`eightTickCadence_eq`

proved

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module: IndisputableMonolith.Unification.RecognitionBandwidth
domain: Unification
line: 54 · github
papers citing: none yet

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IndisputableMonolith.Unification.RecognitionBandwidth on GitHub at line 54.

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depends on

formal source

  51  unfold eightTickCadence τ₀ tick
  52  norm_num
  53
  54theorem eightTickCadence_eq : eightTickCadence = 8 := by
  55  unfold eightTickCadence τ₀ tick
  56  ring
  57
  58/-! ## §2. Recognition Bandwidth -/
  59
  60/-- **DEFINITION**: Recognition bandwidth of a region with boundary area A.
  61
  62    The maximum number of recognition events per unit time that the holographic
  63    bound permits within the region, given that each event costs k_R = ln(φ) bits
  64    and the 8-tick cadence limits processing to one cycle per 8τ₀.
  65
  66        R_max(A) = A / (4ℓ_P² · k_R · 8τ₀)
  67
  68    Units: events per unit time.
  69
  70    This combines three previously disconnected elements:
  71    - Holographic capacity: A/(4ℓ_P²)      [from Quantum.HolographicBound]
  72    - Per-bit cost: k_R = ln(φ)            [from Constants.BoltzmannConstant]
  73    - Processing rate: 8τ₀ per cycle       [from Foundation.EightTick]  -/
  74noncomputable def bandwidth (area : ℝ) : ℝ :=
  75  area / (4 * planckArea * k_R * eightTickCadence)
  76
  77/-- Planck area is positive. -/
  78theorem planckArea_pos : 0 < planckArea := by
  79  unfold planckArea planckLength
  80  positivity
  81
  82/-- The denominator of the bandwidth formula is positive. -/
  83theorem bandwidth_denom_pos : 0 < 4 * planckArea * k_R * eightTickCadence := by
  84  apply mul_pos

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