What the Algebra Decides: Branches, Records, and Scalars (research note, v0.7)

A conjectural research note in the Algebra of Agency program, revised after independent mathematical review whose findings are recorded in the text. The framework takes two structural commitments — the dynamics of a local algebra is the modular flow of its faithful state, and subsystems compose by an admissible join — and the note argues that these decide three things the interpretations of quantum mechanics ordinarily stipulate. (A) Fine-grained Everett branches are not objects of any non-type-I local algebra; branch structure is conjectured to be relative to admissible decompositions, whose admissible objects are abelian or symmetry fixed-point subalgebras, not spatial tensor factors. (B) The abelian pointer algebra of a measurement is admissible exactly when the state is decohered in the pointer basis, and a record is relaxation onto a gapped generator’s fixed-point manifold; a thermostat and an error-corrected quantum memory pass the setpoint test and fail autopoiesis and transfer, and the note says so. (C) In finite dimensions, real and quaternionic matrix algebras with a non-tracial faithful state admit no orthogonal flow generated by their modular data that acts by automorphisms, while complex ones do, with a central complex structure unique up to sign (proved); a type-independent lemma shows the complexification’s modular flow never preserves a real algebra unless trivial; the type III case is left open. Composition over the real tensor product yields ℂ⊕ℂ, and factoriality of the join selects a shared complex structure (a reformulation, so marked). A tautological formulation and a misidentification with the Renou et al. real-quantum-theory experiments in an earlier draft are withdrawn explicitly. The resulting position: no collapse, no hidden variables, no preferred frame, no fundamental branch structure. Finite check in realscalars.py, computational supplement v1.1.0.