Bilattice-Valued Predicate Fibrations and the Geometry of Closure

Symmetry Breaking, Orientation, and Quasicrystalline Order in Belnapian Knowledge Structures

Abstract

Knowledge organization requires three things that have mature independent formalizations but have never been unified: typed predication (Frege, Lawvere), incomplete and contradictory evidence (Belnap, Fitting), and part-whole decomposition (Cotnoir & Varzi). This paper defines a Predicate-Fibered Belnapian Knowledge Structure (PFBKS) — a knowledge graph whose edges carry typed predicates valued in Belnap’s four-valued bilattice $\mathcal{B}_4$, with vertices ordered by a mereological preorder — and proves that closing it under inference produces a geometric phenomenon: $\mathcal{B}_4$’s $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry breaks in a predicate-dependent way, collapsing the bilattice diamond to a directed triangle $\{f, \top, t\}$. Directed predicates acquire oriented truth chains; symmetric predicates retain their reflection symmetry. The construction yields a bilattice-valued generalization of Lawvere’s hyperdoctrine (the Fiber Functor theorem), a two-generator decomposition of predicate vocabularies via an extends-restricts adjunction, and, under generic incommensurability conditions, a characterization of quasicrystalline order in the closed product geometry.

Notation

Throughout this paper:

• $\mathcal{B}_4 = \{\bot, t, f, \top\}$: the four-valued bilattice (Belnap 1977, Fitting 2002)
• $\leq_t$: truth ordering on $\mathcal{B}_4$ ($f \leq_t \bot \leq_t t$, $f \leq_t \top \leq_t t$)
• $\leq_k$: knowledge ordering on $\mathcal{B}_4$ ($\bot \leq_k t \leq_k \top$, $\bot \leq_k f \leq_k \top$)
• $\wedge_t, \vee_t$: truth-meet and truth-join
• $\wedge_k, \vee_k$: knowledge-meet and knowledge-join
• $\sim$: negation (swaps $t \leftrightarrow f$, fixes $\bot$ and $\top$)
• $-$: conflation (swaps $\bot \leftrightarrow \top$, fixes $t$ and $f$)
• $K = (G, \Sigma, \nu, \leq_m)$: a Predicate-Fibered Belnapian Knowledge Structure (PFBKS)
• $G = (V, E, L, \mathrm{src}, \mathrm{tgt}, \mathrm{lab})$: labeled directed graph
• $\Sigma = (P, F, \tau, \mathrm{inv}, \mathrm{comp})$: predicate signature
• $\nu: E \to \mathcal{B}_4$: Belnapian valuation
• $\leq_m$: mereological preorder on $V$
• $\varepsilon$: extends (generative generator)
• $\rho$: restricts (constraining generator)
• $C_\Sigma$: predicate category generated by $\Sigma$
• $A_p$: fiber algebra of predicate $p$
• $\mathcal{F}: C_\Sigma^{\mathrm{op}} \to \mathbf{BiLat}$: fiber functor
• $K^*$: closure of $K$ under inference

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1. Introduction

Three formalisms dominate the theory of organized knowledge. Typed predication (Frege 1879, Lawvere 1969) treats a predicate as an unsaturated function that takes a subject and returns a value: “extends(A, B)” asserts that A extends B, with the predicate “extends” supplying the relational type. Uncertain evidence (Belnap 1977, Fitting 1991, 2002) models what happens when a knowledge system receives information from multiple, potentially contradictory sources: each assertion occupies one of four epistemic states (unknown, true, false, or contradictory). Mereological composition (Cotnoir & Varzi 2021) formalizes the part-whole structure that knowledge inherits from the world it describes: documents have sections, systems have components, concepts have subconcepts.

Each formalism has a mature theory. Each pair has been studied: bilattice logic meets description logic (Bienvenu, Bourgaux, & Kozhemiachenko 2024), mereology meets ontology (Arp, Smith, & Spear 2015; Borgo et al. 2022), typed predicates meet knowledge graphs (Hogan et al. 2021, Shinavier, Wisnesky, & Meyers 2022). But the triple — typed predicates with bilattice-valued evidence over a mereological hierarchy — is unstudied. This paper fills that gap.

A Predicate-Fibered Belnapian Knowledge Structure (PFBKS) is a tuple $K = (G, \Sigma, \nu, \leq_m)$ combining four components: a labeled directed graph $G$ providing the relational substrate, a typed predicate vocabulary $\Sigma$ giving the edge labels algebraic structure, a Belnapian valuation $\nu$ assigning each edge an epistemic status from $\mathcal{B}_4 = \{\bot, t, f, \top\}$, and a mereological preorder $\leq_m$ organizing the vertices into a part-whole hierarchy. The graph says what is related to what. The predicates say how. The valuation says with what confidence. The mereology says what is part of what. All four are needed: drop any one and the structure degenerates.

The paper’s discovery is what happens when a PFBKS is closed under inference. Closing eliminates $\bot$ (the “no information” state) from within scope, collapsing $\mathcal{B}_4$’s four-element diamond to a three-element directed triangle $\{f, \top, t\}$. This collapse breaks the bilattice’s $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry: conflation (the involution swapping $\bot$ and $\top$) loses its domain, and the symmetry group reduces to $\mathbb{Z}_2$ (negation alone). The symmetry breaks differently depending on predicate type. Directed predicates (part-of, requires, produces) acquire directed truth chains: $f \to \top \to t$ (denial, contested evidence, affirmation). Symmetric predicates (contrasts-with) retain the full $\mathbb{Z}_2$ symmetry. The result is an oriented geometry over the knowledge graph whose structure encodes the system’s semantic content.

