IndisputableMonolith.Gravity.CoherenceFall

   1/-
   2  CoherenceFall.lean
   3
   4  Formalizes the insight: "Falling is Coherence/Synchronization".
   5
   6  THE KEY INSIGHT:
   7  In the presence of a gravitational gradient (varying processing density),
   8  an extended object experiences a "refresh rate mismatch" (decoherence).
   9
  10  However, in a frame accelerating at `a`, there is an induced inertial potential
  11  `Φ_acc(z) = a * z` so that the induced inertial force is `F_inertial = -∇Φ_acc = -a`.
  12
  13  The "Falling Condition" is the choice of acceleration `a` such that
  14  the total potential `Φ_total = Φ_grav + Φ_acc` is locally constant (flat).
  15
  16  Flat potential => Uniform refresh rate => Coherence (Zero J-cost).
  17
  18  Part of: IndisputableMonolith/Gravity/
  19-/
  20
  21import Mathlib
  22-- import IndisputableMonolith.Foundation.RecognitionOperator  -- [not in public release]
  23
  24noncomputable section
  25
  26namespace IndisputableMonolith.Gravity
  27
  28-- (no external Foundation import needed for this self-contained module)
  29
  30/-! ## The Setup -/
  31
  32abbrev Position := ℝ
  33
  34structure ProcessingField where
  35  /-- Potential function Φ(h) -/
  36  phi : Position → ℝ
  37
  38structure ExtendedObject where
  39  h_cm : Position
  40  extent : ℝ
  41  extent_pos : extent > 0
  42
  43/-! ## Frame Dependent Refresh Rate -/
  44
  45/-- Total Potential in a frame accelerating with `a` at position `h` (relative to CM).
  46    Φ_tot(z) ≈ Φ_grav(h_cm + z) + a * z
  47    (Linear approximation for local frame)
  48-/
  49def total_potential_in_frame (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) (z : ℝ) : ℝ :=
  50  -- Taylor expand phi around h_cm: phi(h_cm) + phi'(h_cm) * z
  51  let phi_grav := field.phi obj.h_cm + (deriv field.phi obj.h_cm) * z
  52  -- Inertial potential from acceleration a (pointing up? a is vertical acceleration)
  53  -- If object accelerates down (a < 0), inertial force is up.
  54  -- Potential Φ_acc such that F = -∇Φ_acc. F_inertial = -a, hence Φ_acc = a * z.
  55  let phi_acc := a * z
  56
  57  phi_grav + phi_acc
  58
  59/-- Coherence Defect: Variance of the potential across the object.
  60    If Potential is flat, Defect is 0.
  61-/
  62def coherence_defect (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) : ℝ :=
  63  -- Difference in potential between Head (z = +extent) and Feet (z = -extent)
  64  let pot_head := total_potential_in_frame field obj a obj.extent
  65  let pot_feet := total_potential_in_frame field obj a (-obj.extent)
  66  abs (pot_head - pot_feet)
  67
  68/-- Helper: expand coherence_defect into explicit arithmetic (no let bindings). -/
  69private lemma coherence_defect_expand (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) :
  70    coherence_defect field obj a =
  71      abs ((field.phi obj.h_cm + deriv field.phi obj.h_cm * obj.extent + a * obj.extent) -
  72           (field.phi obj.h_cm + deriv field.phi obj.h_cm * (-obj.extent) + a * (-obj.extent))) := by
  73  rfl
  74
  75/-- Closed form for the linearized coherence defect:
  76    `coherence_defect = | 2 * extent * (∂ϕ + a) |`. -/
  77lemma coherence_defect_simplify (field : ProcessingField) (obj : ExtendedObject) (a : ℝ) :
  78    coherence_defect field obj a =
  79      abs (2 * obj.extent * (deriv field.phi obj.h_cm + a)) := by
  80  rw [coherence_defect_expand]
  81  congr 1
  82  ring
  83
  84/-! ## The Theorem -/
  85
  86/-- Falling (Acceleration) Restores Coherence.
  87
  88    Theorem: There exists a unique acceleration `a` that reduces the
  89    linear Coherence Defect to zero.
  90
  91    This `a` is exactly the gravitational acceleration `g = -∇Φ`.
  92-/
  93theorem falling_restores_coherence (field : ProcessingField) (obj : ExtendedObject) :
  94    ∃! a : ℝ, coherence_defect field obj a = 0 := by
  95  -- We want |2 * e * (ϕ' + a)| = 0 ⇒ a = -ϕ' (since e > 0).
  96  -- This is exactly "Falling with acceleration = -Gradient".
  97  use -(deriv field.phi obj.h_cm)
  98  constructor
  99  · -- Existence
 100    -- | 2 * e * (ϕ' + (-ϕ')) | = |0| = 0
 101    simp [coherence_defect_simplify]
 102  · -- Uniqueness
 103    intro a' h_zero
 104    -- Reduce to a product equals zero
 105    have h0 : 2 * obj.extent * (deriv field.phi obj.h_cm + a') = 0 := by
 106      simpa [coherence_defect_simplify, abs_eq_zero] using h_zero
 107    -- From |x| = 0 we get x = 0
 108    -- Since obj.extent > 0, we have 2 * obj.extent ≠ 0
 109    have h_extent_pos : (0 : ℝ) < 2 * obj.extent := by
 110      have htwo : (0 : ℝ) < 2 := by norm_num
 111      exact mul_pos htwo obj.extent_pos
 112    have h_extent_ne : 2 * obj.extent ≠ 0 := ne_of_gt h_extent_pos
 113    -- So (deriv field.phi obj.h_cm + a') = 0
 114    have h2 : deriv field.phi obj.h_cm + a' = 0 := by
 115      have := mul_eq_zero.mp h0
 116      cases this with
 117      | inl h => exact absurd h h_extent_ne
 118      | inr h => exact h
 119    -- Therefore a' = -(deriv field.phi obj.h_cm)
 120    linarith
 121
 122/-! ## Interpretation
 123
 124"Gravity" is the requirement to accelerate in order to maintain
 125a locally constant processing environment (Coherence).
 126
 127In this view:
 128- Standing still (a = 0) in a gravitational field means experiencing a coherence defect
 129- Free-falling (a = -∇Φ) cancels the defect, restoring coherence
 130- This is why free fall feels like nothing: you're in the coherent state
 131-/
 132
 133end IndisputableMonolith.Gravity
 134