Knowledge representation standardly assumes bivalence: every entity determinately satisfies a predicate or determinately does not. Many domains (vague geographic regions, borderline biological taxa, mereologically indeterminate wholes) resist this assumption. The stabilizer formalism of quantum error correction is presented as a vector-based ontology language for classifying and reasoning about indeterminate predicates, with a dispositional dynamics: systems are represented by state vectors, predicates by Pauli operators, indeterminacy as an object-level property, not a meta-level family of precisifications. It complements, rather than replaces, the model-theoretic representation of classical first-order logic. The real (rebit) fragment is efficiently classically simulable (Gottesman-Knill), so no quantum hardware is required. Three determinacy modes (determinacy (D), gappy indeterminacy (G) and correlated indeterminacy (C)) generate, for three predicates, a complete classification hierarchy, its counts established previously, here derived as orbits of nested symmetry-group actions: 120 information kinds, grouped into 49 semantic kinds, 15 determinacy patterns, 7 information forms and 3 correlation classes. Applied to three mereological predicates, a representative subset of kinds is read explicitly, including correlated-indeterminacy patterns beyond bivalent satisfaction. Two results follow. The first is a violation-cost landscape, induced by a chosen dispositional specification, in which the ideal configuration is itself vague; its same-kind transformations are the normalizer of the unsigned landscape group in the group of local Clifford operations and predicate permutations. The second is a centralizer partition of the interaction space, deciding for each query whether it is revelatory or disturbing. The representability boundary lies between the semantic-kind and information-kind levels.
Knowledge representation standardly assumes bivalence: every entity determinately satisfies a predicate or determinately does not. Many domains (vague geographic regions, borderline biological taxa, mereologically indeterminate wholes) resist this assumption. The stabilizer formalism of quantum error correction is presented as a vector-based ontology language for classifying and reasoning about indeterminate predicates, with a dispositional dynamics: systems are represented by state vectors, predicates by Pauli operators, indeterminacy as an object-level property, not a meta-level family of precisifications. It complements, rather than replaces, the model-theoretic representation of classical first-order logic. The real (rebit) fragment is efficiently classically simulable (Gottesman-Knill), so no quantum hardware is required. Three determinacy modes (determinacy (D), gappy indeterminacy (G) and correlated indeterminacy (C)) generate, for three predicates, a complete classification hierarchy, its counts established previously, here derived as orbits of nested symmetry-group actions: 120 information kinds, grouped into 49 semantic kinds, 15 determinacy patterns, 7 information forms and 3 correlation classes. Applied to three mereological predicates, a representative subset of kinds is read explicitly, including correlated-indeterminacy patterns beyond bivalent satisfaction. Two results follow. The first is a violation-cost landscape, induced by a chosen dispositional specification, in which the ideal configuration is itself vague; its same-kind transformations are the normalizer of the unsigned landscape group in the group of local Clifford operations and predicate permutations. The second is a centralizer partition of the interaction space, deciding for each query whether it is revelatory or disturbing. The representability boundary lies between the semantic-kind and information-kind levels.
Ontometry and Supra-Boolean Algebra: Formal Foundations, Geometric Structure, and the Algebra of Intermediate Ontological States This article introduces Ontometry as a formal discipline aimed at modeling ontological states that resist categorical classification within classical Boolean logic. While the mathematical structures employed—Lie-algebraic isomorphisms, tensor decompositions, and operator theory—are standard results of linear algebra and representation theory, their resemantisation constitutes the core contribution of the work. The central object is the supra-Boolean alphabet B* = {0, 1, 0̸, 1̸}, generated by the fundamental predicates of Existence (E) and Manifestation (M). Classical Boolean algebra operates on the phenomenological subset {0,1}, forcing a bivalent choice that fails to capture phenomena exhibiting ontological indetermination, gradual transitions, or essential superposition. The present framework expands this alphabet to four irreducible ontological states—Non-being, Full Being, Pure Potentiality, and Phantasma—and endows the resulting set with algebraic, geometric, and operator-theoretic structure. Key results include: (i) the construction of the Ethereal Operator E, whose involution property and non-commutation with the projectors P_E, P_M establish an ontological uncertainty principle structurally analogous to its quantum counterpart; (ii) the algebraic closure from the discrete set B* to the continuous positive cone R²_+, enabling the modeling of gradual ontological transitions; (iii) the identification of the singlet state |S⟩ as a state of maximum unresolved tension between Existence and Manifestation, emerging inevitably from the decomposition ½ ⊗ ½ = 1 ⊕ 0 and exhibiting maximal entanglement entropy S = ln 2; and (iv) the geometric embedding of the ontometric space T³ into a Lie-algebraic structure isomorphic to so(3), presented here in its static and interpretative dimension. The article does not claim to derive new theorems of linear algebra; rather, it offers an ontological vocabulary for states of essential indetermination that existing formalisms describe mathematically but do not interpret. It is the first instalment of a trilogy that will address Lagrangian dynamics and Noetherian conservation laws (Article II) and applications to artificial intelligence and multi-agent systems (Article III).