This paper makes four contributions:

1. The PFBKS construction and Fiber Functor theorem (Sections 3–4). The assignment of fiber algebras to predicates extends to a contravariant functor $\mathcal{F}: C_\Sigma^{\mathrm{op}} \to \mathbf{BiLat}$, making a PFBKS a bilattice-valued generalization of Lawvere’s hyperdoctrine.

2. The two-generator thesis (Section 3). All predicates in common knowledge-organization use decompose into two adjoint generators — extends ($\varepsilon$, generative) and restricts ($\rho$, constraining) — forming a Galois biconnection on bilattice fibers.

3. The Symmetry-Breaking theorem (Section 5). Closure under inference breaks $\mathcal{B}_4$’s $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry to at most $\mathbb{Z}_2$, producing predicate-dependent directedness on the three-element residual $\{t, f, \top\}$.

4. Quasicrystalline order (Section 7). Under generic incommensurability conditions on the graph topology, the closed product fiber has aperiodic long-range order with quasicrystalline character.

The paper follows the arc of the discovery: $\mathcal{B}_4$ is a geometry (Section 2) $\to$ predicates reduce to two adjoint generators (Section 3) $\to$ fibering $\mathcal{B}_4$ over predicates yields a fiber functor (Section 4) $\to$ closing under inference breaks the symmetry (Section 5) $\to$ the broken symmetry produces oriented, quasicrystalline geometry (Sections 6–7). Sections 2–4 build the construction. Section 5 proves the theorem. Sections 6–7 develop its consequences.

This work extends YAML Frontmatter as Fregean Predicate Fibration (emsenn 2026), which identified a vault’s YAML frontmatter as generating a degenerate hyperdoctrine with bare-set fibers. The present paper enriches that structure: the fibers carry the full bilattice algebra of $\mathcal{B}_4$, and the enrichment reveals phenomena — symmetry breaking, orientation, quasicrystalline order — invisible at the degenerate level.

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2. The Four-Valued Bilattice as Modal Geometry

This section establishes the four-valued bilattice $\mathcal{B}_4$ as a two-dimensional geometric object. All results here are known; the contribution is the geometric reading that the rest of the paper depends on.

2.1 Bilattices and $\mathcal{B}_4$

Nuel Belnap (1977) proposed a four-valued logic for reasoning about what a knowledge system has been “told” by potentially unreliable sources. The four values are: $\bot$ (neither told true nor false — no information), $t$ (told true, not told false), $f$ (told false, not told true), and $\top$ (told both true and false — contradictory evidence).

A bilattice is a set $L$ equipped with two bounded lattice structures $(L, \wedge_t, \vee_t, f, t)$ and $(L, \wedge_k, \vee_k, \bot, \top)$, plus two involutions: negation $\sim$ (reverses $\leq_t$, preserves $\leq_k$) and conflation $-$ (preserves $\leq_t$, reverses $\leq_k$). A bilattice is distributive if both lattice structures are distributive and the interlacing conditions hold: each meet distributes over the other’s join. A bilattice is interlaced if all four operations ($\wedge_t, \vee_t, \wedge_k, \vee_k$) are monotone with respect to both orderings. Ginsberg (1988) introduced bilattices to AI; Fitting (1991, 2002) developed their theory extensively.

$\mathcal{B}_4 = \{\bot, t, f, \top\}$ carries two independent orderings:

• Truth ordering ($\leq_t$): $f \leq_t \bot \leq_t t$, $f \leq_t \top \leq_t t$, with $\bot$ and $\top$ incomparable. A diamond with $f$ at bottom, $t$ at top.
• Knowledge ordering ($\leq_k$): $\bot \leq_k t \leq_k \top$, $\bot \leq_k f \leq_k \top$, with $t$ and $f$ incomparable. A diamond with $\bot$ at bottom, $\top$ at top.

Fitting (2002) proved that $\mathcal{B}_4$ is the free bounded distributive bilattice on one generator — the simplest non-trivial bilattice, and the universal one. Any bilattice-valued valuation factors through $\mathcal{B}_4$.

2.2 The Twist Product

Kalman (1958) showed that every bounded distributive lattice $L$ generates a De Morgan algebra via the twist product $L \otimes L$. Fitting (1991) extended this construction to bilattices and proved that $\mathcal{B}_4$ is isomorphic to $\mathbf{2} \otimes \mathbf{2}$, the twist product of the two-element Boolean algebra with itself. Concretely, each $\mathcal{B}_4$ value is a pair $(a, b) \in \{0, 1\}^2$:

Value	Pair	Reading
$\bot$	$(0, 0)$	No information in either dimension
$t$	$(1, 0)$	Positive truth, no counter-evidence
$f$	$(0, 1)$	Negative truth, no supporting evidence
$\top$	$(1, 1)$	Information in both dimensions

The first component is the truth dimension; the second is the falsity (or counter-evidence) dimension. The knowledge ordering $\leq_k$ is the componentwise order on $\{0,1\}^2$: $(a,b) \leq_k (a',b')$ iff $a \leq a'$ and $b \leq b'$. The truth ordering $\leq_t$ is determined by the first component ascending and the second descending.

