Constants Euler Mascheroni Euler Mascheroni Implies Pos

The positivity theorem

The Euler-Mascheroni constant, usually written γ, is the number you get by subtracting the natural logarithm from the harmonic series as both grow. For the first few terms, 1 + 1/2 + 1/3 + ... grows like ln(n) plus a leftover, and that leftover settles near 0.5772. It appears throughout number theory and physics, but its exact nature remains mysterious; nobody has proved whether it is irrational.

In the Recognition Science framework's machine-checked library of formal theorems, one small result, named euler_mascheroni_implies_pos, takes a bound on γ and extracts its positivity. The bound is 0 < γ < 1, a bundle that the framework's library has already established from a stronger pair of inequalities: 1/2 < γ < 2/3. The theorem itself is a logical step: given the bound, it concludes γ > 0. It also derives γ ≠ 0 as a direct consequence.

The declaration does not prove the bound from first principles. It does not show where γ comes from in the Recognition Science framework, nor does it attempt to derive γ from the golden ratio or zeta values. Those derivations remain open targets, blocked on the Riemann hypothesis and on a ledger-zeta connection that has not been developed. The theorem's role is narrower: it packages the known bounds into a reusable form for later arguments.

What the theorem does establish is a clean, checkable fact: the Euler-Mascheroni constant is positive and nonzero. That may sound trivial, but in formal mathematics, every step must be explicit. The library also records that γ is less than 1 and, more tightly, less than 2/3, using bounds imported from Mathlib. The structural prediction that γ might equal a closed-form function of φ and zeta values is a hypothesis with a named falsifier, not a proved result.

/-- Euler-Mascheroni bound bundle implies positivity of `γ`. -/
theorem euler_mascheroni_implies_pos (h : 0 < gamma ∧ gamma < 1) :
    0 < gamma :=
  h.1

/-- 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⟩

/-- **C-011 Status**: γ is well-defined; RS derivation OPEN.

    γ appears in:
    - Renormalization (QFT)
    - Prime counting (Mertens)
    - Riemann zeta ζ(s)

    Full derivation from RS: BLOCKED on M-001 (Riemann hypothesis)
    and development of ledger–zeta connection. -/
theorem euler_mascheroni_bounds : 0 < gamma ∧ gamma < 1 :=
  ⟨gamma_pos, lt_of_lt_of_le gamma_lt_two_thirds (by norm_num)⟩

/-- 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 theorem does not prove the bounds 1/2 < γ < 2/3 from Recognition Science first principles. The theorem does not derive γ from the golden ratio or from zeta values. The theorem does not establish whether γ is irrational.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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: