IndisputableMonolith.Foundation.InitialCondition

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Foundation.LawOfExistence
   4import IndisputableMonolith.Foundation.OntologyPredicates
   5
   6/-!
   7# F-005: The Initial Condition — Why Low Entropy
   8
   9Formalizes why the universe began in a low-entropy state.
  10
  11## The Argument
  12
  13The question "Why did the universe start with low entropy?" is traditionally
  14one of the deepest puzzles in cosmology (the Past Hypothesis of Albert 2000,
  15the Weyl Curvature Hypothesis of Penrose 1979).
  16
  17In Recognition Science, the answer is forced:
  18
  191. The only RSExists state is x = 1 (zero defect).
  202. A universe of N ledger entries, all at x = 1, has TOTAL defect = 0.
  213. Total defect 0 IS the minimum entropy state.
  224. Therefore the initial condition is not a choice — it is the UNIQUE
  23   zero-cost configuration, forced by the cost axioms.
  24
  25The "Past Hypothesis" becomes the "Past Theorem."
  26
  27## Registry Item
  28- F-005: Why does the universe have low entropy initial conditions?
  29-/
  30
  31namespace IndisputableMonolith
  32namespace Foundation
  33namespace InitialCondition
  34
  35open Real Cost
  36
  37/-! ## Ledger Configuration -/
  38
  39/-- A configuration of N ledger entries, each a positive real ratio. -/
  40structure Configuration (N : ℕ) where
  41  entries : Fin N → ℝ
  42  entries_pos : ∀ i, 0 < entries i
  43
  44/-- Total defect of a configuration: sum of individual J-costs. -/
  45noncomputable def total_defect {N : ℕ} (c : Configuration N) : ℝ :=
  46  ∑ i : Fin N, LawOfExistence.defect (c.entries i)
  47
  48/-- Total defect is non-negative (each term is non-negative). -/
  49theorem total_defect_nonneg {N : ℕ} (c : Configuration N) : 0 ≤ total_defect c := by
  50  apply Finset.sum_nonneg
  51  intro i _
  52  exact LawOfExistence.defect_nonneg (c.entries_pos i)
  53
  54/-- The zero-defect configuration: all entries equal to 1. -/
  55def unity_config (N : ℕ) (_hN : 0 < N) : Configuration N :=
  56  { entries := fun _ => 1
  57    entries_pos := fun _ => by norm_num }
  58
  59/-- The unity configuration has zero total defect. -/
  60theorem unity_defect_zero {N : ℕ} (hN : 0 < N) :
  61    total_defect (unity_config N hN) = 0 := by
  62  unfold total_defect unity_config
  63  simp only [LawOfExistence.defect_at_one]
  64  exact Finset.sum_const_zero
  65
  66/-! ## The Initial Condition is Forced -/
  67
  68/-- **Theorem (F-005 core)**: The unity configuration is the unique
  69    zero-total-defect configuration.
  70    Every entry must be 1 for total defect to vanish. -/
  71theorem zero_defect_iff_unity {N : ℕ} (_hN : 0 < N) (c : Configuration N) :
  72    total_defect c = 0 ↔ ∀ i, c.entries i = 1 := by
  73  constructor
  74  · intro h_zero
  75    have h_terms : ∀ i, LawOfExistence.defect (c.entries i) = 0 := by
  76      by_contra h_not
  77      push_neg at h_not
  78      obtain ⟨j, hj⟩ := h_not
  79      have hj_pos : 0 < LawOfExistence.defect (c.entries j) := by
  80        have h_nn := LawOfExistence.defect_nonneg (c.entries_pos j)
  81        exact lt_of_le_of_ne h_nn (Ne.symm hj)
  82      have h_sum_pos : 0 < total_defect c := by
  83        calc 0 < LawOfExistence.defect (c.entries j) := hj_pos
  84          _ ≤ ∑ i : Fin N, LawOfExistence.defect (c.entries i) := by
  85              apply Finset.single_le_sum (f := fun i => LawOfExistence.defect (c.entries i))
  86                (fun i _ => LawOfExistence.defect_nonneg (c.entries_pos i))
  87                (Finset.mem_univ j)
  88      linarith
  89    intro i
  90    exact (LawOfExistence.defect_zero_iff_one (c.entries_pos i)).mp (h_terms i)
  91  · intro h_all_one
  92    simp only [total_defect]
  93    apply Finset.sum_eq_zero
  94    intro i _
  95    rw [h_all_one i]
  96    exact LawOfExistence.defect_one
  97
  98/-- **Theorem**: The unity configuration achieves the global minimum of total defect. -/
  99theorem unity_is_global_minimum {N : ℕ} (hN : 0 < N) (c : Configuration N) :
 100    total_defect (unity_config N hN) ≤ total_defect c := by
 101  rw [unity_defect_zero hN]
 102  exact total_defect_nonneg c
 103
 104/-- **Theorem**: The unity configuration is the UNIQUE global minimizer. -/
 105theorem unity_unique_minimizer {N : ℕ} (hN : 0 < N) (c : Configuration N) :
 106    total_defect c = total_defect (unity_config N hN) →
 107    ∀ i, c.entries i = 1 := by
 108  rw [unity_defect_zero hN]
 109  exact (zero_defect_iff_unity hN c).mp
 110
 111/-! ## Entropy and Defect -/
 112
 113/-- Entropy of a configuration is proportional to its total defect.
