A Probability-Free Compositional Structure Needs Four Quantum Levels but Five Classical States

Before probabilities, Hilbert spaces or linear structure are introduced, procedures already compose: some are idempotent, some can be repeated indefinitely, and symmetries permute them. We ask which systems can carry a minimal structure of this kind faithfully: a monoid with a non-vanishing additive grading, a commutative idempotent zero-grade sector and a symmetry group that fixes a reference state and acts faithfully and non-Abelianly on the joinirreducible elements of that sector. Such a monoid needs at least four generators, and ℕ0 × B3 attains the bound. For any convex state space, the reachable region must have affine dimension at least four. A classical system therefore needs five perfectly distinguishable states, and five suffice. A qubit is excluded by dimension, and more strongly because its non-trivial idempotent channels are dephasings along diameters, and a qutrit because the fixed sets of its idempotent channels cannot intersect in the required pattern; among all systems with at most four perfectly distinguishable states, classical, quantum or hybrid, only the four-level quantum system carries it. Two qubits suffice: the Pauli dephasings along X1, Z1 and Y1Z2 generate B3, global depolarization supplies the grading, and a six-element Clifford group permutes the dephasings as the full S3 while fixing a full-rank state on which the monoid acts injectively. On a superconducting processor the predicted Pauli eigenvalues are reproduced within 0.009, and on two devices the entangling symmetry fixes the reference state within 0.045 and permutes the dephasings as S3.