Constants Euler Mascheroni Target Gamma Irrational

The unproved target

The Euler-Mascheroni constant, written γ and about 0.5772, is the limit of the difference between the harmonic series and the natural logarithm: γ = lim_n→∞ (1 + 1/2 + 1/3 + ... + 1/n − ln n). It appears across mathematics, in prime counting via Mertens' theorems, in the Riemann zeta function ζ(s), and in quantum field theory's renormalization. Whether γ is rational or irrational is an open question in general mathematics; no proof exists either way.

The machine-checked library of formal statements contains a definition named target_gamma_irrational, which simply states the proposition that γ is irrational. The library proves bounds on γ: it is positive, greater than 1/2, and less than 2/3. These bounds are established results. The definition of the target itself is not a proof; it is a formal statement of the open problem.

In Recognition Science, the framework's own account suggests γ might derive from the same φ-ladder structure as other constants, with a predicted closed-form expression γ = f(φ, ζ(2), ζ(3), ...). This is a structural prediction, not a derivation. The framework's derivation of γ is blocked on the Riemann hypothesis and on an unproven correspondence between a gap-45 structure and zeta zeros. The irrationality target would follow from a unique solvability of ledger harmonic equations, but that development remains open.

What the declaration does not claim is decisive: it does not prove γ irrational, it does not derive γ from first principles, and it does not assert the Riemann hypothesis. It names the target and records the current bounds. A reader should see the definition as a formal placeholder for an unsolved problem, not as a result.

/-- **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

/-- 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

The declaration does not prove that γ is irrational; irrationality remains an open problem in general mathematics. The declaration does not derive γ from Recognition Science first principles; that derivation is blocked. The declaration does not assert the Riemann hypothesis; the framework's derivation is conditional on it.

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: