 128/-! ## Modal Nyquist Theorem -/
 129
 130/-- **8-TICK PERIOD**: The fundamental period of modal evolution. -/
 131def modal_period : ℕ := 8
 132
 133/-- **MODAL NYQUIST LIMIT**: The finest modal resolution is one tick.
 134
 135    Possibilities that differ by less than one tick are "modally equivalent."
 136    This is analogous to the Nyquist sampling theorem:
 137    - Maximum modal frequency = 1/(2τ₀)
 138    - Modal bandwidth = 4 ticks per octave -/
 139def modal_nyquist_limit : ℕ := modal_period / 2
 140
 141/-- Two configs are **modally equivalent** if they differ by less than one tick. -/
 142def modally_equivalent (c1 c2 : Config) : Prop :=
 143  c1.value = c2.value ∧ (c1.time : ℤ) - (c2.time : ℤ) < 1 ∧ (c2.time : ℤ) - (c1.time : ℤ) < 1
 144
 145/-- Modal equivalence is reflexive. -/
 146lemma modally_equivalent_refl (c : Config) : modally_equivalent c c := by
 147  simp [modally_equivalent]
 148
 149/-- Modal equivalence is symmetric. -/
 150lemma modally_equivalent_symm (c1 c2 : Config) :
 151    modally_equivalent c1 c2 ↔ modally_equivalent c2 c1 := by
 152  simp [modally_equivalent]
 153  constructor <;> (intro ⟨h1, h2, h3⟩; exact ⟨h1.symm, h3, h2⟩)
 154
 155/-- **MODAL NYQUIST THEOREM**: The universe cannot distinguish possibilities
 156    finer than one tick.
 157
 158    This is the modal analog of:
 159    - Nyquist sampling (time)
 160    - Heisenberg uncertainty (phase space)
 161    - Gap-45 consciousness threshold (qualia)

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.