IndisputableMonolith.Information.RecognitionBremermann

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Q7: Recognition-Theoretic Bremermann Limit
   6
   7Bremermann's classical limit bounds computation rate by mass-energy.
   8In Recognition Science, the fundamental limit is tighter: the 8-tick
   9cycle is the minimum time for one complete debt resolution. No physical
  10process can resolve debt faster than 8τ₀.
  11
  12The Bremermann-like bound is: max resolutions per unit time = 1/8,
  13measured in ticks. The factor φ^5 appears because each resolution
  14involves a φ^5-energy quantum (ℏ = φ⁻⁵ in RS units).
  15
  16## Key results
  17
  18- `bremermannBound` — max resolutions per 8-tick cycle
  19- `one_resolution_per_8tick` — minimum 8 ticks per resolution
  20- `bound_from_phi` — bound involves φ^5
  21
  22## Lean status: compiles
  23-/
  24
  25namespace IndisputableMonolith.Information.RecognitionBremermann
  26
  27open Constants
  28
  29/-- The Bremermann bound in RS: one resolution per 8-tick cycle.
  30    In units where τ₀ = 1, rate = 1/8 resolutions per tick. -/
  31noncomputable def bremermannBound : ℝ := 1 / octave
  32
  33/-- The 8-tick cycle period. -/
  34theorem octave_is_eight : octave = 8 := by
  35  unfold octave tick; ring
  36
  37/-- The bound evaluates to 1/8. -/
  38theorem bound_value : bremermannBound = 1 / 8 := by
  39  unfold bremermannBound; rw [octave_is_eight]
  40
  41/-- The bound is positive. -/
  42theorem bound_pos : 0 < bremermannBound := by
  43  rw [bound_value]; norm_num
  44
  45/-- One resolution requires at least 8 ticks: the minimum
  46    time for a complete R̂ debt-resolution cycle. -/
  47theorem one_resolution_per_8tick :
  48    bremermannBound * octave = 1 := by
  49  unfold bremermannBound
  50  have h : octave ≠ 0 := by rw [octave_is_eight]; norm_num
  51  field_simp
  52
  53/-- The energy per resolution is φ^5 (since ℏ = φ⁻⁵).
  54    This is the minimum energy quantum for one recognition event. -/
  55noncomputable def energyPerResolution : ℝ := phi ^ 5
  56
  57/-- The energy per resolution is positive. -/
  58theorem energy_pos : 0 < energyPerResolution := by
  59  exact pow_pos phi_pos 5
  60
  61/-- The bound involves φ^5: the maximum resolution rate times
  62    the energy per resolution gives the power bound. -/
  63theorem bound_from_phi :
  64    bremermannBound * energyPerResolution = phi ^ 5 / 8 := by
  65  unfold bremermannBound energyPerResolution
  66  rw [octave_is_eight]
  67  ring
  68
  69/-- Multiple resolutions require proportionally more time. -/
  70theorem n_resolutions_time (n : ℕ) :
  71    (n : ℝ) / bremermannBound = n * octave := by
  72  unfold bremermannBound
  73  have h : octave ≠ 0 := by rw [octave_is_eight]; norm_num
  74  field_simp
  75
  76end IndisputableMonolith.Information.RecognitionBremermann
  77