Improved interval bounds for exceptional abc triples via full six-layer geometry

Abstract

GPT 5.6 Sol found the result and proof presented in this paper. We establish improved upper bounds for the number of exceptional triples associated with the abc conjecture across several explicit ranges of the exponent parameter. The argument refines an existing analytic reduction by retaining more information about the radical constraint and using the full collection of six-layer geometry-of-numbers inequalities. A computer-assisted step verifies eight finite optimization bounds through exact rational certificates, without floating-point arithmetic or reliance on a numerical optimizer. Monotonicity then extends these pointwise bounds to interval results. These estimates improve previously available bounds but do not prove the abc conjecture or establish that the exceptional set is finite.