EnergyProcessingEquivalence

formal source

 105/-! ## 3. The Bridge: Any Energy Source Gravitates -/
 106
 107/-- Structure packaging the energy-processing equivalence. -/
 108structure EnergyProcessingEquivalence where
 109  /-- J-cost at balance is zero (existence is free) -/
 110  balance_zero_cost : Jcost 1 = 0
 111  /-- J-cost away from balance is positive (deviation costs) -/
 112  deviation_positive_cost : ∀ x : ℝ, 0 < x → x ≠ 1 → 0 < Jcost x
 113  /-- J-cost matches kinetic energy in weak field: J(1+ε) = ε²/(2(1+ε)) -/
 114  quadratic_energy_bridge : ∀ ε : ℝ, -1 < ε →
 115    Jcost (1 + ε) = ε ^ 2 / (2 * (1 + ε))
 116  /-- Energy creates processing field -/
 117  energy_sources_processing : ∀ (e : EnergyDistribution) (G : ℝ),
 118    ∃ pf : ProcessingField, pf = energy_to_processing_field e G
 119
 120/-- The energy-processing bridge is proved from RS first principles. -/
 121theorem energy_processing_bridge : EnergyProcessingEquivalence where
 122  balance_zero_cost := by unfold Jcost; simp
 123  deviation_positive_cost := by
 124    intro x hx hx1
 125    have h := Jcost_nonneg x hx
 126    rcases eq_or_lt_of_le h with h_eq | h_pos
 127    · exfalso; exact hx1 ((Jcost_zero_iff_one x hx).mp h_eq.symm)
 128    · exact h_pos
 129  quadratic_energy_bridge := Jcost_one_plus_exact
 130  energy_sources_processing := fun e G => ⟨energy_to_processing_field e G, rfl⟩
 131
 132/-! ## 4. Consequence: Any Energy Source Creates a Processing Field -/
 133
 134/-- An energy distribution with non-zero gradient at position h₀ creates a
 135    non-trivial processing field whose gradient can oppose gravity.
 136    This is the key bridge: energy → processing → gravitational modification. -/
 137theorem energy_distribution_creates_gravity_modifier
 138    (energy : EnergyDistribution) (G_eff : ℝ) (hG : 0 < G_eff)