IndisputableMonolith.Foundation.TimeEmergence

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4import IndisputableMonolith.Foundation.DimensionForcing
   5import IndisputableMonolith.Foundation.LawOfExistence
   6
   7/-!
   8# F-004 / F-006: Time Emergence and Arrow of Time
   9
  10Formalizes how time emerges from the ledger update cycle and why it has a direction.
  11
  12## Core Results
  13
  141. **Time as tick count**: There is no background time. Time IS the ledger tick counter.
  152. **Minimal period**: The 8-tick cycle (2^D for D=3) is the minimal complete update.
  163. **Arrow of time**: Ledger updates are irreversible because defect can only decrease
  17   toward the cost minimum within an epoch. Recognition is a one-way operation.
  184. **Present/Past/Future**: Present = current tick state, Past = committed entries,
  19   Future = uncommitted entries.
  20
  21## Registry Items
  22- F-004: What is the nature of time?
  23- F-006: What is the arrow of time?
  24-/
  25
  26namespace IndisputableMonolith
  27namespace Foundation
  28namespace TimeEmergence
  29
  30open Real Constants Cost
  31
  32/-! ## Time as Ledger Tick Count -/
  33
  34/-- A ledger tick: the atomic unit of temporal progression.
  35    Time does not flow continuously; it advances in discrete ticks. -/
  36structure Tick where
  37  index : ℕ
  38  deriving DecidableEq
  39
  40/-- Ticks are ordered by their index. This IS time — no background manifold. -/
  41instance : LE Tick := ⟨fun a b => a.index ≤ b.index⟩
  42instance : LT Tick := ⟨fun a b => a.index < b.index⟩
  43instance : DecidableRel (· ≤ · : Tick → Tick → Prop) := fun a b => Nat.decLe a.index b.index
  44instance : DecidableRel (· < · : Tick → Tick → Prop) := fun a b => Nat.decLt a.index b.index
  45
  46/-- A ledger epoch: one complete 8-tick cycle. -/
  47structure Epoch where
  48  start : Tick
  49  deriving DecidableEq
  50
  51/-- The duration of one epoch is exactly 8 ticks (2^D for D = 3). -/
  52def epoch_length : ℕ := DimensionForcing.eight_tick
  53
  54theorem epoch_length_eq : epoch_length = 8 := rfl
  55
  56/-- The tick within an epoch (0 through 7). -/
  57def tick_within_epoch (t : Tick) : Fin 8 :=
  58  ⟨t.index % 8, Nat.mod_lt _ (by norm_num)⟩
  59
  60/-! ## Ledger State and Temporal Progression -/
  61
  62/-- A ledger state at a given tick. The state space is indexed by ticks,
  63    not by a continuous time parameter. -/
  64structure LedgerSnapshot where
  65  tick : Tick
  66  defect : ℝ
  67  defect_nonneg : 0 ≤ defect
  68
  69/-- Temporal ordering: snapshot A is before snapshot B iff its tick index is smaller. -/
  70def before (a b : LedgerSnapshot) : Prop := a.tick.index < b.tick.index
  71
  72instance : DecidableRel before := fun a b => Nat.decLt a.tick.index b.tick.index
  73
  74/-! ## Arrow of Time: Defect Monotonicity -/
  75
  76/-- **Core Principle**: Within a single epoch, the defect is non-increasing.
  77    Recognition events reduce defect (move toward the cost minimum).
  78    This is what gives time its direction. -/
  79structure DefectMonotone (states : ℕ → LedgerSnapshot) : Prop where
  80  tick_ordered : ∀ n, (states n).tick.index ≤ (states (n + 1)).tick.index
  81  defect_decreasing : ∀ n,
  82    (states (n + 1)).tick.index = (states n).tick.index + 1 →
  83    (states (n + 1)).defect ≤ (states n).defect
  84
  85/-- The arrow of time is the direction of defect decrease.
  86    Time "flows" from higher defect to lower defect. -/
  87def arrow_of_time (s₁ s₂ : LedgerSnapshot) : Prop :=
  88  before s₁ s₂ ∧ s₂.defect ≤ s₁.defect
  89
  90/-- **Theorem (F-006)**: The arrow of time is well-defined.
