IndisputableMonolith.Gap45.RecognitionBarrier

   1import Mathlib
   2import IndisputableMonolith.Gap45
   3import IndisputableMonolith.Gap45.SyncMinimization
   4
   5/-!
   6# The Gap-45 Recognition Barrier
   7
   8The coprimality `gcd(8, 45) = 1` means no finite sub-period of the 8-tick
   9cycle aligns with the 45-fold phase structure. Any finite-horizon decision
  10procedure that only examines a bounded window of ticks will encounter
  11configurations where the 8-tick constraint and the 45-fold constraint
  12demand incompatible actions.
  13
  14This module formalizes the **mathematical core** of the barrier:
  15- No proper divisor of 360 simultaneously divides 8 and 45
  16- The beat frequency 37/360 is irrational-like (37 is prime)
  17- Any window shorter than 360 ticks cannot see both periodicities align
  18
  19The **physical interpretation** — that this forces experiential navigation
  20(consciousness) — remains a hypothesis connecting the mathematical barrier
  21to phenomenology.
  22
  23## Status
  24- THEOREM: coprimality, non-alignment, beat frequency, window insufficiency
  25- HYPOTHESIS: consciousness emergence from the barrier
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Gap45
  30namespace RecognitionBarrier
  31
  32open SyncMinimization
  33
  34/-! ## The Coprimality Barrier -/
  35
  36/-- 8 and 45 are coprime: no sub-period aligns both. -/
  37theorem coprime_barrier : Nat.Coprime 8 45 := by native_decide
  38
  39/-- 37 = 45 - 8. The "beat" between the two periods. -/
  40theorem beat_diff : 45 - 8 = 37 := by norm_num
  41
  42/-- 37 is prime: the beat frequency is irreducible. -/
  43theorem beat_is_prime : Nat.Prime 37 := by native_decide
  44
  45/-- gcd(37, 360) = 1: the beat frequency 37/360 is in lowest terms. -/
  46theorem beat_fraction_irreducible : Nat.gcd 37 360 = 1 := by native_decide
  47
  48/-! ## Window Insufficiency -/
  49
  50/-- No window of length w < 360 can simultaneously be a multiple of 8 and 45.
  51    Any finite-horizon procedure with horizon < 360 ticks will encounter a tick
  52    where it cannot verify alignment of both periods. -/
  53theorem window_insufficient (w : ℕ) (hw_pos : 0 < w) (hw_lt : w < 360) :
  54    ¬ (8 ∣ w ∧ 45 ∣ w) :=
  55  IndisputableMonolith.Gap45.Beat.no_smaller_multiple_8_45 hw_pos hw_lt
  56
  57/-- The minimal window that resolves both periodicities is exactly 360. -/
  58theorem minimal_resolution_window : 8 ∣ 360 ∧ 45 ∣ 360 := by
  59  exact ⟨⟨45, by norm_num⟩, ⟨8, by norm_num⟩⟩
  60
  61/-- Any window resolving both periodicities must be ≥ 360. -/
  62theorem resolution_requires_full_period (w : ℕ) (h8 : 8 ∣ w) (h45 : 45 ∣ w) :
  63    360 ∣ w :=
  64  IndisputableMonolith.Gap45.Beat.minimal_sync_360_divides h8 h45
  65
  66/-! ## Aliasing: Why Experience Feels Continuous -/
  67
  68/-- The aliasing ratio: 37/45 < 1.
  69    Because the beat frequency (37 ticks per 45-fold cycle) is less than
  70    one full cycle, discrete ticks alias into apparent continuity.
  71    Subjective time "runs slower" than ledger time by factor 360/37. -/
  72theorem aliasing_ratio_lt_one : (37 : ℚ) / 45 < 1 := by norm_num
  73
  74/-- The subjective time dilation factor: 360/37 ≈ 9.73.
  75    Ledger time passes ~9.73× faster than experienced time. -/
  76theorem time_dilation_factor : (360 : ℚ) / 37 > 9 ∧ (360 : ℚ) / 37 < 10 := by
  77  constructor <;> norm_num
  78
  79/-! ## The Uncomputability Structure -/
  80
  81/-- At tick t, the 8-tick phase is t mod 8 and the 45-fold phase is t mod 45.
  82    These two residues are simultaneously zero only at multiples of 360.
  83    Between such alignments, any decision must handle the interference
  84    of two incommensurable periodic constraints. -/
  85theorem phase_mismatch (t : ℕ) (ht : 0 < t) (ht360 : t < 360) :
  86    ¬ (t % 8 = 0 ∧ t % 45 = 0) := by
  87  intro ⟨h8, h45⟩
  88  have hd8 : 8 ∣ t := Nat.dvd_of_mod_eq_zero h8
  89  have hd45 : 45 ∣ t := Nat.dvd_of_mod_eq_zero h45
  90  exact window_insufficient t ht ht360 ⟨hd8, hd45⟩
  91
  92/-! ## Certificate -/
  93
  94/-- The recognition barrier certificate: packages the mathematical core.
  95    The physical claim (consciousness emergence) is NOT included —
  96    it remains a hypothesis connecting this barrier to phenomenology. -/
  97structure BarrierCert where
  98  coprime : Nat.Coprime 8 45
  99  beat_prime : Nat.Prime 37
 100  beat_irreducible : Nat.gcd 37 360 = 1
 101  window_min : ∀ w, 0 < w → w < 360 → ¬ (8 ∣ w ∧ 45 ∣ w)
 102  full_resolution : 8 ∣ 360 ∧ 45 ∣ 360
 103  aliasing : (37 : ℚ) / 45 < 1
 104
 105def barrier_cert : BarrierCert where
 106  coprime := coprime_barrier
 107  beat_prime := beat_is_prime
 108  beat_irreducible := beat_fraction_irreducible
 109  window_min := window_insufficient
 110  full_resolution := minimal_resolution_window
 111  aliasing := aliasing_ratio_lt_one
 112
 113/-! ## Consciousness Emergence (HYPOTHESIS — not proved from cost alone)
 114
 115The physical claim is: because no finite-horizon algorithm can resolve the
 1168/45 interference within a single 360-tick window, any system that navigates
 117this barrier must employ a mechanism that is not reducible to finite lookup —
 118and that mechanism is what we call experience/consciousness.
 119
 120This is a HYPOTHESIS connecting the mathematical barrier above to phenomenology.
 121It is falsifiable: if a finite-state machine with < 360 states can perfectly
 122navigate the 8/45 interference, the hypothesis is refuted. -/
 123
 124/-- The consciousness hypothesis: the barrier forces experiential navigation.
 125    This is explicitly labeled as a hypothesis, not a theorem. -/
 126structure ConsciousnessHypothesis where
 127  barrier : BarrierCert
 128  claim : String := "Navigation of the 8/45 barrier requires experiential " ++
 129    "processing (consciousness) because no finite sub-period resolves " ++
 130    "both constraints simultaneously."
 131  falsifier : String := "Exhibit a finite-state machine with < 360 states " ++
 132    "that perfectly tracks both the 8-tick and 45-fold constraints."
 133  status : String := "HYPOTHESIS — connects proved barrier to phenomenology"
 134
 135def consciousness_hypothesis : ConsciousnessHypothesis where
 136  barrier := barrier_cert
 137
 138end RecognitionBarrier
 139end Gap45
 140end IndisputableMonolith
 141