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.
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.
Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable. Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ and a binary operator d(x,y) that are generalizations of the binary relation C and the commutator c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ and ↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ-blocks with respect to Boolean subalgebras and distributive subposets they include.
We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence “classical”) substructures of the overall quantum logic. 1
This is Paper 142 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops a finite mechanical account of quantum entanglement within pq-spacetime. Entanglement is identified with the persistence of a non-factorable family of admissible future histories. Two measurement regions can therefore appear as separate local systems in emergent spacetime while still remaining projections of one joint finite closure problem. Bell inequality violation follows because the joint outcomes are not generated by two independent local completion families. No-signalling is nevertheless preserved because complete changes to a distant measurement boundary do not alter the local marginal outcome weights. The paper establishes the Entanglement Closure Theorem and distinguishes microscopic non-factorability from observable entanglement after coarse-graining into stable measurement records. It also gives an explicit minimal singlet construction using two opposed history fibers carrying the relevant four-phase weights. The resulting finite ledger is antisymmetric, nonseparable, and exactly balanced at each local wing. Complete changes of basis on the distant side preserve those local marginals. Using the previously derived Hilbert-space and beta-Dirac bridges of pq-spacetime, the paper recovers the familiar spinor overlap law, the quantum singlet correlation, the maximal Bell-CHSH value, monogamy of maximal entanglement, and decoherence through environmental boundary enlargement. The analysis also explains why no preferred measurement ordering is required and why entanglement remains compatible with relativistic causality. The resulting picture is direct. Entangled systems do not exchange superluminal messages, carry classical answer sheets, or require a force acting across space. Their local outcomes arise from one shared finite admissibility structure whose joint correlations are nonlocal while its controllable local statistics remain causal. Entanglement is therefore physically real, nonclassical, and nonlocal in correlation, but not a signalling mechanism.
This paper is Paper 002 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops Binary Mechanics as the local dynamical layer generated by the eight-form kernel derived from the unresolved axiom-space construction of Anaxiomatic Mechanics. The paper shows how the no-fence and no-knob constraints force a single-bit admissibility criterion for elementary local continuations. Under this criterion, the localized eight-form kernel closes uniquely into a four-class cyclic structure represented by Z4. This Z4 residue is not introduced as a Hilbert-space postulate or as an additional phase register. Instead, it is the bookkeeping of admissible contrast traversal. The central ontology of Binary Mechanics is a finite resolved polarized graph whose vertices are pq-bits, or primitive quantum bits. A pq-bit is not a Hilbert-space qubit; it is a pre-Hilbert local contrast unit prior to amplitudes, basis choice, normalization, tensor products, and measurement structure. Realized exterior graph states contain no hidden local phase labels. Phase enters only through admissible finite histories generated from a resolved graph. The paper also clarifies local polarity-resolution. Elementary substitutions may temporarily expose raw local graph presentations, but the exterior ontology and successor fan live in resolved graph space. Adjacent same-polarity pq-bits coalesce into a single resolved pq-bit, so Binary Mechanics is not unconstrained graph growth. The finite amplitude law sums over admissible finite histories of resolved successors, with the Z4 residue induced by the underlying substitution structure. The result is a finite graph-history amplitude calculus: definite resolved graph states, admissible one-bit local updates, finite future-history trees, and Z4-weighted amplitude sums over coarse history classes. The paper does not yet assume Hilbert space, the Born rule, particles, fields, continuum spacetime, or general relativity. Those are left as later reconstruction problems. Its purpose is to establish the substrate-level mechanics on which the later pq-Spacetime construction is built.
The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved. Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]