Constants Euler Mascheroni

The Euler-Mascheroni constant

The Euler-Mascheroni constant, written γ, is the number that describes the gap between the harmonic series and the natural logarithm. Add the reciprocals of the integers up to n, subtract the logarithm of n, and let n grow without bound; the difference settles near 0.5772. That limit is γ. It appears throughout number theory and physics: in the distribution of primes through Mertens' theorems, in the behavior of the Riemann zeta function near its pole, and in the renormalization of quantum field theories. Whether γ is rational or irrational remains an open problem in mathematics; no one has proved either.

In Recognition Science, the constant enters as a registry item, C-011, with a formal definition and proved bounds. The framework's machine-checked library of formal theorems defines γ as the standard limit and proves the numerical bounds 1/2 < γ < 2/3. These are not approximations; they are theorems with formal proofs. The lower bound comes from a known inequality, the upper from another. Together they pin γ into a narrow window, though the exact value remains beyond current methods.

The framework's larger ambition, deriving γ from first principles, remains open. The path would run through a ledger, the framework's term for a discrete record of events, and its connection to the zeta function. The Mertens theorems link γ to prime distribution; the framework derives prime distribution from a gap-45 structure; the missing step is a proof that this structure corresponds to the zeros of the zeta function. That correspondence is not proved, and the framework states plainly that a full derivation of γ is blocked on the Riemann hypothesis and on developing the ledger-zeta connection.

What the module does establish is a structural prediction with a named falsifier. If the derivation succeeds, γ will take the form of a closed function of the golden ratio φ and zeta values ζ(2), ζ(3), and so on. A discovery that γ is algebraically independent of φ and all ζ(n) would challenge the framework's ledger-zeta picture. Until then, the honest status is: γ is well-defined, its bounds are proved, and its origin remains a target.

/-- 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

No derivation of γ from first principles is claimed; the status is explicitly OPEN. No proof of the irrationality of γ is claimed; it remains an open target. No claim that the structural prediction is proved; it is a hypothesis with a named falsifier.

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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