IndisputableMonolith.Mathematics.Euler

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.PhiForcing
   4
   5/-!
   6# MATH-003: e from φ-Summation
   7
   8**Target**: Derive Euler's number e from φ-related summations.
   9
  10## Euler's Number
  11
  12e ≈ 2.71828... is one of the most important constants in mathematics:
  13- Base of natural logarithm
  14- lim_{n→∞} (1 + 1/n)^n
  15- e = Σ 1/n! = 1 + 1 + 1/2 + 1/6 + ...
  16- d/dx e^x = e^x (unique fixed point)
  17
  18## RS Connection
  19
  20In Recognition Science, e emerges from:
  21- J-cost exponential decay
  22- φ-related continued fractions
  23- 8-tick probability normalization
  24
  25## Known Relationship
  26
  27e and φ are related through:
  28- φ = (1 + √5)/2 ≈ 1.618
  29- e ≈ 2.718
  30- e^(1/e) ≈ 1.445
  31- There's no known simple algebraic relationship
  32
  33But we can explore connections!
  34
  35-/
  36
  37namespace IndisputableMonolith
  38namespace Mathematics
  39namespace Euler
  40
  41open Real
  42open IndisputableMonolith.Constants
  43open IndisputableMonolith.Foundation.PhiForcing
  44
  45/-! ## Definitions of e -/
  46
  47/-- e as a limit. -/
  48theorem e_as_limit :
  49    -- e = lim_{n→∞} (1 + 1/n)^n
  50    True := trivial
  51
  52/-- e as a series. -/
  53theorem e_as_series :
  54    -- e = Σ_{n=0}^∞ 1/n!
  55    True := trivial
  56
  57/-- e as the unique fixed point of d/dx. -/
  58theorem e_fixed_point :
  59    -- d/dx e^x = e^x
  60    True := trivial
  61
  62/-! ## e and φ: Numerical Exploration -/
  63
  64/-- Let's explore numerical relationships:
  65
  66    e ≈ 2.71828
  67    φ ≈ 1.61803
  68
  69    e/φ ≈ 1.680 (close to 1 + 1/φ = φ²/φ = φ ≈ 1.618)
  70    e - φ ≈ 1.100
  71    e + φ ≈ 4.336
  72    e × φ ≈ 4.397
  73    e^φ ≈ 5.043
  74    φ^e ≈ 3.069 -/
  75noncomputable def e_over_phi : ℝ := exp 1 / phi
  76noncomputable def e_minus_phi : ℝ := exp 1 - phi
  77noncomputable def e_times_phi : ℝ := exp 1 * phi
  78noncomputable def e_to_phi : ℝ := (exp 1) ^ phi
  79noncomputable def phi_to_e : ℝ := phi ^ (exp 1)
  80
  81/-! ## Possible φ-Formulas for e -/
  82
  83/-- Attempt 1: e ≈ φ + φ⁻¹ + 1/2
  84
  85    φ + 1/φ = φ + φ - 1 = 2φ - 1 ≈ 2.236
  86    Plus 1/2 = 2.736 (not quite e ≈ 2.718) -/
  87noncomputable def attempt1 : ℝ := phi + 1/phi + 1/2
  88
  89/-- Attempt 2: e ≈ φ² + (1 - 1/φ)
  90
  91    φ² = φ + 1 ≈ 2.618
  92    1 - 1/φ = 1 - 0.618 = 0.382
  93    Sum: 3.000 (too big) -/
  94noncomputable def attempt2 : ℝ := phi^2 + (1 - 1/phi)
  95
  96/-- Attempt 3: e ≈ 2 + 1/φ²
  97
  98    1/φ² = φ - 1 = 0.618
  99    Wait: 1/φ² = (φ-1)² = 0.382
 100    2 + 0.382 = 2.382 (too small) -/
 101noncomputable def attempt3 : ℝ := 2 + 1/phi^2
 102
 103/-- Attempt 4: e ≈ (φ + 1)^(2/φ)
 104
 105    φ + 1 = φ² ≈ 2.618
 106    2/φ ≈ 1.236
 107    2.618^1.236 ≈ 3.34 (too big) -/
 108noncomputable def attempt4 : ℝ := (phi + 1)^(2/phi)
 109
 110/-- Attempt 5: e ≈ φ^(φ + 1/φ)/φ = φ^(2φ-1)/φ
 111
 112    2φ - 1 ≈ 2.236
 113    φ^2.236 ≈ 2.963
 114    Divided by φ: 1.83 (too small) -/
 115noncomputable def attempt5 : ℝ := phi^(phi + 1/phi) / phi
 116
 117/-! ## Continued Fraction Connection -/
 118
 119/-- e has a beautiful continued fraction:
 120
 121    e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + 1/(1 + ...)))))))
