module module moderate

`IndisputableMonolith.Constants.EulerMascheroni`

The EulerMascheroni module supplies the definition of the Euler-Mascheroni constant γ together with basic positivity, bound, and conjecture lemmas inside the Recognition Science constants layer. Researchers extracting numerical values for physical constants from the phi-ladder would cite these facts when checking consistency with the alpha band or mass formula. The module is built from definitions and short lemmas imported from the base Constants module.

claimThe Euler-Mascheroni constant is given by $\gamma = \lim_{n\to\infty} (H_n - \ln n) ≈ 0.5772$, equipped with positivity, an upper bound below $2/3$, numerical bounds, and an irrationality conjecture in the RS constants setting.

background

The module imports IndisputableMonolith.Constants, whose doc-comment states that the fundamental RS time quantum is $\tau_0 = 1$ tick. It therefore operates inside the Recognition Science derivation of all physics from a single functional equation, with constants expressed in native units where $c=1$, $\hbar=\phi^{-5}$, and $G=\phi^5/\pi$. The supplied doc-comment records the classical limit definition of $\gamma$ and places it among the numerical constants examined for RS predictions.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

The module supplies the Euler-Mascheroni constant to the parent Constants framework for use in numerical constant analysis. It supports the T0-T8 forcing chain by furnishing a concrete mathematical constant whose bounds and conjectures can be compared against the phi-ladder and the alpha inverse band (137.030, 137.039). No downstream theorems are recorded yet.

scope and limits

Does not derive $\gamma$ from the J-uniqueness relation or the Recognition Composition Law.
Does not connect $\gamma$ to the eight-tick octave or spatial dimension $D=3$.
Does not prove the listed irrationality conjecture.
Does not supply a closed-form expression in terms of $\phi$.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (12)

structure or L16
abbrev gamma L28
theorem gamma_pos L31
theorem gamma_lt_two_thirds L35
theorem gamma_numerical_bounds L39
theorem euler_mascheroni_bounds L53
theorem euler_mascheroni_implies_pos L57
theorem euler_mascheroni_implies_ne_zero L62
theorem gamma_irrational_conjecture L75
theorem gamma_bounds_optimal L81
theorem gamma_rs_prediction L92
theorem gamma_gap_analysis L103