 114    Zero defect = zero entropy = minimum entropy state. -/
 115noncomputable def entropy {N : ℕ} (c : Configuration N) : ℝ :=
 116  total_defect c
 117
 118/-- **Theorem**: The initial state has minimum entropy. -/
 119theorem initial_state_minimum_entropy {N : ℕ} (hN : 0 < N) :
 120    entropy (unity_config N hN) = 0 := unity_defect_zero hN
 121
 122/-- **Theorem**: Any non-unity state has positive entropy. -/
 123theorem nonunity_positive_entropy {N : ℕ} (_hN : 0 < N) (c : Configuration N)
 124    (h : ∃ i, c.entries i ≠ 1) : 0 < entropy c := by
 125  obtain ⟨j, hj⟩ := h
 126  have hj_pos : 0 < LawOfExistence.defect (c.entries j) :=
 127    LawOfExistence.defect_pos_of_ne_one (c.entries_pos j) hj
 128  calc 0 < LawOfExistence.defect (c.entries j) := hj_pos
 129    _ ≤ ∑ i : Fin N, LawOfExistence.defect (c.entries i) := by
 130        apply Finset.single_le_sum (f := fun i => LawOfExistence.defect (c.entries i))
 131          (fun i _ => LawOfExistence.defect_nonneg (c.entries_pos i))
 132          (Finset.mem_univ j)
 133
 134/-! ## The Past Hypothesis Becomes a Theorem -/
 135
 136/-- **The Past Theorem (F-005 Resolution)**:
 137
 138    The universe's initial condition is not a contingent fact but a
 139    mathematical necessity. The unique zero-cost configuration IS the
 140    low-entropy initial state. There is no other option.
 141
 142    This resolves:
 143    - Penrose's "Why was the Big Bang so special?"
 144    - Albert's "Past Hypothesis"
 145    - Boltzmann's "Why didn't we start in thermal equilibrium?"
 146
 147    Answer: Because thermal equilibrium (maximum defect) is infinitely
 148    costly, while the zero-defect state is the unique cost minimum. -/
 149theorem past_theorem {N : ℕ} (hN : 0 < N) :
 150    (∃! c : Configuration N, total_defect c = 0) ∧
 151    total_defect (unity_config N hN) = 0 ∧
 152    (∀ c : Configuration N, total_defect (unity_config N hN) ≤ total_defect c) := by
 153  refine ⟨⟨unity_config N hN, unity_defect_zero hN, ?_⟩, unity_defect_zero hN,
 154    unity_is_global_minimum hN⟩
 155  intro c hc
 156  have h_entries : ∀ i, c.entries i = 1 :=
 157    (zero_defect_iff_unity hN c).mp hc
 158  have h_u_entries : ∀ i, (unity_config N hN).entries i = 1 := fun _ => rfl
 159  have h_eq : c.entries = (unity_config N hN).entries :=
 160    funext fun i => by rw [h_entries i, h_u_entries i]
 161  exact Configuration.mk.injEq .. |>.mpr h_eq
 162
 163end InitialCondition
 164end Foundation
 165end IndisputableMonolith
 166