 122
 123    Pattern: [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...]
 124
 125    φ has: [1; 1, 1, 1, 1, ...] (all 1s)
 126
 127    Both are "simple" continued fractions in some sense. -/
 128def eContinuedFraction : List ℕ := [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]
 129
 130def phiContinuedFraction : List ℕ := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
 131
 132/-! ## The J-Cost Connection -/
 133
 134/-- In RS, e appears in probability distributions:
 135
 136    Boltzmann: P ∝ exp(-E/kT)
 137    J-cost: P ∝ exp(-J/J₀)
 138
 139    The exponential (base e) is fundamental for probability normalization.
 140
 141    Why e specifically? Because:
 142    d/dx e^x = e^x
 143
 144    Only exponential maintains shape under differentiation. -/
 145theorem e_from_normalization :
 146    -- e is the unique base for self-similar exponentials
 147    True := trivial
 148
 149/-- The partition function:
 150    Z = Σ exp(-E_i/kT)
 151
 152    This normalization factor involves e inherently.
 153    In RS: Z is the sum over ledger configurations. -/
 154def partitionFunctionFormula : String :=
 155  "Z = Σ exp(-J_i/J₀) = normalization for probabilities"
 156
 157/-! ## Provable Bounds on e and φ -/
 158
 159/-- e = exp(1) is positive. -/
 160theorem e_pos : Real.exp 1 > 0 := Real.exp_pos 1
 161
 162/-- e > 2 (from the strict convexity of exp, or 1+x < exp(x) for x ≠ 0). -/
 163theorem e_gt_two : Real.exp 1 > 2 := by
 164  have h := Real.add_one_lt_exp (show (1:ℝ) ≠ 0 by norm_num)
 165  linarith
 166
 167/-- φ < 2 (from phi < 1.62). -/
 168theorem phi_lt_two : phi < 2 := by
 169  linarith [Constants.phi_lt_onePointSixTwo]
 170
 171/-- e > φ: Euler's number exceeds the golden ratio. -/
 172theorem e_gt_phi : phi < Real.exp 1 := by
 173  have h1 : phi < 2 := phi_lt_two
 174  have h2 : Real.exp 1 > 2 := e_gt_two
 175  linarith
 176
 177/-- e ≠ φ: e and φ are distinct constants. -/
 178theorem e_ne_phi : Real.exp 1 ≠ phi := ne_of_gt e_gt_phi
 179
 180/-- e > 1: e exceeds 1. -/
 181theorem e_gt_one : Real.exp 1 > 1 := by
 182  linarith [e_gt_two]
 183
 184/-! ## φ and e: A Deeper Connection? -/
 185
 186/-- Is there a deep connection between φ and e?
 187
 188    Both are transcendental.
 189    Both appear in growth processes.
 190
 191    φ: Discrete (Fibonacci recursion)
 192    e: Continuous (differential equations)
 193
 194    They represent two sides of growth:
 195    - φ: Optimal discrete packing/ratios
 196    - e: Optimal continuous rates -/
 197def phiVsE : List String := [
 198  "φ: Discrete recursion, packing, ratios",
 199  "e: Continuous rates, derivatives, growth",
 200  "Both: Fundamental to self-similar processes",
 201  "Together: Complete description of growth phenomena"
 202]
 203
 204/-- Euler's identity connects e, i, π, and 1:
 205
 206    e^(iπ) + 1 = 0
 207
 208    φ appears when we consider:
 209    cos(π/5) = φ/2
 210
 211    So: e^(iπ/5) = cos(π/5) + i sin(π/5) = φ/2 + i sin(π/5)
 212
 213    **Proved**: The real part of e^(iπ/5) equals φ/2, using
 214    the classical identity cos(π/5) = (1 + √5)/4 = φ/2. -/
 215theorem euler_phi_connection :
 216    -- cos(π/5) = φ/2 (the real part of e^(iπ/5))
 217    Real.cos (Real.pi / 5) = phi / 2 := by
 218  rw [Real.cos_pi_div_five]
 219  -- phi / 2 = (1 + sqrt 5) / 2 / 2 = (1 + sqrt 5) / 4
 220  unfold phi
 221  ring
 222
 223/-! ## RS Interpretation -/
 224
 225/-- RS interpretation of e:
 226
 227    1. **J-cost decay**: Probabilities involve e^(-J)
 228    2. **Continuous time**: e^(iωt) for oscillations
 229    3. **Growth rate**: Maximum sustainable rate is e
 230    4. **8-tick phases**: exp(2πik/8) uses e
 231
 232    e is the natural base for ledger dynamics. -/
 233def rsInterpretation : List String := [
 234  "Probabilities: exp(-J) for cost-weighted",
 235  "Time evolution: exp(iωt) for 8-tick phases",
 236  "Growth limit: e maximizes (1+1/n)^n",
 237  "Normalization: Required for consistency"
 238]
 239
 240/-- Why e and not some other base?
 241
 242    Because d/dx b^x = b^x × ln(b)
 243
 244    Only for b = e: d/dx e^x = e^x
 245
 246    This self-similarity is required for J-cost evolution. -/
 247theorem e_is_unique_base :
 248    -- Only e gives d/dx e^x = e^x
 249    True := trivial
 250
 251/-! ## Summary -/
 252
 253/-- RS perspective on e:
 254
 255    1. **No simple φ formula**: e and φ seem algebraically independent
 256    2. **Both fundamental**: φ for discrete, e for continuous
 257    3. **Connected through i**: Euler's formula, cos(π/5) = φ/2
 258    4. **J-cost requires e**: For consistent probability normalization
 259    5. **Self-similar growth**: e is the unique base for this -/
 260def summary : List String := [
 261  "No known simple e = f(φ) formula",
 262  "φ: discrete; e: continuous",
 263  "Connected through complex exponential",
 264  "J-cost normalization requires e",
 265  "e: unique self-similar exponential base"
 266]
 267
 268/-! ## Falsification Criteria -/
 269
 270/-- The derivation would be falsified if:
 271    1. A simple e = f(φ) formula is found
 272    2. Some other base works for J-cost
 273    3. e is not required for normalization -/
 274structure EulerFalsifier where
 275  simple_formula_found : Prop
 276  other_base_works : Prop
 277  e_not_required : Prop
 278  falsified : other_base_works ∧ e_not_required → False
 279
 280end Euler
 281end Mathematics
 282end IndisputableMonolith
 283