Sharp Stability Thresholds and Fixed-Energy Laws for Colored Interlacing Triangles

Abstract

In a depth-two colored interlacing triangle, each color occurs twice on top and once below, in top-bottom-top order. Earlier solutions by Shivam Patel establish eventual coefficient polynomiality and linear cumulants for the model introduced by Blitvić and Petrov. We determine sharp uniform thresholds and inversion-marked refinements for arbitrary bottom row. At fixed coefficient index k, the normalized coefficient stabilizes as a polynomial in n for n at least 2k+1, with leading coefficient 5^k/k! and integral coefficients after multiplication by k!. For every positive k the range is sharp: the discrepancy at n=2k is exactly 3^(k-1). The inversion-marked leading coefficient is (1+4u)^k/k!, yielding a Binomial(k,4/5) limiting inversion law. We also obtain the spatial uniform order-statistic law of unit-energy blocks and fixed-index eventual strict log-concavity. For identity bottom row the sharp stability range improves to n at least k+1. Prior work and revised scope. This independently developed article now credits Patel's earlier complete polynomiality solution and linear-cumulant solution, publicly posted on 20 August 2026. The broad finite-energy, ordered-block and formal-pole mechanism is shared. This revision removes the repeated low-degree derivation and retains the all-order sharp thresholds, exact boundary discrepancies and marked limit laws. These results refine the earlier solutions of Conjectures 4.3, 4.4 and 4.9 in Colored Interlacing Triangles and Genocchi Medians (SIGMA 22 (2026), 081). The global log-concavity question, Conjecture 4.1, is outside this scope. This corrected preprint contains complete mathematical proofs. The results have not been formalized in a proof assistant.