Purely Combinatorial Quantum Mechanics: Working at Finite Depth First

We present a purely combinatorial approach to quantum mechanics using the Tree ofContinua. Every expression is written explicitly at finite depth D, using only algebraicnumbers—roots of unity, finite sums, and rational combinations. No transcendentalfunctions (sin, cos, exp) appear as primitives; no π appears as a number. All ana-lytic shorthand is expanded into its finite-depth combinatorial meaning. At depth D,everything is finite, exact, and algebraic. The IPG reading at ∞ recovers the exactquantum values in the continuum limit, but is never computed directly—we compute atfinite D and take D large enough for any desired precision. This paper is exploratory:it demonstrates the feasibility of the combinatorial approach and lays groundwork fora systematic combinatorial construction of Hilbert space and quantum observables.