ln n) ≈ 0.5772. It appears across mathematics and physics, from the growth of prime numbers in Mertens' theorems to the renormalization of quantum field theories. A standard question about any constant is whether it is zero, positive, or negative.

The Recognition Science framework's machine-checked library of formal theorems proves that γ is strictly between 1/2 and 2/3. From that bound, the declaration euler_mascheroni_implies_ne_zero derives the immediate consequence that γ is not equal to zero. The proof is a one-step logical move: if a number is greater than zero, then it is not zero. This is a theorem in the library, with no assumptions beyond the standard axioms of the underlying logic.

The framework does not claim to derive γ from first principles. Its own status notes mark the derivation as OPEN, blocked on the Riemann hypothesis and on a ledger-zeta connection that is not yet developed. The library also defines an OPEN target for whether γ is irrational, noting that this is not proved in general mathematics either. The framework's structural prediction, that γ would be a closed-form function of φ and zeta values if derived, comes with a named falsifier: discovering that γ is algebraically independent of φ and all ζ(n) would challenge the framework's ledger-zeta approach.

What the declaration establishes is narrow but solid: given the proved numerical bounds, γ cannot be zero. What it does not establish is any formula for γ, any link to the framework's golden ratio ladder, or any resolution of the constant's deeper open questions. The bound is the result; the rest remains a target.

/-- γ = Euler-Mascheroni constant = lim_{n→∞} (H_n - ln n) ≈ 0.5772. -/
noncomputable abbrev gamma : ℝ := Real.eulerMascheroniConstant

/-- Numerical bounds: 1/2 < γ < 2/3. -/
theorem gamma_numerical_bounds : (1/2 : ℝ) < gamma ∧ gamma < 2/3 :=
  ⟨Real.one_half_lt_eulerMascheroniConstant, Real.eulerMascheroniConstant_lt_two_thirds⟩

/-- Euler-Mascheroni bound bundle excludes `γ = 0`. -/
theorem euler_mascheroni_implies_ne_zero (h : 0 < gamma ∧ gamma < 1) :
    gamma ≠ 0 := by
  exact ne_of_gt (euler_mascheroni_implies_pos h)

What this page does not claim

The declaration does not prove any formula for γ in terms of φ or zeta values. The declaration does not prove that γ is irrational. The declaration does not establish the ledger-zeta correspondence that the framework's derivation would require.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/EulerMascheroni.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: