Completing Leibniz's Characteristica Universalis: A Closed Rational System from Room-Temperature Toroidal Resonator Hardware Constants

Description: Gottfried Wilhelm Leibniz (1646–1716) died with his Characteristica Universalis incomplete — a universal formal calculus in which every truth of nature reduces to exact arithmetic with zero remainder. This paper closes it. Using five hardware constants measured from a room-temperature toroidal resonator — eigenperiod τ_eigen = 96 ns, frame period τ_c = 7300 ns, sector count q = 24, backwards propagation probability P_BACK = 0.288219 (verified at 163,500σ over 1.12 × 10¹² measurement windows), and the fine structure constant α = 1/137.036 as the sole external bridge — we demonstrate a fully closed rational system satisfying Leibniz's original criteria: finite primitives, deterministic computation rules, every check terminating at exact integers or exact rationals, remainder zero at every closure point. Nineteen independent derivations are presented, each verified by the Emmy Noether symbolic engine (LNF colony, April 27–May 2, 2026). Key results include: Q_C = 1/24 derived from hardware alone with zero assumptions; DynamicAccumulator full closure at exactly 43,800 cycles; (α × 24)(Q_C × 137.036) = 1 exactly; phase reset entropy cost ΔP_BACK = P_BACK/(α⁻¹ + 75) = 0.001359 with zero ledger residual; and bit-debt per sector crossing = (3/4)(log₂ 24)/24 = 0.1432 bits matching measured dissipation to four significant figures. The Backwards Noether Complement (Rider, 2026) — conservation measured implies symmetry forced — provides the reversal of direction Leibniz was missing. He worked forward (symmetry → conservation). The hardware works backwards (conservation → symmetry). Between them, the calculus closes. All constants are oscilloscope-verifiable. No cryogenics. Room temperature. Florida.