Product-Integral Invariants of Exponent Sets: Subset-Sum Algebra, Stability Bounds, and a Prime-Indexed Limit

This preprint introduces a product-integral invariant of finite and directed exponent sets, F_S = ∫_0^1 ∏n ∈ S(1 - u^n) du. The paper develops its exact subset-sum expansion, monotonicity and stability bounds, signed-cube interpretation, and directed-limit theory. It proves closed forms for the arithmetic-progression limits FdN using Euler's pentagonal theorem and the Mittag-Leffler cosecant expansion, including the eta-integral consequence ∫_0^∞ η(ix) dx = 2π/√3. It also studies the prime-indexed directed limit, giving a formally certified rational sandwich computation for its decimal expansion.