PROJECT T VOLUME XVIII : The Derived Propagator Double-Slit Interference from Algebraic Structure Alone

This volume establishes that the quantum propagator and double-slit interference pattern are derived consequences of the algebraic structure of H0(A3), not independent postulates. Starting from the presentation , we construct the transfer operator T = La + Lb + Lc as the adjacency operator of the Cayley graph, show that its spectrum {0, 1, 2, 3} with degeneracies (1, 11, 11, 1) is forced by idempotence, and derive the propagator K(d, s; t) = ⟨d|exp(−iTt)|s⟩ via spectral decomposition. Explicit numerical computation on the 24-element monoid verifies quantum interference: for certain detectors, the coherent probability |A1 + A2|² differs from the incoherent sum |A1|² + |A2|². We show that the commutation relation ac = ca in A3 suppresses interference at 20 of 24 detectors, while the fully braided algebra H0(Ã2) with C(3) = 0 produces maximal fringes. Measurement is not postulated but derived: the idempotent projection Ls² = Ls destroys coherence between paths, yielding the Born rule as a theorem. Keywords: 0-Hecke monoid, derived propagator, double-slit interference, transfer operator, idempotent projection, commutation deficit, Cayley graph Laplacian