Integers as Output: Deriving the Standard Model Vocabulary from Torus-Knot Topology (NWT Paper 14)

The companion Paper 13 reproduced eighty Standard Model observables from a single mass-scale anchor and fifteen topological integers, achieving a median residual of 0.1% against PDG values. This paper proves the integers were not fitted: every integer is the output of the torus-knot crossing algebra, forced by topology at each step of a deductive chain. Starting from the abelian Higgs Lagrangian at the BPS self-dual point, we show that the trefoil T(2,3) is forced by minimality, its crossings generate su(3) via braid closure, and the cinquefoil T(2,5) is the trefoil's own icosahedral completion: (+1)-Dehn surgery on the trefoil produces the Poincare homology sphere with fundamental group A_5, whose 5 irreducible representations fix q=5 without reference to the measured value of alpha. The seven-step Aharonov-Bohm construction yields 1/alpha = 25*pi*sqrt(3) + 1 = 137.035 at 7.6 ppm, with every factor traced to the trefoil. v2: Dehn surgery derivation of q=5 (two independent paths converge); explicit su(3) commutator closure in appendix; expanded bibliography (13 refs); statistical significance analysis.