IndisputableMonolith.Foundation.Breath1024

   1import Mathlib
   2
   3/‑!
   4# 1024‑Tick Generative↔Radiative Interchange (scaffold)
   5
   6Defines a minimal oscillator model over 1024 ticks (2¹⁰) with an eight‑tick
   7periodic micro‑structure, sliding‑8 neutrality, and a flip at tick 512.
   8Includes a fixed phase‑lag predicate for diagnostics/falsifiers.
   9-/
  10
  11namespace IndisputableMonolith
  12namespace Foundation
  13namespace Breath1024
  14
  15open scoped BigOperators Real
  16
  17/‑ Time index in ticks. -/
  18abbrev T := ℕ
  19
  20/‑ Fundamental periods. -/
  21def period8 : ℕ := 8
  22def period1024 : ℕ := 1024
  23def flipTick : ℕ := 512
  24
  25/‑ Sliding sum over 8 ticks. -/
  26def sum8 (x : T → ℝ) (t0 : T) : ℝ :=
  27  (Finset.range period8).sum (fun k => x (t0 + k))
  28
  29/‑ Eight‑window neutrality predicate. -/
  30def neutral8 (x : T → ℝ) (t0 : T) : Prop :=
  31  sum8 x t0 = 0
  32
  33/‑ Oscillator record (display‑level): generative g and radiative r streams
  34   over ticks, assumed period‑8 and period‑1024 periodic respectively. -/
  35structure Osc where
  36  g : T → ℝ
  37  r : T → ℝ
  38
  39/‑ Flip@512 predicate: radiative stream equals generative shifted by 512. -/
  40def flipAt512 (O : Osc) : Prop :=
  41  ∀ t, O.r (t + flipTick) = O.g t
  42
  43/‑ Fixed phase‑lag predicate (diagnostic): r leads g by π/4 on each octave.
  44   Here represented at the level of a simple sinusoidal probe (display‑only). -/
  45def phaseLagPiOver4 (ω : ℝ) (O : Osc) : Prop :=
  46  ∀ t : T,
  47    O.r t = Real.sin (ω * (t : ℝ) + Real.pi / 4) ∧
  48    O.g t = Real.sin (ω * (t : ℝ))
  49
  50end Breath1024
  51end Foundation
  52end IndisputableMonolith
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