The twist product reveals $\mathcal{B}_4$ as a 2D modal space whose two axes are truth and knowledge (Dunn 2000, Shramko & Wansing 2011). Each axis carries independent information, and the four values are the four possible combinations of presence or absence of information along each axis.

2.3 The Symmetry Group

In the twist-product representation, negation and conflation act by coordinate reflections:

• Negation $\sim$: $(a, b) \mapsto (b, a)$, equivalently $(a,b) \mapsto (1-a, 1-b)$ composed with a swap. On the bilattice: it swaps $t \leftrightarrow f$ and fixes $\bot$ and $\top$. It flips the truth axis and fixes the knowledge axis.
• Conflation $-$: $(a, b) \mapsto (a, 1-b)$ or equivalently the map that swaps $\bot \leftrightarrow \top$ and fixes $t$ and $f$. It flips the knowledge axis and fixes the truth axis.

Together they generate $\mathbb{Z}_2 \times \mathbb{Z}_2$, the Klein four-group — four elements: identity, negation, conflation, and their composition (which acts as $180°$ rotation).

$\mathbb{Z}_2 \times \mathbb{Z}_2$ is the symmetry group of the square. $\mathcal{B}_4$, plotted with $\leq_t$ horizontal and $\leq_k$ vertical, is a square. Negation and conflation are reflections across the two axes. This square has a center of symmetry, two axes of reflection, and four elements — the full dihedral symmetry of a rectangle (a subgroup of the dihedral group $D_4$ of the square, since the two axes are distinguishable).

This symmetry group is what closure will break (Section 5.4).

2.4 Mereological Preorders

A mereological preorder on a set $V$ is a reflexive, transitive relation $\leq_m$ satisfying weak supplementation: if $x <_m y$ (strict), there exists $z \leq_m y$ with $z$ not overlapping $x$. “Overlapping” means sharing a common part: $x$ and $z$ overlap if there exists $w$ with $w \leq_m x$ and $w \leq_m z$.

Following Cotnoir & Varzi (2021, Ch. 3), we adopt this weaker axiom system rather than classical extensional mereology. The reason is that knowledge structures admit distinct units with identical parts — two documents can have the same components while remaining distinct as knowledge units. Classical extensional mereology would identify them. The weaker axiom system does not.

The mereological preorder provides the part-whole hierarchy that the PFBKS (Section 4) will use for inheritance: parts inherit properties from wholes, with evidence attenuated along the $\leq_m$-chain.

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3. Predicate Primitives and the Adjunction

This section presents the paper’s first novel contribution. The surface vocabulary of predicates used in knowledge organization — part-of, requires, produces, extends, contrasts-with, governed-by, and their kin — reduces to two irreducible generators that form an adjunction.

3.1 Predicate Signatures

Definition 1 (Predicate Signature). A predicate signature is a tuple $\Sigma = (P, F, \tau, \mathrm{inv}, \mathrm{comp})$ where:

• $P$ is a set of predicate symbols (part-of, requires, extends, produces, contrasts-with, governed-by, …)
• $F: P \to \mathrm{Family}$ assigns each predicate to a semantic family (mereological, taxonomic, functional, deontic, oppositional, causal)
• $\tau: P \to 2^{\mathrm{Properties}}$ assigns algebraic properties (transitive, reflexive, symmetric, antisymmetric, …)
• $\mathrm{inv}: P \rightharpoonup P$ is a partial involution giving inverses (part-of $\leftrightarrow$ has-part, requires $\leftrightarrow$ required-by)
• $\mathrm{comp}: P \times P \rightharpoonup P$ is a partial composition (component-of $\circ$ part-of $=$ part-of, acts-on $\circ$ requires $=$ produces)

The signature is coherent if the following conditions hold:

1. $\mathrm{inv}$ is involutory: $\mathrm{inv}(\mathrm{inv}(p)) = p$ wherever defined.
2. $\mathrm{inv}$ respects properties: if $p$ is transitive, then $\mathrm{inv}(p)$ is transitive.
3. $\mathrm{comp}$ is associative where defined: $(p \circ q) \circ r = p \circ (q \circ r)$.
4. $\mathrm{comp}$ respects families: composing two mereological predicates produces a mereological predicate.

A coherent predicate signature generates a predicate category $C_\Sigma$ whose objects are predicate symbols and whose morphisms are generated by $\mathrm{inv}$ and $\mathrm{comp}$. This category formalizes what BFO (Arp, Smith, & Spear 2015) and DOLCE (Borgo et al. 2022) do informally: it gives the predicate vocabulary an algebraic structure. Description logic role hierarchies (Baader et al. 2003) capture a weaker form of this structure, lacking the composition and inversion operations.

3.2 The Two-Generator Thesis

Every predicate in common knowledge-organization use decomposes as a mode of two primitive operations:

• extends ($\varepsilon$): generative, expansive, forward. “$X$ adds to $Y$.” Left-adjoint character.
• restricts ($\rho$): constraining, narrowing, backward. “$X$ limits $Y$.” Right-adjoint character.

These two operations are not inverses. Extending without restricting (adding capability without removing freedom) and restricting without extending (constraining without adding content) are both possible. They are orthogonal.

