Euler-Mascheroni Constant Band

1. Setting

Euler-Mascheroni Constant Band is anchored in Constants.EulerMascheroni. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y) $$

Recognition Composition Law.

3. Prediction or structural target

• Euler-Mascheroni gamma: predicted in (0, 2/3) (dimensionless); empirical 0.5772156649. Source: Classical

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Constants.EulerMascheroni..gamma_pos.

/-- γ is positive (γ > 1/2). -/
theorem gamma_pos : 0 < gamma :=
  lt_trans (by norm_num : (0 : ℝ) < 1/2) Real.one_half_lt_eulerMascheroniConstant

/-- 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 :=

5. What is inside the Lean module

Key theorems:

• gamma_pos
• gamma_lt_two_thirds
• gamma_numerical_bounds
• euler_mascheroni_bounds
• euler_mascheroni_implies_pos
• euler_mascheroni_implies_ne_zero
• gamma_irrational_conjecture
• gamma_bounds_optimal
• gamma_rs_prediction
• gamma_gap_analysis

Key definitions:

• or

6. Derivation chain

• gamma_pos - gamma > 0
• gamma_lt_two_thirds - gamma < 2/3
• gamma_numerical_bounds - Numerical bounds
• euler_mascheroni_bounds - Euler-Mascheroni bounds
• gamma_rs_prediction - RS prediction
• gamma_irrational_conjecture - Irrationality conjecture (open)

7. Falsifier

An exact algebraic identity for gamma that places it outside (0, 2/3) refutes the RS bound; an irrationality proof would resolve gamma_irrational_conjecture.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit euler-mascheroni, start with the primary Lean anchor Constants.EulerMascheroni.gamma_pos. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.