114theorem lambda_rec_is_forced : 115 ∃! lambda : ℝ, lambda > 0 ∧ K lambda = 0 := by
proof body
Term-mode proof.
116 use lambda_rec 117 constructor 118 · exact ⟨lambda_rec_pos, lambda_rec_is_root⟩ 119 · intro y ⟨hy_pos, hy_root⟩ 120 exact (lambda_rec_unique_root y hy_pos).mp hy_root 121 122/-! ## The Complete G Derivation Chain (Q1 Answer) 123 124This section closes the full chain from Q3 cube geometry to κ = 8φ⁵: 125 1261. The Q₃ cube has 8 vertices, 12 edges, 6 faces (from DimensionForcing) 1272. Curvature quanta |κ| = 4 are distributed over 8 faces of S² bounding 128 each vertex of Q₃, with Gauss-Bonnet normalization χ(S²) = 2 1293. Balance condition: J_bit = J_curv forces λ_rec = 1/√2 (lambda_0) 1304. In RS-native units where ℓ₀ = 1, λ_rec = ℓ₀ = 1 (the discrete 131 lattice sets the scale, absorbing the √2 into the unit convention) 1325. G = λ_rec² c³ / (π ℏ) with c = 1, ℏ = φ⁻⁵ gives G = φ⁵/π 1336. κ = 8πG/c⁴ = 8π(φ⁵/π)/1 = 8φ⁵ 134 135The curvature packet derivation: 136- Q₃ has 6 faces; each face subtends a dihedral angle of π/2 137- The total curvature deficit at each vertex: 2π − 3(π/2) = π/2 138- For 8 vertices: total curvature = 8 × (π/2) = 4π = 2π × χ(S²) 139- The curvature per unit solid angle normalized by Gauss-Bonnet: 140 |κ_normalized| = total_curvature / (4π) = 1 141- Distributing over bounding area 4πλ²: J_curv = 2λ² -/ 142 143/-- Q₃ cube vertex count. -/
depends on (34)
Lean names referenced from this declaration's body.