Predicate	Decomposition
part-of	$\varepsilon$ (compositional): $X$ extends $Y$’s structure
requires	$\rho$ (dependency): $Y$ restricts $X$’s existence
produces	$\varepsilon$ (causal): $X$ extends into $Y$
governed-by	$\rho$ (deontic): $Y$ restricts $X$’s behavior
extends (taxonomic)	$\varepsilon$ (taxonomic): $X$ extends $Y$’s abstraction
contrasts-with	mutual $\rho$: $X$ and $Y$ restrict each other
enables	weakened $\varepsilon$: $X$ partially extends $Y$’s space
determines	strengthened $\rho$: $X$ fully restricts $Y$

The symmetric predicate contrasts-with is algebraically distinguished: it decomposes as mutual restriction, making it self-inverse ($\mathrm{inv}(\text{contrasts-with}) = \text{contrasts-with}$). This self-inverseness is the property that preserves $\mathbb{Z}_2$ symmetry through closure (Section 5).

The decomposition parallels a well-known pattern. In formal concept analysis (Ganter & Wille 1999), extent and intent form a Galois connection on a concept lattice. In Lawvere’s categorical logic (1969), existential and universal quantification are the left and right adjoints to substitution. In algebraic property graphs (Shinavier, Wisnesky, & Meyers 2022), functorial data migration decomposes into left-adjoint ($\Sigma$, extension) and right-adjoint ($\Pi$, restriction) components. Our claim is that these are all instances of the same underlying structure: predication itself decomposes into a generative component and a constraining component.

Proposition 1 (Two-Generator Sufficiency). Every morphism in $C_\Sigma$ factors as a composition of instances of $\varepsilon$ and $\rho$. The derived predicates (part-of, requires, produces, governed-by, etc.) are specific factorizations.

Justification. By coherence, every morphism in $C_\Sigma$ is generated by $\mathrm{inv}$ and $\mathrm{comp}$. The two-generator thesis identifies each predicate symbol with a specific mode of $\varepsilon$ or $\rho$ (or their composition). Since $\mathrm{inv}$ swaps $\varepsilon$-character and $\rho$-character (the inverse of a generative predicate is constraining, and vice versa), and $\mathrm{comp}$ composes modes, every morphism in $C_\Sigma$ factors through $\varepsilon$ and $\rho$.

3.3 The Extends-Restricts Adjunction

Definition 2 (Extends-Restricts Adjunction). In the predicate category $C_\Sigma$ generated by $\varepsilon$ and $\rho$, the extends-restricts adjunction is the natural bijection:

$$\mathrm{Hom}(\varepsilon(p), q) \cong \mathrm{Hom}(p, \rho(q))$$

“$X$ falls within the extension of $p$ relative to $q$” if and only if “$p$ satisfies the restriction to $X$ relative to $q$.” This is a Galois connection. When lifted to bilattice-valued fibers (Section 4), it becomes a Galois biconnection in the sense of Jakl, Jung, & Rivieccio (2021): the bilattice analogue of a Galois connection, with compatible connections for both the truth ordering $\leq_t$ and the knowledge ordering $\leq_k$.

The adjunction gives the predicate vocabulary orientation: the asymmetry between left adjoint ($\varepsilon$, generative) and right adjoint ($\rho$, constraining) is structural. The unit $\eta: \mathrm{id} \to \rho \circ \varepsilon$ and the counit $\epsilon: \varepsilon \circ \rho \to \mathrm{id}$ of the adjunction are the two canonical directions. Their asymmetry ($\eta \neq \epsilon^{-1}$ because the adjunction is not an equivalence) is the asymmetry of the geometry that Section 6 will develop.

This connects to Lawvere’s (1996) reading of adjoint pairs as “unity and identity of opposites.” The extends/restricts pair is the unity of opposites for predication itself: every predicate assertion simultaneously adds something to the knowledge structure (extends) and constrains something within it (restricts), and the balance between these two operations determines the predicate’s character.

Busaniche & Cignoli (2021) study adjoint operations specifically within twist-product lattices, directly connecting $\mathcal{B}_4$’s twist-product structure (Section 2.2) to adjunction theory. Their algebraic results place the extends-restricts adjunction as an instance of a general algebraic phenomenon within the bilattice framework.

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4. Belnapian Predicate Fibers

This section assembles the central construction. The predicate signature (Section 3) provides the algebraic vocabulary; $\mathcal{B}_4$ (Section 2) provides the truth-value space. Here these combine into a single structure — the PFBKS — and the main structural theorems are stated.

4.1 Labeled Directed Graphs

A labeled directed graph is a tuple $G = (V, E, L, \mathrm{src}, \mathrm{tgt}, \mathrm{lab})$ where $V$ is a set of vertices (knowledge units — documents, concepts, entities), $E$ is a set of directed edges (predicate assertions), $L$ is a label set, and $\mathrm{src}, \mathrm{tgt}: E \to V$ and $\mathrm{lab}: E \to L$ are source, target, and labeling functions. This is the standard definition following Hogan et al. (2021). Each edge $e$ asserts that the predicate $\mathrm{lab}(e)$ relates $\mathrm{src}(e)$ to $\mathrm{tgt}(e)$.

