For consecutive integers (a, a+1), the SL(2,R) transfer matrix with eigenvalues a/(a+1) and (a+1)/a has trace k = 2 + 1/L where L = a(a+1) is the pronic product. The sequence of pronic levels forms a trace ladder converging to the parabolic boundary k = 2. These eigenvalues are algebraically identical to the superparticular ratios of the harmonic series: the octave (2/1), perfect fifth (3/2), perfect fourth (4/3), and so on. A dimensional scaling law links the ladder to spatial embedding: for a one-dimensional chain in D spatial dimensions, degree-of-freedom counting yields the modulation depth µ = 1/(D+1), the matrix size √ D+1 × √ D+1, and the master identity k2 − Θ2 = D+1. At D = 3: µ = 1/4 (measured 0.245 in the protein backbone, 2% off), SL(2,R) (2 ×2), k = 2 (parabolic boundary), and the silver mean σ2 = 1 + √ 2 yields the mutation enrichment OR = σ2 − 1 = √ 2 (measured 1.416, 0.15% off). These matches are post-hoc observations that constrain the model, not confirmations. Two substrates are tested. The protein backbone clusters at pronic level L = 42 (eigenvalues 6/7 and 7/6; p = 0.003 against null). The exoplanet orbital-spacing chain clusters at L = 90 (eigenvalues 9/10 and 10/9; p < 10−4 ). The FM channel is independently confirmed in both substrates by shuffle tests (p < 10−4 ). The excess coverage ratio between substrates is 9.4/6.7 = 1.40 ≈ √ 2 (0.8% off). The ladder is shared; the rung is substrate-specific.