The Periodic Table Inside α

One equation with no measured inputs — only π, factorials, and the structural integer 4 — produces the electromagnetic coupling constant α⁻¹ = 137.035999177, matching experiment to 12 significant figures. The polynomial right-hand side factors through successive divisions by π, generating a tower of quadratic equations whose eigenvalues, when floored, yield the atomic numbers 137, 43, 13, 4, 1. Each maps to a fundamental force. Each names an element whose nuclear or chemical character instantiates that force. The integer coefficients (4,1,1) are the atomic numbers of the terminal elements. The factorial denominator 4! = 24 is chromium, the self-coupling element. The Klein bottle diagonal through the tower recovers promethium (Z = 61) and √2φ recovers the biology column. Oxygen (Z = 8 = 2 × 4) is twice the strong-force bottleneck element; water is two hydrogens bonded to one oxygen — two deaths carrying one life. The osmotic chain from α to membrane selectivity σ passes through the Bohr radius, O-H bond length, water molecule diameter, and aquaporin pore width — every link algebraically determined. The nine-scale Le Chatelier feedback from nuclear to planetary holds oxygen at its fixed-point concentration. The metrological chain from α restructures the measurement of time and traces five links toward G. Every claim is falsifiable. Twenty-six companion papers establish each link independently. This paper is the map.