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

An information state, in the companion framework, is a deductively closed partial assignment of answers to attributes; unanswered attributes are read as superposition, and inconsistent structure is tagged invalid by a consistency-pruning rule — a tag on a standing element, never a deletion. The companion on counterfactuals develops the modal face of these states. This paper develops the algebraic one: the exact object they form. The construction has two layers. The intrinsic layer takes the universe U of unweighted states, closed under parentage and joint assertion, and shows it to be a commutative semiring, with informational union as addition, joint assertion as multiplication, and distributivity stated as an explicit hypothesis. The extrinsic layer builds the weighted algebra T of measure-valued finite combinations over U, quotiented only by same-state additive collapse and zero-term deletion. Its main result is that last-common-ancestor factorization, imposed as a clause in earlier formulations, is derived: it is an identity of the bilinear multiplication, the carrier being the monoid semiring M[U], and it is through this identity that U’s parent–child graph re-enters, every weighted sum admitting the factored form. On this object the proper-child and reduced-form apparatus is defined and the consistency postulate is stated as pruning that bottoms out at the additive identity. Three further results follow. The semiring-not-ring constraint splits by layer, forced by idempotency at the unweighted layer and carried by one guard axiom, zerosumfreeness of the measure type, at the weighted one. Admissible probability assignments are characterized as congruence-respecting semiring homomorphisms, with the real-weights base case discharged. And a fixed-point lemma identifies the postulate’s recursive content. The selection of a distinguished homomorphism — the Born question — is deferred to the physics-face companion.