DHR Superselection Structure from the MEG Axioms and the Completion of the Born Rule Dissolution

The four axioms of Modular Entropic Gravity (MEG) implicitly contain the Doplicher–Haag–Roberts (DHR) superselection structure of algebraic quantum field theory. This paper establishes the result through a four-link chain: Axiom 1 (modular Hamiltonians of local regions) commits the theory to the Haag–Kastler framework; Axiom 3 (Bisognano–Wichmann and the PLES functional) provides a mass gap with exponential clustering; Axiom 2 (JLMS) restricts physical states to a class satisfying the Buchholz–Fredenhagen exponential localisation criterion, as proved for an explicitly defined admissible state class; and the DHR theorem then implies superselection sectors labelled by representations of a compact group, with block decomposition of regional operator algebras and block-diagonal physical states.

Part I establishes the DHR chain as a standalone result beyond gauge theory. The four links proceed from Axiom 1 through the Haag–Kastler framework, the mass gap and Fredenhagen exponential clustering from Axiom 3, the exponential localisation of JLMS-admissible states from Axiom 2, to the DHR/BF superselection theorem. Each link's hypotheses are necessitated by the axioms rather than merely consistent with them.

Part II completes the Born rule dissolution established in the companion paper on convex field representations (Zenodo DOI: 10.5281/zenodo.19600153). That paper demonstrated that within a single superselection sector, the Born rule is a structural feature of the convex entropy field representation rather than a separate measurement postulate. What it left open were three questions: why sectors exist, why cross-sector superpositions are forbidden, and why cross-sector branches are automatically distinguishable.

The DHR superselection structure answers all three from the axioms. Sectors exist because the DHR theorem reconstructs them from the Haag–Kastler framework with mass gap and exponential localisation. Cross-sector superpositions are forbidden because JLMS-admissible states are block-diagonal — a theorem of the DHR framework, not a postulate. Cross-sector branches are automatically distinguishable because the block-diagonal constraint forces vanishing cross-sector matrix elements of local observables, guaranteeing that the readout faithfulness condition is satisfied without any requirement on the measurement apparatus. A qualitative argument establishes that sectors with distinct Casimir eigenvalues must have distinct entropy fields S_π ≠ S_π'; a quantitative lower bound on the H¹ separation is identified as an open step.

The Born rule is furthermore shown to be the unique probability rule compatible with the entropy-field representation. Within a superselection sector, the combination of Axiom 1's linearity (the entropy field respects convex mixtures: S_ρ = λS₁ + (1−λ)S₂) and Axiom 4's restriction of observables to readouts of S(x) forces the measurement map to be affine on the probability simplex. Any affine map on the simplex that fixes the pure-state vertices and respects permutation symmetry is necessarily the identity, uniquely selecting the Born rule P(i) = |αᵢ|². The key insight is that Axiom 4's restriction of observables to S(x) is what forces the affine property: if additional degrees of freedom beyond S(x) were accessible to the measurement apparatus, the affine condition would not hold. This result is the MEG analogue of Gleason's theorem, with the logical structure parallel but the inputs different — Gleason assumes Hilbert space structure, while the present result derives uniqueness from the entropy-field representation.

The synthesis gives the complete Born rule and superselection structure from the MEG axioms: the sector decomposition from DHR, the Born rule within each sector from convex dissolution, the uniqueness of the Born rule from affine readout, and the absence of cross-sector interference from the block-diagonal constraint. No independent probability postulate is required, and no alternative probability rule is consistent with the axioms.

Changes in v2: Added Section 7.3 establishing the uniqueness of the Born rule through the affine readout property forced by Axioms 1 and 4. Updated abstract, key result box, logical chain, and conclusion to incorporate the uniqueness result. Removed an incomplete bibliography entry.