Operator Theory from Three Primitives

We develop operator theory entirely from the three primitives — same, different,opposite — working inside the Tree of Continua C. At finite depth d, a matrix is alabeling of ordered pairs of cylinder sets with values in the cyclotomic field Q(ωN ), whereN = (k + 1)d . Every operation of matrix algebra — addition, multiplication, adjoint,trace, determinant, commutator, eigenvalue — is finite arithmetic on periodic orbits in C,requiring no analysis, no limits, and no axioms beyond the three primitives.The special classes of matrices — self-adjoint, unitary, projection — correspondrespectively to observables, dynamics, and measurement; each is forced by a specificprimitive. Self-adjointness is forced by real-valued labelings (same/different). Unitarity isforced by preservation of the counting measure (same/different). The adjoint operation isforced by the chiral involution φ(s) = −s (opposite) combined with index transposition(same/different). The commutator measures non-commutativity — the difference betweentwo orderings of the same pair of maps — and the imaginary unit in the canonicalcommutation relation X, P = iħI is chirality: the opposite primitive.The IPG reading at ∞ of the compatible family of finite-depth matrix algebras givesthe full operator theory on Hilbert space. All of operator theory is finite arithmetic onlabelings of pairs of cylinder sets, taken to its IPG limit. No axioms are imported. Nostructures are assumed. Three primitives suffice.