energy_creates_processing_gradient
Energy distributions with non-zero density gradients induce non-zero gradients in the associated processing field. Physicists modeling weak-field gravity via J-cost in Recognition Science cite this to connect energy concentrations directly to gravitational sourcing. The tactic proof unfolds the linear definition of the processing field, applies the constant-multiple derivative rule, and concludes via the non-zero product lemma.
claimLet $ρ : ℝ → ℝ$ be the density function of an energy distribution. For $G_{eff} ≠ 0$ and position $h_0$ where $ρ$ is differentiable with $ρ'(h_0) ≠ 0$, the processing field $Φ(h) = G_{eff} ⋅ ρ(h)$ satisfies $Φ'(h_0) ≠ 0$.
background
EnergyDistribution is a structure carrying a non-negative density function from Position (the reals) to the reals, identified with J-cost density. The definition energy_to_processing_field constructs the processing field by direct scaling: phi(h) := G_eff * density(h). This models the weak-field Newtonian potential in one dimension, where the Poisson relation is axiomatized rather than derived. Position and Energy are reals in RS-native units; the result depends on differentiability from Mathlib and the upstream density structure.
proof idea
The tactic proof begins by simp-unfolding energy_to_processing_field to expose the linear scaling. It then applies the lemma deriv_const_mul to obtain deriv(G_eff * density) = G_eff * deriv(density). A rewrite step reduces the goal to showing the product is non-zero, which follows immediately from mul_ne_zero using the supplied hypotheses hG and h_grad.
why it matters in Recognition Science
This theorem supplies the direct link from energy density gradients to non-trivial processing fields inside the EnergyProcessingBridge module. It is invoked by the downstream result energy_distribution_creates_gravity_modifier, which lifts the gradient non-vanishing to an opposing gravitational modification. In the Recognition Science setting it realizes the T⁰⁰ = J-cost density identity that lets energy concentrations source gravity through the processing potential in the weak-field limit.
scope and limits
- Does not derive the three-dimensional Poisson equation.
- Does not incorporate relativistic or curvature corrections.
- Does not fix the numerical value of G_eff from RS constants.
- Does not construct explicit energy distributions.
Lean usage
exact energy_creates_processing_gradient energy G_eff hG h0 h_diff h_gradformal statement (Lean)
93theorem energy_creates_processing_gradient
94 (energy : EnergyDistribution) (G_eff : ℝ) (hG : G_eff ≠ 0)
95 (h0 : Position)
96 (h_diff : DifferentiableAt ℝ energy.density h0)
97 (h_grad : deriv energy.density h0 ≠ 0) :
98 deriv (energy_to_processing_field energy G_eff).phi h0 ≠ 0 := by
proof body
Tactic-mode proof.
99 simp only [energy_to_processing_field]
100 have : deriv (fun h => G_eff * energy.density h) h0 = G_eff * deriv energy.density h0 := by
101 exact deriv_const_mul G_eff h_diff
102 rw [this]
103 exact mul_ne_zero hG h_grad
104
105/-! ## 3. The Bridge: Any Energy Source Gravitates -/
106
107/-- Structure packaging the energy-processing equivalence. -/