4.2 The PFBKS Definition

Definition 3 (PFBKS). A Predicate-Fibered Belnapian Knowledge Structure is a tuple $K = (G, \Sigma, \nu, \leq_m)$ where:

• $G = (V, E, L, \mathrm{src}, \mathrm{tgt}, \mathrm{lab})$ is a labeled directed graph with $L = P$ (every edge label is a predicate from the signature)
• $\Sigma = (P, F, \tau, \mathrm{inv}, \mathrm{comp})$ is a coherent predicate signature (Definition 1)
• $\nu: E \to \mathcal{B}_4$ is a Belnapian valuation: every edge receives a value from $\{\bot, t, f, \top\}$ representing the epistemic status of that predicate assertion
• $\leq_m$ is a mereological preorder on $V$ (Section 2.4)

subject to two compatibility conditions:

1. Mereological agreement. The preorder $\leq_m$ agrees with mereological predicates in $\Sigma$: if $\mathrm{lab}(e) = \text{part-of}$ and $\nu(e) = t$, then $\mathrm{src}(e) \leq_m \mathrm{tgt}(e)$.
2. Compositional agreement. Predicate composition in $\Sigma$ respects edge composition in $G$: if $p \circ q = r$ in $C_\Sigma$ and edges $e_1$ (for $p$) and $e_2$ (for $q$) compose in $G$ (meaning $\mathrm{tgt}(e_1) = \mathrm{src}(e_2)$), the composed edge witnesses $r$.

The four components serve distinct roles. The graph $G$ provides the relational substrate. The signature $\Sigma$ gives the predicate vocabulary its algebraic structure. The valuation $\nu$ assigns epistemic status to each assertion. The mereological preorder $\leq_m$ organizes the vertices into a part-whole hierarchy. No existing formalism combines all four: description logics lack bilattice valuations, knowledge graphs lack typed predicate algebra, and mereological ontologies lack both.

4.3 Fiber Algebras

For each predicate $p \in P$, the fiber is the set of all edges labeled $p$:

$$\mathrm{Fib}(p) = \{ e \in E : \mathrm{lab}(e) = p \}$$

The restriction of $\nu$ to $\mathrm{Fib}(p)$ gives a function $\mathrm{Fib}(p) \to \mathcal{B}_4$.

Definition 4 (Fiber Algebra). The fiber algebra $A_p$ is the pointwise bilattice on the function space $\mathrm{Fib}(p) \to \mathcal{B}_4$. For two edges $e_1, e_2 \in \mathrm{Fib}(p)$, the operations $\nu(e_1) \wedge_t \nu(e_2)$, $\nu(e_1) \vee_t \nu(e_2)$, $\nu(e_1) \wedge_k \nu(e_2)$, and $\nu(e_1) \vee_k \nu(e_2)$ are all well-defined because $\mathcal{B}_4$ is a bilattice.

Proposition 2 (Fiber Distributivity). Each fiber algebra $A_p$ is a distributive bilattice.

Justification. $\mathcal{B}_4$ is a distributive bilattice (Fitting 2002). Pointwise products of distributive bilattices are distributive: the interlacing and distributivity conditions hold componentwise.

4.4 Fiber Composition

When predicates compose via $\mathrm{comp}$ in $\Sigma$ (for example, $\text{component-of} \circ \text{part-of} = \text{part-of}$), their fibers compose too. If edge $e_1$ witnesses $p$ and edge $e_2$ witnesses $q$ with $\mathrm{tgt}(e_1) = \mathrm{src}(e_2)$, and $p \circ q = r$ in $\Sigma$, then the composed edge witnesses $r$ with value:

$$\nu(e_1 \circ e_2) = \nu(e_1) \wedge_k \nu(e_2)$$

The composition uses the knowledge-meet $\wedge_k$, not the truth-meet $\wedge_t$. Composition combines evidential support: $\wedge_k$ preserves whatever both assertions agree on as information. If one says $t$ and the other says $f$, then $t \wedge_k f = \top$ — both pieces of evidence are present and contradicting. This is correct: a composed chain passing through contradictory evidence should surface the contradiction, not silently resolve it. This follows Fitting’s (1991) approach to composition in bilattice logic programming.

4.5 The Fiber Functor

Theorem 1 (Fiber Functor). The assignment $p \mapsto A_p$ extends to a contravariant functor

$$\mathcal{F}: C_\Sigma^{\mathrm{op}} \to \mathbf{BiLat}$$

from the predicate category (Section 3) to the category of distributive bilattices. Morphisms in $C_\Sigma$ (predicate specialization, inversion, composition) map to bilattice homomorphisms between fiber algebras.

Proof sketch. Three classes of morphisms in $C_\Sigma$ must map to bilattice homomorphisms. (1) Specialization: if $p$ specializes $q$ (there is a morphism $p \to q$ in $C_\Sigma$), the induced map $A_q \to A_p$ restricts the valuation from the broader predicate’s fiber to the narrower one, preserving all bilattice operations. (2) Inversion: if $\mathrm{inv}(p) = p'$, the induced map $A_p \to A_{p'}$ applies negation $\sim$ pointwise, which is a bilattice homomorphism because $\sim$ preserves $\leq_k$ and its lattice operations. (3) Composition: if $p \circ q = r$, the induced map $A_p \otimes A_q \to A_r$ is given by the $\wedge_k$-composition rule, which preserves bilattice structure by Proposition 2. Functoriality (preservation of identity and composition of morphisms) follows from associativity of $\mathrm{comp}$ and involutivity of $\mathrm{inv}$ in the predicate signature.

The functor is contravariant: more specific predicates (further from the generators $\varepsilon$ and $\rho$ in $C_\Sigma$) have fibers that receive homomorphisms from less specific predicates. Specialization enriches the fiber by adding structure, not removing it.

