Minima Scaling of Prime Palindromic Recurrences near Exceptional Points

We investigate four linear homogeneous recurrences with characteristic polynomials χ1 = (x + 1)^2, χ2 = (x−1)(x+1)^2, χ3 = (x+1)^3, and χ4 = (x−1)(x+1)^3, conjectured to be the only ones whose modular palindromic weight satisfies W_m = 1 if and only if m is prime. Physically, their companion matrices are defective transfer matrices carrying a Jordan block of size N ∈ {2, 3} at the degenerate root λ = −1, corresponding to exceptional points of order N (EP_N ). At that unperturbed point (γ = 0), the probability distribution P (n) that a random initialized sequence has its absolute minimum at position n exhibits heavy-tailed algebraic decay (n−2) for all four recurrences, admitting an exact closed form for χ1. Under a coefficient perturbation of strength γ ≠ 0, the Jordan block unfolds with fractional Puiseux scaling ∝ γ^(1/N) . For N = 2 this separates a hyperbolic phase with localized minima (whose tail mass numerically collapses onto a single curve) from an elliptic phase with oscillating solutions and delocalized minima. For N = 3 a complex root pair is always present, and the tail mass decays in steps whose positions are set by the rotation angle φ of this pair.