The Operator-Algebraic Bridge Principle in MEG: Z3 Sector Indistinguishability as a Theorem of the MEG Axioms

The derivation of U = exp(iS) in the MEG programme rests on a coarse-grained vacuum-sector indistinguishability principle: within a coherence volume of size l0, configurations differing only by passage between degenerate Z3-related vacuum sectors are indistinguishable to the effective theory. In the paper deriving U = exp(iS), this principle was stated as a physically well-motivated bridge, with its operator-algebraic derivation identified as the one remaining open step. This paper closes that step by deriving the principle from the MEG axioms alone.

The derivation uses five ingredients, all internal to the MEG framework. The Bogomolny bound establishes that the three Z3 vacuum sectors have equal PLES cost. The vacuum distinguishability kernel is the Hessian of relative entropy, and the PLES functional is its local quasi-static reduction — so equal PLES cost means equal relative entropy with respect to the reference vacuum. The JLMS identity (Axiom 2) then gives equal modular Hamiltonian expectation values across the three sectors, using only differences as JLMS provides, with no absolute identification required. Axiom 1 identifies the entropy field S(x) as the smeared modular Hamiltonian expectation value, so equal expectation values give identical entropy fields across the three sectors. Axiom 3 identifies coarse-grained observables with PLES functionals of S(x), so no observable can distinguish sectors that produce identical S(x).

No external quantum-algebraic machinery is required. In particular, the Tomita-Takesaki theorem is not used as a foundation. The MEG derivation arrives at the same conclusion by a more direct route: from the concrete physical content of the axioms rather than from abstract operator algebra. The Tomita-Takesaki theorem, where applicable, is corroborated by the MEG derivation rather than the other way around.

This reflects the general relationship between MEG and quantum mechanics. MEG is the more fundamental framework, and standard quantum-algebraic results should emerge from it in the appropriate regime rather than being imported as foundations. The bridge principle provides a concrete example: the same physical conclusion that Tomita-Takesaki gives via operator algebra follows in MEG from the axioms directly, without operator-algebraic input. Where the two agree — as they do here — Tomita-Takesaki is a consistency check. Where they might differ, MEG takes precedence.

The bridge principle is therefore an exact theorem of the MEG axioms. With it established, the complete chain from the MEG vacuum structure to U = exp(iS) and the Z3 cosine potential V(S) = A(1 - cos(3S)) has no remaining open steps. Every arrow in the derivation is now an established result.