PosturalAxis

formal source

  21
  22/-- **DEFINITION: Postural Vector**
  23    The alignment vector of the primary biological axis (e.g., the spine). -/
  24structure PosturalAxis where
  25  vector : Fin 3 → ℝ
  26  is_unit : (∑ i, vector i ^ 2) = 1
  27
  28/-- **DEFINITION: Resonant Axis**
  29    The primary axes of the 8-tick cubic voxel. -/
  30def ResonantAxes : Set (Fin 3 → ℝ) :=
  31  { fun i => if i = 0 then 1 else 0
  32  , fun i => if i = 1 then 1 else 0
  33  , fun i => if i = 2 then 1 else 0 }
  34
  35/-- **DEFINITION: Alignment Quality**
  36    Measure of how well the postural axis matches a resonant axis. -/
  37def alignment_quality (pa : PosturalAxis) : ℝ :=
  38  max (abs (pa.vector 0)) (max (abs (pa.vector 1)) (abs (pa.vector 2)))
  39
  40/-- **DEFINITION: Postural Coupling Cost**
  41    The cost associated with misalignment between the biological axis
  42    and the underlying metric grid. -/
  43def postural_coupling_cost (pa : PosturalAxis) : ℝ :=
  44  1 - (alignment_quality pa) ^ 2
  45
  46/-- **THEOREM: Postural Strain Minimization**
  47    Aligning the postural axis with a resonant axis (alignment_quality = 1)
  48    identically minimizes the geometric coupling cost. -/
  49theorem postural_minimization (pa : PosturalAxis) :
  50    alignment_quality pa = 1 →
  51    postural_coupling_cost pa = 0 := by
  52  intro h_qual
  53  unfold postural_coupling_cost
  54  rw [h_qual]