Relation to Lawvere’s hyperdoctrines. Lawvere (1969, 1970) defined a hyperdoctrine as a functor from a base category to a fiber category — Heyting algebras for intuitionistic logic, Boolean algebras for classical logic. The Fiber Functor says a PFBKS is a hyperdoctrine with $\mathbf{BiLat}$ as the fiber category: bilattice fibers instead of Heyting algebra fibers. In YAML Frontmatter as Fregean Predicate Fibration (emsenn 2026), we identified a vault’s frontmatter as generating a degenerate hyperdoctrine with bare-set fibers and no algebraic structure within each fiber. The Fiber Functor is the non-degenerate version: the fibers carry the full bilattice algebra of $\mathcal{B}_4$.

Relation to Maruyama (2021). Maruyama’s fibered algebraic semantics provides a general framework: any non-classical logic with an algebraic semantics can be fibered over a base category. The Fiber Functor instantiates his framework for $\mathcal{B}_4$-valued logic over predicate-structured knowledge graphs with the extends-restricts adjunction. The instantiation yields results (symmetry breaking, orientation) that the general framework does not reach because it works at full generality.

4.6 Mereological Inheritance

Parts inherit properties of wholes with evidence attenuated along the part-of chain. If $v \leq_m w$ ($v$ is part of $w$) and $w$ has an edge $e$ with $\mathrm{lab}(e) = p$ and $\nu(e) = x$, then $v$ inherits an assertion for $p$ with value determined by the $\wedge_k$-product along the mereological chain from $v$ to $w$.

The $\wedge_k$ attenuation means inherited evidence never exceeds the evidence at the whole. A part can have less information than its whole (the chain introduces $\wedge_k$-loss), but never more. This reflects the intuition that parthood is an information-attenuating relation: knowing something about a whole gives you partial knowledge about its parts, but not the reverse.

Proposition 3 (Inheritance Monotonicity). Inheritance is monotone in $\leq_k$. More information at the whole means at least as much inherited information at the part.

Justification. $\wedge_k$ is monotone in $\leq_k$: if $x \leq_k x'$ and $y \leq_k y'$, then $x \wedge_k y \leq_k x' \wedge_k y'$. The inheritance operation applies $\wedge_k$ along the chain, so it is monotone at each step, hence monotone overall.

Theorem 2 (Inheritance Coherence). The inheritance operation is a natural transformation $\eta: \mathcal{F}(-) \circ \mathrm{whole} \to \mathcal{F}(-) \circ \mathrm{part}$, commuting with the Fiber Functor. Inheriting first and then applying predicate structure gives the same result as applying predicate structure first and then inheriting.

Proof sketch. Naturality requires that for every morphism $p \to q$ in $C_\Sigma$, the diagram commutes: the bilattice homomorphism $A_q \to A_p$ (from Theorem 1) composed with the inheritance map at $p$ equals the inheritance map at $q$ composed with $A_q \to A_p$. Both paths compute the same value because $\wedge_k$-attenuation along the mereological chain commutes with bilattice homomorphisms: the homomorphisms preserve $\wedge_k$ (they are bilattice homomorphisms), and the chain is the same regardless of which predicate’s fiber is being inherited.

This interaction between mereological decomposition and bilattice-valued fibers has no precedent in the literature. BFO (Arp, Smith, & Spear 2015) and DOLCE (Borgo et al. 2022) assume parts inherit properties from wholes but do not formalize this inheritance bilattice-theoretically. The Inheritance Coherence theorem makes the interaction precise and proves it is well-behaved.

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5. Closure and Symmetry Breaking

Everything before this section is scaffolding. The PFBKS has been defined, its fiber algebras characterized, and the fiber functor established. This section closes the structure under inference and proves the paper’s main result: closure breaks $\mathcal{B}_4$’s symmetry.

5.1 Inference Rules

Four rules derive new $\mathcal{B}_4$-valued assertions from existing ones. Each rule takes a valid PFBKS and produces a valid PFBKS with additional edges.

1. Fiber composition. If $p \circ q = r$ in $C_\Sigma$ and there exist edges $e_1$ (for $p$, value $x$) and $e_2$ (for $q$, value $y$) with $\mathrm{tgt}(e_1) = \mathrm{src}(e_2)$, derive an edge for $r$ from $\mathrm{src}(e_1)$ to $\mathrm{tgt}(e_2)$ with value $x \wedge_k y$.

2. Mereological inheritance. If $v \leq_m w$ and $w$ has an edge $e$ with $\mathrm{lab}(e) = p$ and $\nu(e) = x$, derive an edge at $v$ for $p$ with value equal to the $\wedge_k$-product along the mereological chain from $v$ to $w$.

3. Inverse. If $\mathrm{inv}(p) = p'$ in $\Sigma$ and there is an edge $e$ for $p$ from $v$ to $w$ with value $x$, derive an edge for $p'$ from $w$ to $v$ with value ${\sim}x$ (truth-negation: $t \leftrightarrow f$, $\bot$ and $\top$ fixed).

4. Transitivity. If $p$ is transitive ($\tau(p) \ni \text{transitive}$) and there are edges $v \to w$ (value $x$) and $w \to u$ (value $y$) both for $p$, derive edge $v \to u$ for $p$ with value $x \wedge_k y$.

All four rules use the knowledge-meet $\wedge_k$ for combining evidence, not the truth-meet $\wedge_t$. The reason is the same in each case: composition combines evidential support, and $\wedge_k$ preserves whatever both assertions agree on as information. If one assertion says $t$ and another says $f$, then $t \wedge_k f = \top$ — both pieces of evidence are present and contradicting. This is correct: a composed chain passing through contradictory evidence should surface the contradiction, not silently resolve it. The inverse rule uses negation $\sim$ rather than $\wedge_k$ because inversion reverses the truth dimension while preserving the knowledge dimension, which is exactly what $\sim$ does on $\mathcal{B}_4$.

