Constants Euler Mascheroni Gamma Pos

The positivity bound

The Euler-Mascheroni constant γ is the limit, as n grows, of the difference between the nth harmonic number (1 + 1/2 + 1/3 + ... + 1/n) and the natural logarithm of n. It appears throughout number theory and analysis, for instance in the growth rate of the harmonic series and in the Mertens theorems on prime distribution. Its decimal expansion begins 0.5772156649, and it is not known whether this number is rational or irrational.

A machine-checked library of formal theorems contains a declaration, gamma_pos, which proves that γ is greater than zero. The proof is short: it combines the known fact that 1/2 is positive with a standard bound showing that 1/2 < γ. The same library also proves the companion bound γ < 2/3, so together these give 1/2 < γ < 2/3. These are rigorous results, but they only establish where γ lies; they say nothing about why γ has this value.

In Recognition Science, the framework models physical constants as forced by a discrete record of recognition events. Its library has started to formalize γ, but the derivation of its value from first principles remains an open target. The framework's own status notes state plainly that the derivation is blocked on the Riemann hypothesis and on developing a connection between the ledger and the Riemann zeta function. The positivity bound is a theorem; the deeper origin of γ is not.

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

/-- γ is positive (γ > 1/2). -/
theorem gamma_pos : 0 < gamma :=
  lt_trans (by norm_num : (0 : ℝ) < 1/2) Real.one_half_lt_eulerMascheroniConstant

/-- γ < 2/3 (Mathlib bound). -/
theorem gamma_lt_two_thirds : gamma < 2/3 :=
  Real.eulerMascheroniConstant_lt_two_thirds

/-- 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⟩

What this page does not claim

The declaration gamma_pos does not claim that γ is irrational, which remains an open problem in general mathematics. This page does not claim that Recognition Science has derived the value of γ from its forcing chain; the derivation is explicitly blocked. The positivity bound does not explain why γ appears in physics, such as in renormalization or prime counting.

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

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