The Measure That Was Never Built: Feynman's Path Integral, Priced

The most famous integral in quantum mechanics integrates against a measure that does not exist. Feynman's 1948 space-time formulation writes the amplitude for a particle to travel between two points as a sum over all paths, made concrete by cutting time into N slices and multiplying one factor of 1/A into every slice, with A = (2πiħε/m)^{1/2} for slice width ε. No measure on the space of paths is constructed in the paper — the sliced product is the definition — and the one concrete constant in the construction, A, is fixed by demanding that the construction reproduce the free-particle propagator it was built to reproduce. This paper prices that joint, and only that joint. The count is one assumed object with zero derivations. The normalization diverges as the slicing refines: at N = 100 slices the prefactor already stands at log₁₀|(1/A)^N| = +60.09, and at N = 1000 at +1100.91, a divergence cancelled only by oscillation, in a limit no measure controls. Cameron proved in 1960 that the missing object cannot be supplied: a countably additive complex measure of this form would carry infinite total variation. The measure exists in exactly one place — imaginary time, where the phase is absent and Wiener's construction gives a probability measure of total mass exactly 1. The formalism's truth is not graded here; a century of use confirms it as an instrument. What is priced is the derivation deficit at its load-bearing joint: a postulate that works is still a postulate. Grade: ONE ASSUMED OBJECT, ZERO DERIVATIONS — PRICED, STILL TURNING. This paper stands outside my work on the vacuum catastrophe; it draws nothing from that work and repairs nothing in it, and it touches no claim of that program — quantum gravity stands not solved in my books, and nothing here moves it.