Constants Euler Mascheroni Gamma Numerical Bounds

The proved bounds

The Euler-Mascheroni constant, usually written γ, is the number that measures how much the harmonic series 1 + 1/2 + 1/3 + ... outruns the natural logarithm as both grow. Formally, γ is the limit as n grows of (H_n minus ln n), where H_n is the nth harmonic number. Its value is about 0.5772. The constant appears across mathematics and physics: in the renormalization of quantum field theory, in Mertens' prime-counting theorems, and in the Riemann zeta function ζ(s).

What the Recognition Science library establishes is a pair of strict inequalities. The declaration gamma_numerical_bounds proves that 1/2 < γ < 2/3. The lower bound comes from the fact that γ is greater than one half, and the upper bound from Mathlib's result that γ is less than two thirds. The proof is machine-checked in the framework's library of formal theorems, meaning the two inequalities are not approximations but exact statements about the real number γ.

In Recognition Science, the framework models constants as emerging from a ledger, a discrete record of recognition events, and its cost structure. For γ, the framework's status is explicitly STARTED, not complete. The bound theorem is proved, but a full derivation of γ from first principles remains blocked on the ledger-zeta development, which itself depends on the Riemann hypothesis. The framework's docstring states plainly: RS derivation OPEN.

What the declaration does not claim is equally important. It does not claim to derive γ from the framework's forcing chain; it only formalizes known bounds. It does not claim γ is irrational; that remains an OPEN target, with a definition target_gamma_irrational but no proof. And it does not claim the bounds are tight in any physical sense; the docstring notes that any tighter bound requires deeper understanding of the zeta-ledger connection. The structural prediction that γ would equal a closed-form function of φ and zeta values carries a named falsifier: discovery that γ is algebraically independent of φ and all ζ(n).

The practical upshot is modest but real. Anyone working with γ now has a machine-checked guarantee that the constant sits in a narrow interval, and the framework has a clear, honest record of where its derivation stands. The bounds themselves are classical; the contribution is the formal verification and the explicit map of what remains open.

/-- 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⟩

/-- 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 does not derive γ from the framework's first principles. The declaration does not prove γ is irrational. The declaration does not claim the bounds are optimal in a physical sense.

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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