Finite Operator-Algebraic Representation of Phase-Marked SF/SOO Relations and Tomita–Takesaki Applicability in Entanglement-Algebraic Spacetime

This paper is archived as a speculative research work. We construct a mark-preserving operator-algebraic representation of a phase-marked second-order relational structure arising in the scalar-field/second-order-operation (SF/SOO) framework. The primitive data are not assumed to be globally deterministic. Each phase relation is represented by its orthogonal graph projection, together with designated source, range, phase, point-report, primitive/adjoint, and direct/composite marks. Raw internal association and stiffness operators are retained in the exact relations, while only their bounded transforms and the marked projections are used to generate the tracial completion. On specified deterministic compressions, and globally only for a cyclically compatible family, phase steps admit polar decompositions separating partial-isometric transport from positive deformation. The resulting SOO-generated von Neumann algebra is finite, while the full unlimited ambient matrix hull is a type-II_1 factor. A relation-frame density, obtained as the sum of the three graph projections, survives the trace-2-norm quotient by a uniform normalized-rank estimate and defines a normal state faithful on its support. Every faithful normal semifinite weight on the finite completion has inner Tomita–Takesaki modular flow and therefore trivial outer modular class. Actual represented automorphisms and relational coupling-support corner isomorphisms satisfy exact Connes-cocycle covariance. Three independent countermodels establish separate selection obstructions: positivity and phase advancement do not uniquely determine a transfer; under an explicit mark-preservation premise, the stated phase and normalization conditions do not uniquely determine a transfer or its normalized Perron–Frobenius state; and a fixed phase-marked graph correspondence does not select between the Toeplitz–Pimsner and full Cuntz–Pimsner covariance quotients. These countermodels do not by themselves exhibit one frozen SF/SOO datum carrying two complete transfer-state-completion triples. No change of density or weight inside the finite completion can produce type III. The result establishes finite operator-algebraic representability of the phase-marked datum and applicability of Tomita–Takesaki theory. It does not identify SOO transport with modular flow, interpret the modular parameter as physical time, or assert that the actual SOO algebra is a factor.