Difference posets and the histories approach to quantum theories

Ferdinand Chovanec, Františk Kôpka

Ferdinand Chovanec, Mária Jurečkova

A new probability model based on the theory of fuzzy sets is presented. In thismodel, a difference of comparable fuzzy sets is the primary operation. The idea ofa difference of fuzzy sets (fuzzy events) is simple: If we have two comparable eventsa and b ( a ≤ b ), then our knowledge on a and b entails the complete knowledge ofthe complement of a in b , i. e., b ⊖ a .The new algebraic structure of fuzzy sets is called a difference poset (a D-poset) offuzzy sets. Some properties of a lattice ordered D-poset of fuzzy sets (a D-lattice offuzzy sets) are studied. An MV-algebra of fuzzy sets (a Bold algebra) is characterizedin the D-poset of fuzzy sets set-up. The sufficient and necessary conditions for a Dlatticeof fuzzy sets to be a Bold algebra are given. The basic notions of the quantumlogic theory - a state and an observable are defined in D-posets of fuzzy sets.

Petr Sramek

This paper develops a finite-dimensional record-algebraic account of decoherence, erasure, boundary records, threshold recovery, islands, entropy ledgers, and modular flow on causal posets. Record creation correlates histories with marker degrees of freedom, and ignoring those markers implements a completely positive trace-preserving dephasing channel. In a fixed history basis the channel is a Schur product, rho -> G ∘ rho, where G is the Gram matrix of marker overlaps. The paper packages this mechanism in causal-poset language: history bases are defined from chains or admissible paths; observer record algebras define branch quotients; conditional eraser instruments restore off-diagonal terms in selected sectors; and boundary record algebras yield observer-indexed coherence kernels. The manuscript treats horizon, Page-time, island, replica, and modular-flow terminology as a conditional dictionary built on finite theorem statements plus explicit continuum or code-subspace assumptions. It includes a claim-proof ledger, a continuum-assumption register, and toy-model or experiment proposals including a two-history MARK/ERASE circuit, a tunable boundary-kernel model, a three-fragment threshold-recovery ledger, and a modular-covariance recovery test. The paper is part of the DAGI observer-theory/foundations track and is version DAGI-OBS-REC-001 v0.3.

Marcelo Losada, Roberto Laura

Marcelo Losada, Roberto Laura

NANJIE MA

The incomparability polarity of a finite poset P yields a complete ortholattice L(P) of polarity-closed sets. The map from P to L(P) is the STANDARD closed-set / polarity-closed ortholattice of an orthogonality space, with incomparability as orthogonality. It is not a new causal logic. What is new is the covering-specific image of that functor: Boolean on tree-orders, the non-modular non-orthomodular hexagon after a merge-induced zigzag, and no injective ortholattice embedding into the projection lattice of any Hilbert space when L(P) is not orthomodular. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. Covering birth is not Rideout-Sorkin sequential growth. Companion Paper 30 is a different exclusion theorem (sprinkling versus covering support and Rideout-Sorkin CSG); cite 10.5281/zenodo.23385450. This note does not derive quantum mechanics, Hilbert space, spacetime, or a unique sequential growth law. New child record. Never is-new-version-of the parent programme manuscript.