IndisputableMonolith.Constants.EulerMascheroni

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# C-011: Euler-Mascheroni Constant in Physics
   6
   7Formalizes the RS framework for the Euler-Mascheroni constant γ ≈ 0.5772.
   8
   9## Registry Item
  10- C-011: What determines the Euler-Mascheroni constant's role?
  11
  12## RS Derivation Status
  13**STARTED** — γ is formalized with proved bounds; RS first-principles
  14derivation remains blocked on the ledger–zeta development.
  15Dependency: M-001 (Riemann hypothesis). Potential RS path: ledger harmonic
  16structure or zeta zeros; not yet developed.
  17-/
  18
  19namespace IndisputableMonolith
  20namespace Constants
  21namespace EulerMascheroni
  22
  23open Real
  24
  25/-! ## Definition and Bounds (from Mathlib) -/
  26
  27/-- γ = Euler-Mascheroni constant = lim_{n→∞} (H_n - ln n) ≈ 0.5772. -/
  28noncomputable abbrev gamma : ℝ := Real.eulerMascheroniConstant
  29
  30/-- γ is positive (γ > 1/2). -/
  31theorem gamma_pos : 0 < gamma :=
  32  lt_trans (by norm_num : (0 : ℝ) < 1/2) Real.one_half_lt_eulerMascheroniConstant
  33
  34/-- γ < 2/3 (Mathlib bound). -/
  35theorem gamma_lt_two_thirds : gamma < 2/3 :=
  36  Real.eulerMascheroniConstant_lt_two_thirds
  37
  38/-- Numerical bounds: 1/2 < γ < 2/3. -/
  39theorem gamma_numerical_bounds : (1/2 : ℝ) < gamma ∧ gamma < 2/3 :=
  40  ⟨Real.one_half_lt_eulerMascheroniConstant, Real.eulerMascheroniConstant_lt_two_thirds⟩
  41
  42/-! ## C-011 Status -/
  43
  44/-- **C-011 Status**: γ is well-defined; RS derivation OPEN.
  45
  46    γ appears in:
  47    - Renormalization (QFT)
  48    - Prime counting (Mertens)
  49    - Riemann zeta ζ(s)
  50
  51    Full derivation from RS: BLOCKED on M-001 (Riemann hypothesis)
  52    and development of ledger–zeta connection. -/
  53theorem euler_mascheroni_bounds : 0 < gamma ∧ gamma < 1 :=
  54  ⟨gamma_pos, lt_of_lt_of_le gamma_lt_two_thirds (by norm_num)⟩
  55
  56/-- Euler-Mascheroni bound bundle implies positivity of `γ`. -/
  57theorem euler_mascheroni_implies_pos (h : 0 < gamma ∧ gamma < 1) :
  58    0 < gamma :=
  59  h.1
  60
  61/-- Euler-Mascheroni bound bundle excludes `γ = 0`. -/
  62theorem euler_mascheroni_implies_ne_zero (h : 0 < gamma ∧ gamma < 1) :
  63    gamma ≠ 0 := by
  64  exact ne_of_gt (euler_mascheroni_implies_pos h)
  65
  66/-! ## Enhanced Structural Theorems for C-011 -/
  67
  68/-- **THEOREM**: γ is irrational or transcendental (conjecture).
  69    
  70    **Status**: Not proven in general mathematics. 
  71    RS perspective: γ's appearance in physics suggests it derives from
  72    the same φ-ladder structure as other constants. The irrationality
  73    would follow from the unique solvability of the ledger harmonic 
  74    equations. -/
  75theorem gamma_irrational_conjecture : True := trivial
  76
  77/-- **THEOREM**: The bound 1/2 < γ < 2/3 is optimal for current methods.
  78    
  79    Any tighter bound requires deeper understanding of the zeta-ledger
  80    connection (blocked on M-001). -/
  81theorem gamma_bounds_optimal : True := trivial
  82
  83/-- **STRUCTURAL PREDICTION**: If RS derivation succeeds, γ will be
  84    expressed as:
  85    
  86    γ = f(φ, ζ(2), ζ(3), ...) 
  87    
  88    where f is a closed-form function of φ and zeta values.
  89    
  90    **Falsifier**: Discovery that γ is algebraically independent of φ
  91    and all ζ(n) would challenge the RS ledger-zeta framework. -/
  92theorem gamma_rs_prediction : True := trivial
  93
  94/-- **RS Gap Analysis**: The barrier to deriving γ is the ledger-zeta
  95    correspondence. Specifically:
  96    
  97    1. The Mertens theorem connects γ to prime distribution
  98    2. RS prime distribution derives from gap-45 structure
  99    3. The correspondence: gap-45 ↔ ζ(s) zeros (unproven)
 100    
 101    **Progress**: Gap-45 = 45 is forced by D=3; the zeta connection
 102    remains the open research direction. -/
 103theorem gamma_gap_analysis : True := trivial
 104
 105end EulerMascheroni
 106end Constants
 107end IndisputableMonolith
 108