The successful application of quantum probability in cognitive science has been persistently challenged by the foundational 'why quantum?' question. In this paper, a novel answer is proposed through the development of an ontological grounding for the quantum-like structures observed in human judgment. It is argued that these cognitive structures may be a direct reflection of an objective, information-theoretic vagueness inherent in the macroscopic world. To formalize this, 'Quantized Mereology' is developed - a framework inspired by Rovelli's Relational Quantum Mechanics (RQM), where determinate mereological facts about vague entities like forests and clouds are established relationally through an act of judgment. The framework uses a real-valued Hilbert space as a precise object language, which is given meaning by a partition-theoretic meta-language. A key contribution is a defense of the real-valued structure, in which it is argued that the physical constraints necessitating complex numbers in microphysics constitute a category error when applied to the conceptual systems modeled herein. The result is a generative model for objective vagueness that provides a candidate ontological foundation for quantum cognition.
Qualitative representations of mereological structures in Artificial Intelligence are generally based on a set of mutually exclusive and collectively exhaustive crisp (RCC5) relations. The amount of information carried by each element in this set of (crisp) relations is determined by the number of bits (N) required to distinguish all members of the set. This paper posits that mereological vagueness can arise if mereological relations are limited in the amount of information they can carry. Consequently, a comprehensive set of vague mereological relations is introduced, where each relation in this set can only carry 0 < n < N bits of information. In this framework, mereological information is treated as quantum information, with quantum bits (qubits) distinguishing relations subject to ontic vagueness. The main contribution is the thorough examination of mereological relations within a 3 qubit quantum system. Quantum bits are not represented as vectors in a Hilbert space but as cells in (granular) partitions. This paper is the companion to the author's quantized-mereology framework and develops its entangled-state layer: the combination of three simultaneous elementary questions, the families of entangled information states, the probability distribution for possible observations and the dynamics induced by sequences of non-simultaneous questions.
Abstract Traditional philosophy has long enshrined "determinacy" as the essential attribute of Being, while relegating "indeterminacy" to the status of a temporary defect of human cognition, a vague state awaiting elimination. The revolutionary development of quantum mechanics has thoroughly shattered this millennia-old cognitive presupposition: the superposition state of microscopic particles is not an "undetermined intermediate state," but an intrinsic property of Being itself; wave-particle duality is not a random mutation of particles, but the determinate manifestation of the same indeterminate ontological substance, emerging within different observational relations. This paper proposes a fundamentally new ontological principle: determinacy is the skeleton of Being that emerges from and is anchored by indeterminacy; indeterminacy, in turn, is the primordial matrix from which determinacy is generated, described, and manifested. These two dimensions do not exist in a relation of sequential dependence or hierarchical subordination, but constitute a reciprocal generative structure in which they define each other in opposition and emerge simultaneously in dynamic interaction. Taking quantum mechanical superposition and wave-particle duality as its core empirical evidence, this paper demonstrates the universal penetrative power of this ontological principle, and points out that eliminating indeterminacy is not the core task of philosophy; rather, modern philosophy must urgently break free from the shackles of traditional paradigms and return to the dynamic, authentic state of Being itself in order to reconstruct cognition. Keywords: Determinacy; Indeterminacy; Reciprocal Emergence in Opposition; Emergence; Manifestation; Desire/Contradiction; Ontology
Many stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes. Ordinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next. We equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered. Finally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.
This project hosts the foundational Article I of Ontometry: "Ontometry and Supra-Boolean Algebra: Formal Foundations, Geometric Structure, and the Algebra of Intermediate Ontological States." While the mathematical structures employed—Lie-algebraic isomorphisms, tensor decompositions, and operator theory—are classical results of linear algebra and representation theory, their resemantisation constitutes the core contribution of the work. The central object is the supra-Boolean alphabet B* = {0, 1, 0̸, 1̸}, generated by the fundamental predicates of Existence (E) and Manifestation (M). Classical Boolean algebra operates on the phenomenological subset {0,1}, forcing a bivalent choice that fails to capture phenomena exhibiting ontological indetermination, gradual transitions, or essential superposition. The present framework expands this alphabet to four irreducible ontological states—Non-being, Full Being, Pure Potentiality, and Phantasma—and endows the resulting set with algebraic, geometric, and operator-theoretic structure. Key results include: (i) the Ethereal Operator E, whose involution and non-commutation with the projectors P_E, P_M establish an ontological uncertainty principle; (ii) algebraic closure from the discrete set B* to the continuous positive cone R²_+, enabling the modeling of gradual ontological transitions; (iii) the singlet state |S⟩ as a state of maximum unresolved tension between Existence and Manifestation, emerging from the decomposition ½ ⊗ ½ = 1 ⊕ 0 with maximal entanglement entropy S = ln 2; and (iv) the geometric embedding of the ontometric space T³ into a Lie-algebraic structure isomorphic to so(3), presented here in its static and interpretative dimension. This Article I does not derive new physical predictions; it offers an ontological vocabulary for states of essential indetermination that existing formalisms describe mathematically but do not interpret. It is the first instalment of a trilogy that will address Lagrangian dynamics and Noetherian conservation laws (Article II) and applications to artificial intelligence and multi-agent systems (Article III). The precursor work, "Axiometry: Prolegomena to a Metric Science of Value" (Zenodo, 2026), introduced the set-theoretic intuitions that are here elevated into rigorous algebraic and operator-theoretic form. Published version with DOI: https://doi.org/10.5281/zenodo.22062034