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 no stable isotopes. Technetium's instability is entirely weak-force mediated: every isotope decays by β± or electron capture. The weak force is the only fundamental interaction that violates parity. Parity violation is the physical signature of non-orientability. The Level 1 surface — the Klein bottle, formed by closing the Möbius strip's boundary — is non-orientable and requires embedding in four dimensions. The chain: Möbius (half-twist, Level 0) → electromagnetic coupling → α. Klein bottle (non-orientable closure, Level 1) → weak-force shadow → β⁻¹ ≈ 43.6. The tower was always inside the polynomial. The periodic table was always inside α.
Jay Andrew Carpenter (Sun,) studied this question.