Constants Euler Mascheroni Gamma Lt Two Thirds

The proved bound

The Euler-Mascheroni constant γ is defined as the limit, as n grows, of the difference between the nth harmonic number H_n = 1 + 1/2 + ... + 1/n and the natural logarithm of n. Its value is approximately 0.5772. It appears throughout mathematics and physics: in the growth rate of the harmonic series, in the distribution of prime numbers via Mertens' theorems, and in renormalization in quantum field theory. A classical, framework-neutral fact about γ is that it is bounded between 1/2 and 2/3, a result known to mathematicians for over a century.

In Recognition Science, the declaration gamma_lt_two_thirds is a formal statement in the framework's machine-checked library. It states, in the language of real numbers, that γ < 2/3. The derivation is direct: it invokes a corresponding bound already established in Mathlib, the standard mathematical library. Alongside the companion statement gamma_pos, which states 0 < γ, the framework assembles the full numerical bounds: 1/2 < γ < 2/3. These are not empirical measurements and not new mathematics; they are formalized, verified restatements of known classical facts, recorded so the framework can use them as building blocks.

What the declaration does not claim is more interesting. It does not claim that γ is irrational, nor that it is transcendental, nor that it has any closed-form expression in terms of the golden ratio φ or values of the Riemann zeta function. The framework's own documentation marks the irrationality of γ as an OPEN target, not a settled result, and notes that a full first-principles derivation of γ from Recognition Science is BLOCKED on the ledger–zeta development, which itself depends on the Riemann hypothesis. The bound is a step, not a destination.

Within the framework, the significance of γ is tied to a structural prediction: if a derivation from first principles ever succeeds, γ will be expressed as a closed-form function of φ and zeta values such as ζ(2), ζ(3), and so on. The falsifier for this prediction is explicit: discovering that γ is algebraically independent of φ and all ζ(n) would challenge the framework's ledger-zeta correspondence. For now, the proved bound 1/2 < γ < 2/3 is what the framework can assert with certainty, and it is a small but solid piece of the larger, unfinished project.

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

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

/-- 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⟩

/-- **OPEN target**: γ is irrational.

    Status: not proved in general mathematics (or here).
    RS perspective: γ's appearance in physics suggests it derives from
    the same φ-ladder structure as other constants. The irrationality
    would follow from the unique solvability of the ledger harmonic
    equations. -/
def target_gamma_irrational : Prop := Irrational gamma

What this page does not claim

This declaration does not prove that γ is irrational or transcendental. This declaration does not derive γ from first principles within the framework. This declaration does not provide a closed-form expression for γ in terms of φ and zeta values.

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: