Planar Disk Filters and Real Chromatic Roots at Beraha Numbers
We construct simple plane triangulations whose chromatic polynomials have real roots accumulating at every noninteger Beraha number $B_\ell=4\cos^2(π/\ell)$; the integer values $0,1,2,3$ are roots of $K_4$. The construction rests on the observation that a triangulated disk with four boundary vertices acts, inside a larger triangulation, as a transfer operator lying in a commutative three-parameter algebra. Rotating such disks by a quarter turn and composing them in series moves their eigenvalue triples around this algebra in a controllable way, and for every fixed $q\in(B_{10},4)$ one can manufacture a finite disk whose two competing local eigenvalues are strictly dominated by the third -- a physical filter. Closing a grid of such disks between two poles produces a sphere triangulation, and a fused Temperley-Lieb trace identity accounts for its chromatic polynomial completely, sector by sector, with no appeal to semisimplicity at the root of unity itself -- generic semisimplicity enters only through polynomial identities continued from $δ>2$ -- and no singular higher Jones-Wenzl projectors. In the one-cap sector a Krein-type argument shows that every sweep word has exactly one expanding eigenvalue; Gram-form inertia controls every other sector uniformly in the width. At odd Beraha indices the trace coefficient of the expanding eigenvalue vanishes; at even indices two copies of that eigenvalue cancel between adjacent sectors, and a short real-variable lemma converts either situation into genuine real roots converging to the target. Indices below eleven are treated by elementary families, exact outward-rounded integer certificates, and one explicit disk at $B_{10}$ combined with Tutte's golden identity. All graphs are ordinary, simple, unweighted plane triangulations.