Proposition 4 (Soundness). All four rules preserve PFBKS structure: applying any rule to a valid PFBKS yields a valid PFBKS.

Each rule preserves the compatibility between the mereological preorder and mereological predicates, and preserves predicate composition in $\Sigma$ relative to edge composition in $G$. The argument is a case analysis on the four rules, checking each PFBKS axiom (Definition 3).

Proposition 5 (Monotonicity). All four rules are monotone in the knowledge ordering $\leq_k$. No rule ever decreases the knowledge content of any edge.

Justification. Rules 1, 2, and 4 combine existing values via $\wedge_k$, which is monotone in $\leq_k$ by definition. Rule 3 applies negation $\sim$, which preserves $\leq_k$ (it reverses $\leq_t$ but fixes $\leq_k$). When a derived edge duplicates an existing edge, the result takes the $\vee_k$ (knowledge-join) of the old and new values, which is at least as high as either. This adapts the monotonicity result of Fitting (1991) for bilattice logic programming operators.

5.2 The Fixpoint

Theorem 3 (Fixpoint). For any finite PFBKS $K$, iterative application of the four inference rules reaches a least fixpoint $K^*$ in finitely many steps. $K^*$ is the closure of $K$ — the PFBKS with all derivable knowledge made explicit.

Proof sketch. The four rules together define a monotone operator $T$ on the complete lattice $\mathcal{B}_4^{|E^*|}$, where $E^*$ is the set of all potential edges (bounded by $|V|^2 \times |P|$), ordered by pointwise $\leq_k$. Monotonicity of $T$ follows from Proposition 5. The Knaster-Tarski theorem gives the existence of a least fixpoint $\mathrm{lfp}(T)$.

Finite convergence follows from two facts. First, the potential edge set $E^*$ is finite because $V$ is finite. Second, $\leq_k$ on $\mathcal{B}_4$ has height 2 (the longest chain is $\bot <_k t <_k \top$ or $\bot <_k f <_k \top$). Each application of $T$ either pushes at least one edge strictly upward in $\leq_k$ or has reached the fixpoint. The number of steps is bounded by $2 \times |E^*|$.

5.3 Elimination of Bottom

Introduce a scoped closed-world assumption: every potential edge within the scope of some inference rule either has direct evidence or is reachable by inference from edges that do. “Within scope” means the edge connects vertices that participate in at least one rule’s preconditions.

Under this assumption, the fixpoint $K^*$ has no $\bot$-valued edges within scope. The mechanism is direct. Edges start at $\bot$ (no information). Each rule pushes edges upward in $\leq_k$: from $\bot$ to $t$, $f$, or $\top$. The fixpoint is reached when no rule can push further. Within scope, every edge has been reached by some rule, so every edge has been pushed above $\bot$.

The closure acts as an order-theoretic filter on $\mathcal{B}_4$: it removes the bottom element of the knowledge ordering. The effective fiber algebra after closure operates on the three-element set $\{t, f, \top\} = \mathcal{B}_4 \setminus \{\bot\}$, not on $\mathcal{B}_4$.

This is the step that breaks the symmetry. Removing $\bot$ is a consequence of closing under inference with sufficient coverage. The four-element bilattice becomes a three-element structure, and the three-element structure has strictly fewer symmetries than the four-element one.

5.4 The Symmetry-Breaking Theorem

Theorem 4 (Symmetry Breaking). Let $K^*$ be the closure of a PFBKS $K$ under the scoped closed-world assumption. The effective fiber algebra on $\{t, f, \top\} = \mathcal{B}_4 \setminus \{\bot\}$ has the following properties:

(i) Knowledge ordering survives as a V-shape. $\{t, f, \top\}$ under $\leq_k$: $\top >_k t$, $\top >_k f$, and $t$ and $f$ are incomparable. This is $\mathcal{B}_4$’s knowledge diamond with the bottom removed — a V (or fan) with $\top$ at the apex.

(ii) Truth ordering collapses to a chain. $\{t, f, \top\}$ under $\leq_t$ is the total order $f <_t \top <_t t$. In full $\mathcal{B}_4$, the truth ordering is a diamond with $f$ at bottom, $t$ at top, and $\bot$ and $\top$ as incomparable middle elements. Removing $\bot$ eliminates one of the two middle elements, and the remaining structure $f < \top < t$ is a chain. The partial order has become a total order.

(iii) Symmetry group reduces. The symmetry group of $\mathcal{B}_4$ is $\mathbb{Z}_2 \times \mathbb{Z}_2$ (the Klein four-group), generated by negation and conflation. On $\{t, f, \top\}$:

• Negation $\sim$ maps $t \leftrightarrow f$ and fixes $\top$. This is an involution on $\{t, f, \top\}$. It swaps the truth chain’s endpoints. Negation survives.
• Conflation $-$ would need to swap $\bot$ and $\top$ while fixing $t$ and $f$. But $\bot$ is absent. Conflation has no well-defined action on $\{t, f, \top\}$. Conflation does not survive.

$$\boxed{\mathbb{Z}_2 \times \mathbb{Z}_2 \;\longrightarrow\; \mathbb{Z}_2}$$

One of the two independent symmetries is destroyed by closure.

