An Improved Upper Bound for the Length of Pierce Remainder Chains

Fix an integer n >= 2 and iterate a_(j+1) = n mod a_j from a positive starting value a_0 <= n until the first zero occurs. Let P(n) be the maximum number of positive terms over all starting values. This preprint presents a self-contained elementary argument for P(n) = O_epsilon(n^(3/10 + epsilon)) for every epsilon > 0. The exponent is smaller than 19/59 in the upper bound of Chase and Pandey (arXiv:2211.08374). The proof uses four consecutive positive remainders to obtain a positive integer curvature parameter and an exact elimination identity. In a long dyadic block, sufficiently many short windows have three small integer parameters. Once these parameters are fixed, a strictly increasing quadratic expression in the initial quotient must divide a fixed nonzero integer. A divisor count gives the central estimate at delta = 1/30; elementary outer-range estimates and an explicit finite-range argument complete the proof. The release includes an English manuscript, a methodological supplement, a deterministic exact-integer verifier, optional symbolic checks and a dated source-search record. The computations are finite consistency checks, not a substitute for the proof. The work received AI-assisted internal checking but has not undergone external peer review or formal proof-assistant verification. Absolute novelty and prize eligibility are not established.