The Klein Bottle Eigenvalue: Technetium, Parity Violation, and the Tower Inside α

The fine structure constant identity α⁻¹ + Sα = 4π³ + π² + π factors as π(4π² + π + 1). The inner polynomial 4π² + π + 1 ≈ 43.620 serves as the right-hand side of a second-level equation with the same structure: β⁻¹ + Sβ = 4π² + π + 1, yielding β⁻¹ = 43.619 053. Dividing again by π produces a third level: γ⁻¹ + Sγ = 4π + 1 + 1/π, yielding γ⁻¹ = 13.882. The identity contains a tower. Each level is the previous level divided by π, exactly. The eigenvalue ratios differ from π by the overshoot constant S/RHS², which grows by π² per level. The tower has at most 5 real levels before the discriminant goes negative. The floor of the Level 1 eigenvalue is 43 — the atomic number of technetium, the lightest element with…