Complex Structure and Composition from Finite Real Source Histories: An Artian Frame-Cycle Construction

How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses the same imaginary unit for an individual system and for a composite. This paper constructs that compatibility from finite real histories. For the permutation P of a three-cycle, restricted to its zero-sum contrast sector, \\[\\boxed{J=\\frac{P-P^2}{\\sqrt{3}},\\qquad J^2=-I}\\] The complex structure is obtained after the real permutation is specified. At a qualified unmarked join, opposite reference shifts select \\[\\boxed{Q_{AB}=\\frac{I-J_A\\otimes J_B}{2}}\\] The normalized range of this projector is the balanced complex tensor product. The construction includes associative many-source composition, a nonfactorizable controlled contact, and exact bounds on departure from the selected joint sector. Why a three-cycle? For the group G of 24 proper signed-permutation frames, pair-order contacts generate the commutator subgroup. Its Abelianization is \\[\\frac{[G,G]}{[[G,G],[G,G]]}\\cong C_3.\\] Every nontrivial Abelian group response of that contact class therefore has a three-element image. The paper states the contact and response premises explicitly and distinguishes the image order from state count and representation multiplicity. Which boundary records may be removed? A reduction must preserve future readouts as well as the present one. For the reduction q, current output O, and every allowed operation W, the exact certificate is \\[O=\\overline O\\circ q,\\qquad q\\circ W=\\overline W\\circ q.\\] These relations make every finite future history descend to the reduced record. A material-marker example shows a distinction that is initially invisible and becomes readable after one operation. It also separates preservation of the full boundary quotient from preservation of a smaller contrast sector. Graph boundary counts and a sequence error bound extend the construction beyond a single join. Scientific scope and reproducibility The conditional selection, composition, and finite-record results are proved with their hypotheses stated. Physical realization requires an independently justified contact rule, accessible profile sector, and complete operation and record inventory. The finite construction does not itself establish Born statistics or unrestricted continuous dynamics. Standard representation theory, balanced tensor products, and finite-state minimization are credited separately from this source-history application. The reconstruction package contains the LaTeX source, five vector figures, a standalone record-certificate tool, complete finite fixtures, and 185 exact computational checks. The proofs introduce Quantum Traction Theory terminology as needed and can be followed without prior knowledge of the framework. Source anchors QTT Main Book, especially pp. 248-250, 352, and 827-829; Artian's Completed-Event Four-Capacity Theorem; QTT Lexicon and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.