homogeneous_minimizes_cost
Homogeneous density configurations have strictly lower J-cost than those with a 1% density contrast. Cosmologists using Recognition Science to address the horizon problem would cite this to show uniformity is selected by cost minimization. The proof is a direct algebraic reduction that unfolds the cost definition, applies the unit and squared-form lemmas for Jcost, and verifies positivity by norm_num.
claimLet $J(x) = (x-1)^2/(2x)$ for $x>0$. Then $J(1) < J(1.01)$.
background
The Cosmology.HorizonProblem module frames the horizon problem as resolved by 8-tick synchronization providing universal coherence without light-speed contact. costOfInhomogeneity(δρ) is defined as Jcost(1 + |δρ|), so the theorem compares the cost at zero contrast to a small nonzero contrast. Upstream lemmas establish Jcost(1) = 0 and the explicit squared form Jcost(x) = (x-1)^2/(2x) for x ≠ 0; tick supplies the fundamental RS time quantum.
proof idea
The term proof unfolds costOfInhomogeneity, simplifies absolutes and zeros, rewrites the left side to zero via Jcost_unit0, applies Jcost_eq_sq with a non-zero guard, simplifies the absolute value on the right, and closes with norm_num to confirm the positive difference.
why it matters in Recognition Science
The result supports the module's claim that J-cost minimization favors homogeneous initial conditions, complementing inflation by explaining why uniformity is selected in the first place. It fills the cost-minimization step in the RS solution to the horizon problem and aligns with J-uniqueness in the forcing chain. No downstream uses are recorded, leaving its integration with the full 8-tick octave and D=3 structure open.
scope and limits
- Does not prove homogeneity is the global minimum over all contrasts.
- Does not derive relaxation dynamics or time evolution.
- Does not invoke the 8-tick period or spatial dimension count.
- Does not address observational bounds on CMB contrasts.
formal statement (Lean)
153theorem homogeneous_minimizes_cost :
154 costOfInhomogeneity 0 < costOfInhomogeneity 0.01 := by
proof body
Term-mode proof.
155 unfold costOfInhomogeneity
156 simp only [abs_zero, add_zero]
157 -- J(1) < J(1.01) because J(1) = 0 and J(1.01) > 0
158 rw [Jcost_unit0]
159 -- Need: 0 < Jcost(1 + |0.01|) = Jcost(1.01)
160 rw [Jcost_eq_sq (by norm_num : (1 : ℝ) + |0.01| ≠ 0)]
161 -- (1.01 - 1)² / (2 × 1.01) = 0.0001 / 2.02 > 0
162 simp only [abs_of_pos (by norm_num : (0.01 : ℝ) > 0)]
163 norm_num
164
165/-! ## Combining Inflation and RS -/
166
167/-- Inflation and RS are complementary:
168
169 - Inflation: Explains HOW uniform regions got stretched
170 - RS: Explains WHY uniformity was favored in the first place
171
172 Together:
173 1. J-cost minimization selected homogeneous initial conditions
174 2. Inflation stretched one homogeneous patch to observable universe
175 3. 8-tick synchronization maintained coherence during expansion -/