The Fractal Kitaev Model K k , constructed in the companion paper [1] by decorating the hierarchical graph G k into a maximum-degree-3 tree, has an isotropic Majorana spectrum [2] that was previously accessed only by brute-force diagonal-ization, limited to k ≤ 6 by memory. This note establishes four results. First, an exact recursive factorization of the characteristic polynomial of an isolated depth-d gadget subtree, A d (x) = A d−1 (x) · R d (x) with deg R d = 10 · 6 d−1 , proved in full generality by induction and independently veried by exact-arithmetic polynomial division, together with an exact assembly formula for the full tree. Second, an eigenvalue-counting algorithm requiring only O(k) recursive arithmetic steps exponentially fewer than the |K k | ∼ 6 k dimension of the matrix itself built from the same recursive structure via the standard pivot-counting form of Sylvester's law of inertia, veried to exact agreement against brute-force spectra for k = 2,. .. , 6 and used here to reach k = 15. Third, extending that same pivot to complex argument and identifying it exactly as the reciprocal of a diagonal resolvent entry, we nd its real part numerically at −x/2 to ∼ 10 −7 precision on the real axis, and, at nite k = 15, a macroscopic counting-function jump at a value consistent with a root of R 1 , backed by the exact divisibilities R 1 | R 3 and R 1 | R 4 , though without yet a general sharing rule between generations or a proof that this persists as k → ∞. Fourth, resolving the evidential core of that open question: working entirely inside the exact quotient ring Z[x]/(R 1 (x)) which avoids the combinatorially infeasible full polynomials (deg R 6 = 77760, with coecients hundreds of digits long already at d = 4) we exactly verify R 1 | R d for every tested d = 3,. .. , 40, and that the exact power of R 1 dividing R d grows sharply: e(3) = 5, e(4) = 30, e(5) = 180, all exact (not extrapolated), matching a conjectured scaling law e(d) = 5 · 6 d−3 at three points but not proved beyond them. Since R 1 is irreducible over Q, its ten roots are Galois conjugates and so share, provably, the same exact multiplicity wherever R 1 divides turning the assembly formula, conditional on that scaling law and a second conjectured law for the full tree, into a quantitative prediction for the mass at the single root x ≈ 0.469: 1.3889% at k = 15, numerically close to the ≈ 1.4% found empirically but conditional on both conjectural exponent laws. Both scaling laws remain open beyond the exact points checked.