Jcost_one_plus_exact

formal source

  57
  58/-- J-cost exact identity: J(1 + ε) = ε²/(2(1+ε)) for ε > -1.
  59    This is the bridge between J-cost and the Hamiltonian (kinetic energy ≈ ε²/2). -/
  60theorem Jcost_one_plus_exact (ε : ℝ) (hε : -1 < ε) :
  61    Jcost (1 + ε) = ε ^ 2 / (2 * (1 + ε)) := by
  62  unfold Jcost
  63  have h1ε : (0 : ℝ) < 1 + ε := by linarith
  64  have h1ε_ne : (1 + ε) ≠ 0 := ne_of_gt h1ε
  65  field_simp
  66  ring
  67
  68/-- For small ε, J(1+ε) ≈ ε²/2. Specifically, the ratio approaches 1. -/
  69theorem Jcost_quadratic_ratio (ε : ℝ) (hε_neg : -1 < ε) (hε_pos : 0 < ε) :
  70    Jcost (1 + ε) ≤ ε ^ 2 / 2 := by
  71  rw [Jcost_one_plus_exact ε hε_neg]
  72  apply div_le_div_of_nonneg_left (sq_nonneg ε) (by positivity) (by nlinarith)
  73
  74/-! ## 2. Energy Density = Processing Potential -/
  75
  76/-- An energy distribution over space creates a processing field.
  77    In RS, energy IS J-cost, and J-cost IS the processing potential that sources gravity.
  78    This is the identity T⁰⁰ = J-cost density from EFE emergence. -/
  79structure EnergyDistribution where
  80  density : Position → ℝ
  81  density_nonneg : ∀ h, 0 ≤ density h
  82
  83/-- The Newtonian potential sourced by an energy distribution.
  84    In weak-field RS: ∇²Φ = 4πG·ρ, where ρ = J-cost density = energy density.
  85    We model the 1D version: Φ(h) = -G ∫ ρ(h') |h - h'|⁻¹ dh' (schematic).
  86    For the formal proof, we axiomatize the Poisson relation. -/
  87def energy_to_processing_field (energy : EnergyDistribution) (G_eff : ℝ) : ProcessingField where
  88  phi h := G_eff * energy.density h
  89
  90/-- ANY energy concentration creates a non-trivial processing field.