PROJECT T Volume XXXVIII : Seeds and Trees The Constructible Universe of H₀(Ã₂)

This volume establishes the structural bridge between the combinatorial seed catalogue of H₀(Ã₂) (Volumes I–III, XIX) and the spectral geometry of symmetric group Cayley graphs (Volumes XXXII–XXXVII). We prove that the seed bound l ≤ 6 coincides exactly with the quadratic surd boundary in the number field tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √5) ⊂ ℚ(√2, √3, √5), where the algebraic degree of 2cos(π/n) remains at most 2 for n ≤ 6 and jumps to 3 at n = 7. The 33 seeds distribute according to a Fibonacci sequence, a pattern structurally forced by the cluster algebra of type A₂ (Fomin–Zelevinsky). The 9 extensions beyond the seed bound equal the commutation deficit C(6) = 6 × 3 / 2 = 9. All claims are computationally verified. No free parameters are introduced.