Research & Formal Writing

TreeKernelSum: Fast Approximate All-Pairs Kernel Summation on Weighted Trees

Addresses the all-pairs kernel summation problem on weighted trees — computing, for every node, a kernel-weighted sum over all other nodes. Using centroid decomposition and a sum-of-exponentials (SOE) approximation, the algorithm runs in O(r n log n) time, achieving a 972× speedup over the naïve O(n²) baseline at n = 50,000 nodes. Supports Gaussian, Ornstein-Uhlenbeck, Matérn, and inverse multiquadric kernels. Validated on real phylogenetic datasets spanning 5,000+ mammalian species, achieving R² = 0.955 in Nadaraya-Watson body mass prediction. Scales to 1 million nodes in approximately 12 seconds.

Longest Increasing Subsequences of Collatz Trajectories: A Null Model Decomposition

Uses the longest increasing subsequence (LIS) as a pseudorandomness metric for Collatz trajectories. The mean LIS-to-trajectory ratio is 0.567 for starting values up to 10⁶ — substantially below the 0.812 ratio observed in random sequences of comparable length. Value shuffling restores the ratio to ~0.813, isolating temporal ordering as the driver of the deviation. A hierarchical null model analysis attributes roughly one-third of the disparity to the 31.6% up-step fraction, with the descending suffix responsible for a further reduction. The remaining unexplained variance of ~0.05 persists across generalized 3n+b variants, pointing to deeper structural geometry in Collatz dynamics.

Unified p-Adic Bounds and Computation for the Erdős–Moser Equation

A Mahler-Interpolation Framework and Exhaustive Computational Survey

Introduces a closed-form p-adic valuation formula for odd-prime moduli and a Mahler-interpolation lemma for even exponents, together forming a unified framework for the p-adic valuations of power sums S_k(m). Computational verification finds no counterexamples for odd primes 3 ≤ p < 200 with even exponents 2 ≤ k ≤ 20. An exhaustive survey of the Erdős–Moser ratio over 2 ≤ m ≤ 2000 and 2 ≤ k ≤ 12 identifies no solutions beyond the classical case (m, k) = (3, 1). Includes two Python validation scripts and a full PDF manuscript.