The Weight Algebra of Information States: A Two-Layer Commutative Semiring for Superposition and Pruning

An information state, in the framework of the companion papers, is a deductively closed partial assignment of answers to attributes, with unanswered attributes read as superposition and inconsistent structure tagged invalid by a consistency-pruning rule — tags on standing elements, not deletions. The companion extraction on counterfactuals develops the modal face of these states; this paper develops the complementary algebra face: the exact algebraic object they form. The construction has two layers. The intrinsic layer builds a universe set U of unweighted states, closed under parentage and joint assertion, as a commutative semiring — addition the informational union, multiplication the joint assertion, with distributivity an explicit hypothesis rather than a declaration. The extrinsic layer builds the weighted algebra T of measure-valued finite combinations over U, quotiented by an equivalence relation — its ancestor decomposition given coarsest-solution semantics, existence and the congruence obligation booked under the distributivity hypothesis — that re-imports U ’s parent–child graph, so weighted sums always reduce through last-common-ancestor factorization. On this object the proper-child and reduced-form apparatus is defined and the consistency postulate stated as pruning bottoming out at the additive identity. The semiring-not-ring constraint splits by layer: forced outright at the unweighted layer by idempotency, and one guard axiom — zerosumfreeness of the measure type — at the weighted layer. Admissible probability assignments are characterized as congruence-respecting semiring homomorphisms, a fixed-point lemma identifies the postulate’s recursive content, and the selection of a distinguished homomorphism — the Born question — is deferred to the physics-face companion.