We show that the flat-connection equation FA = 0 on the complement of a knot K ⊂ S3, combined with the AJ conjecture of Garoufalidis relating the A-polynomial AK(L, M) to the colored Jones polynomial Jn(K; q), gives rise to a one-dimensional q-difference operator whose classical limit reduces to the Klein–Gordon equation (□ + mK2)Φ = 0. Identifying the mass eigenvalue mK2 as a topological invariant of K, and taking the non-relativistic limit, we obtain the standard time-independent Schrödinger equation ĤΨ = EΨ. The mass eigenvalue is computable from the roots of AK(L, M) on the unit torus. For the trefoil 31 calibrated to the electron rest mass, the resulting spectrum of the next four prime knots (41, 51, 52, 61) agrees with leptonic and meson mass ratios at the order of magnitude. Main results: Five-step derivation of the Schrödinger equation from topological data: FA = 0 → AK → mK → Klein–Gordon → Schrödinger. Theorem 1: the non-relativistic limit of the Klein–Gordon equation derived from the AJ conjecture yields the time-independent Schrödinger equation. Explicit mass spectrum for 7 prime knots (01, 31, 41, 51, 52, 61, 62), with the unknot having zero mass and successively more complex knots having strictly larger eigenvalues. The construction does not assume the Schrödinger equation: it derives it as a corollary of well-known properties of SU(2) Chern–Simons gauge theory and the AJ conjecture. The AJ conjecture is verified explicitly for the 19 prime knots up to crossing number 10 in the accompanying open-source implementation. Falsifiable predictions: The mass spectrum of elementary particles is discrete and indexed by knot topology (no continuous mass families). Masses are monotonically increasing in crossing number for prime knots: m(31) < m(41) < m(51) < … Massive excitations require non-trivial topology; the unknot mass eigenvalue is exactly zero. Natural mass cutoff around c ≈ 30 where eigenvalues approach the Planck scale. Companion code reproducing the AJ conjecture verification and the mass spectrum is available at github.com/CA9-sas/TQNT (scripts: theory/a_polynomial.py, theory/chern_simons.py). The wider physical framework is documented in the V5 monograph (doi:10.5281/zenodo.19131538).