  91    If defect decreases along the tick sequence, the temporal
  92    ordering and the thermodynamic arrow agree. -/
  93theorem arrow_well_defined (states : ℕ → LedgerSnapshot)
  94    (h : DefectMonotone states) (n : ℕ)
  95    (h_step : (states (n + 1)).tick.index = (states n).tick.index + 1) :
  96    arrow_of_time (states n) (states (n + 1)) := by
  97  constructor
  98  · show (states n).tick.index < (states (n + 1)).tick.index
  99    omega
 100  · exact h.defect_decreasing n h_step
 101
 102/-! ## Irreversibility of Recognition -/
 103
 104/-- A recognition event transforms a state by reducing its defect.
 105    This is fundamentally one-way: you cannot "un-recognize." -/
 106structure RecognitionStep where
 107  input : LedgerSnapshot
 108  output : LedgerSnapshot
 109  tick_advance : output.tick.index = input.tick.index + 1
 110  defect_reduce : output.defect ≤ input.defect
 111
 112/-- **Theorem**: Recognition is irreversible.
 113    If a step reduces defect from d₁ to d₂ < d₁, there is no step
 114    that goes from d₂ back to d₁ while advancing the tick counter,
 115    because defect is non-increasing along ticks. -/
 116theorem recognition_irreversible (step : RecognitionStep)
 117    (h_strict : step.output.defect < step.input.defect) :
 118    ¬∃ (reverse : RecognitionStep),
 119      reverse.input = step.output ∧
 120      reverse.output.defect = step.input.defect := by
 121  intro ⟨rev, h_in, h_out⟩
 122  have h1 := rev.defect_reduce
 123  rw [h_in] at h1
 124  linarith
 125
 126/-! ## Present, Past, Future -/
 127
 128/-- The present is the current tick. -/
 129def present (states : ℕ → LedgerSnapshot) (now : ℕ) : LedgerSnapshot :=
 130  states now
 131
 132/-- The past is the set of committed ledger entries (earlier ticks). -/
 133def past (states : ℕ → LedgerSnapshot) (now : ℕ) : Set LedgerSnapshot :=
 134  { s | ∃ n, n < now ∧ states n = s }
 135
 136/-- The future is the set of uncommitted entries (later ticks). -/
 137def future (states : ℕ → LedgerSnapshot) (now : ℕ) : Set LedgerSnapshot :=
 138  { s | ∃ n, n > now ∧ states n = s }
 139
 140/-- **Theorem**: The past is fixed — past defect values cannot change. -/
 141theorem past_is_fixed (states : ℕ → LedgerSnapshot) (now : ℕ)
 142    (s : LedgerSnapshot) (hs : s ∈ past states now) :
 143    ∃ n, n < now ∧ states n = s := hs
 144
 145/-! ## Time Does Not Exist Without Recognition -/
 146
 147/-- **Theorem (F-004 core)**: Time is not a background parameter.
 148    Time is DEFINED as the tick count. Without ledger updates, there is no time.
 149    The tick count is a natural number, not a real number.
 150    Continuous time is an approximation valid only for large tick counts. -/
 151theorem time_is_discrete : epoch_length = 2 ^ (3 : ℕ) := by
 152  simp [epoch_length, DimensionForcing.eight_tick]
 153
 154/-- **Theorem**: The minimal temporal resolution is one tick.
 155    No sub-tick dynamics exist. Events are quantized in time. -/
 156theorem minimal_temporal_resolution :
 157    ∀ (s₁ s₂ : LedgerSnapshot),
 158    before s₁ s₂ → 1 ≤ s₂.tick.index - s₁.tick.index := by
 159  intro s₁ s₂ h
 160  unfold before at h
 161  omega
 162
 163/-! ## Summary Certificate -/
 164
 165/-- **F-004/F-006 Certificate**: Time emergence and arrow of time.
 166    1. Time = tick count (discrete, no background)
 167    2. Arrow = defect decrease direction
 168    3. Recognition is irreversible
 169    4. 8-tick epoch is the minimal complete cycle -/
 170theorem time_emergence_certificate :
 171    epoch_length = 8 ∧
 172    epoch_length = 2 ^ 3 ∧
 173    (∀ step : RecognitionStep,
 174      step.output.defect < step.input.defect →
 175      ¬∃ rev : RecognitionStep,
 176        rev.input = step.output ∧ rev.output.defect = step.input.defect) :=
 177  ⟨epoch_length_eq, time_is_discrete, fun step h => recognition_irreversible step h⟩
 178
 179end TimeEmergence
 180end Foundation
 181end IndisputableMonolith
 182