Step 1 —” Hard Topological Theorems
For a compact, oriented, 4-dimensional Calabi—”Yau surface (with $c_1(TX) = 0$ and spin structure):
$$\chi(X) = -\tfrac{1}{2}\int_X p_1(TX) = -\tfrac{3}{2}\,\sigma(X).$$
According to Rokhlin's theorem, the signature $\sigma(X)$ is always a multiple of 16:
$$\sigma(X) \equiv 0 \pmod{16}.$$
This directly implies:
$$\chi(X) \in 24\mathbb{Z}.$$
Consequence: The Euler characteristic of such surfaces can only take values —¦, −72, −48, −24, 0, 24, 48, 72, —¦
Step 2 —” Elimination of Alternatives
- —¢ χ = 0
χ = 0 corresponds to Tâ´. Note: Tâ´ is complex, KΩ¤hler, and Ricci-flat. We exclude Tâ´ by π₠≠0 (not simply-connected) within our admissible class of simply-connected manifolds.
- —¢ χ < 0
Heuristic (model-dependent under R-minimization): Leads to inconsistent spectra (negative index, no chiral multiplets).
- —¢ χ > 24
Heuristic (model-dependent under R-minimization): Within our selection principle, larger $|\chi|$ increases complexity/instability; thus χ=24 is our default minimal positive choice in this class.
Step 3 —” Minimal Positive Solution
The minimal positive value that is allowed is
$$\chi = 24.$$
This corresponds to the K3 surface, the unique compact, Ricci-flat, K٤hler, simply connected surface in this class.
Speculative conjecture
Since χ is always a multiple of 24, we have χ/8 ∈ 3ℤ. For K3, χ/8 = 3 numerically matches the observed generation count, though this connection remains speculative.
Summary
- Topological constraint: χ ∈ 24ℤ (hard topology)
- Physical elimination: negative and large values excluded
- Minimal consistent choice: χ = 24
- This selects K3 within our stated admissible class
- Conjectural: χ/8 = 3 elegantly links to the number of generations