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

An Aethus, in the companion framework, is a deductively closed partial assignment of answers to attributes together with the superposition structure over its unanswered attributes, which may be restricted rather than full; 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 Aethae. 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 Aethae, 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, (D), which the unrestricted intrinsic closure does not satisfy and which the companion proves over carved unions, so that the intrinsic layer is a semiring only under (D); the weighted layer’s monoid semiring M[U] is built on the multiplicative monoid alone and needs no such hypothesis. The extrinsic layer builds the weighted algebra T of measure-valued finite combinations over U, quotiented only by same-Aethus additive collapse and zero-term deletion. Its main result is that last-common-ancestor factorization is not a clause to be imposed but a derived identity: 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 weight valuations are characterized as congruence-respecting normalized semiring valuations, distinguished from empirical probability measures, with the real-weights base case discharged. And a fixed-point lemma identifies the postulate’s recursive content. With each attribute’s regime stated as content, a filling declaring nothing until a carving or an irreducibility marker is intersected with it, in models with finitely many attributes the doomed carriers are exactly the tagged ones and form a multiplicative ideal of the weighted monoid semiring, so that there the validity quotient is an exact semiring quotient. The selection of a distinguished valuation — the Born question — is deferred to the physics-face companion; it cannot be a semiring homomorphism, which sends every idempotent carrier to 0 or 1. The consistency rule runs over a declared descent, the splittings each Aethus admits, distinct from both the lattice order and weighted segmentation, and a finite model works its three cases through.