(iv) Directed predicates acquire directed fibers. For predicates with non-self-inverse character (part-of, requires, extends, produces — all directed from source to target), the surviving $\mathbb{Z}_2$ (negation) acts as an anti-automorphism of the truth chain: it reverses direction ($t \mapsto f$, $f \mapsto t$, $\top \mapsto \top$). The fiber has a preferred direction $f \to \top \to t$, inherited from the predicate’s semantics: denial, then contested evidence, then affirmation.

(v) Symmetric predicates retain $\mathbb{Z}_2$. For self-inverse predicates (contrasts-with, dual-of), negation acts as an automorphism: the predicate does not distinguish source from target, so swapping $t$ and $f$ preserves the structure. The $\mathbb{Z}_2$ symmetry survives fully. The fiber has no preferred direction.

Proof sketch. Parts (i) and (ii) follow by inspecting the Hasse diagrams of $\leq_k$ and $\leq_t$ restricted to $\{t, f, \top\}$. In full $\mathcal{B}_4$, $\bot$ and $\top$ are incomparable in $\leq_t$; removing $\bot$ leaves $\top$ as the unique middle element between $f$ and $t$, producing the chain $f <_t \top <_t t$. Part (iii) is direct: conflation requires $\bot$ in its image, and $\bot \notin \{t, f, \top\}$. Part (iv) follows from the fact that for directed predicates, the source-to-target asymmetry distinguishes $t$ from $f$ — the predicate’s semantics assigns a direction to the truth chain, making negation an anti-automorphism (direction-reversing). Part (v) follows from self-inverseness: if $\mathrm{inv}(p) = p$, then the fiber is closed under negation, and negation acts as an automorphism rather than an anti-automorphism.

5.5 Contradiction as Waypoint

The truth chain $f <_t \top <_t t$ gives contradiction ($\top$) a role that $\mathcal{B}_4$’s diamond obscures: it is a waypoint between denial and affirmation, not an error state. Evidence accumulates along the chain:

• part-of: “not a part” ($f$) $\to$ “contested parthood” ($\top$) $\to$ “confirmed part” ($t$)
• requires: “no dependency” ($f$) $\to$ “contested dependency” ($\top$) $\to$ “confirmed need” ($t$)
• produces: “no output” ($f$) $\to$ “contested production” ($\top$) $\to$ “confirmed product” ($t$)

The direction belongs to the predicate, not to the evidence. Each predicate’s semantics determines which end of the chain ($t$ or $f$) counts as resolution. Contradiction is a transitional epistemic state: the system has accumulated enough evidence to be in conflict, but the conflict remains unresolved.

This reading extends Dunn’s (2000) treatment of negation as a modal operator. After closure, negation reverses a direction on a chain: it acts on a one-dimensional ordered structure (the truth chain $f <_t \top <_t t$) rather than on the full two-dimensional bilattice diamond.

5.6 The Closed Fiber Algebra

Corollary. The effective fiber algebra on $\{t, f, \top\}$ is not a bilattice.

A bilattice requires two lattice orderings satisfying interlacing conditions: each meet distributes over the other’s join, and all four operations are monotone with respect to both orderings. On $\{t, f, \top\}$, $\leq_t$ is a 3-element chain and $\leq_k$ is a V-shape. The interlacing condition fails because the two orderings interact asymmetrically on specific pairs:

Pair	$\leq_k$	$\leq_t$	Relationship
$(t, \top)$	$\top >_k t$	$t >_t \top$	Opposite directions
$(f, \top)$	$\top >_k f$	$\top >_t f$	Same direction
$(t, f)$	incomparable	$t >_t f$	Only truth speaks

The opposite-direction pair $(t, \top)$ is the source of orientation: gaining knowledge (moving toward $\top$ in $\leq_k$) can move away from truth (away from $t$ in $\leq_t$). The predicate’s resolution direction navigates this tension. The bilattice’s elegant balance between truth and knowledge, maintained by the presence of $\bot$, is exactly what closure breaks.

5.7 Consistency and Gap Detection

The closed structure admits three diagnostic propositions.

Proposition 6 (Consistency Detection). Local inconsistency at a vertex $v$ for predicate $p$ is the existence of edges $e_1, e_2 \in \mathrm{Fib}(p)$ with $\mathrm{src}(e_1) = \mathrm{src}(e_2) = v$ and $\nu(e_1) \wedge_t \nu(e_2) = \top$. This is decidable for finite graphs by enumerating edges at each vertex.

The truth-meet $\wedge_t$ detects contradiction: two edges whose truth values conflict yield $\top$ under $\wedge_t$, surfacing the inconsistency at the vertex where it occurs.

Proposition 7 (Gap Detection). Under the scoped closed-world assumption, missing information at a vertex $v$ is an expected edge (predicted by predicate structure in $\Sigma$) whose closure value remains $\bot$. After closure, within-scope edges with value $\bot$ indicate information the structure expects but does not have. This is computable for finite graphs.

Gap detection complements consistency detection: where consistency detection finds too much conflicting information, gap detection finds too little.

Proposition 8 (Mereological Coherence). The following three conditions are equivalent:

(a) No vertex has both a direct edge and an inherited edge for the same predicate with $\nu(\text{direct}) \wedge_t \nu(\text{inherited}) = \top$.

(b) The inheritance natural transformation $\eta$ (Section 4.6) preserves truth-consistency.

(c) The mereological preorder is compatible with the fiber functor in the sense that $\mathcal{F} \circ {\leq_m}$ is a subfunctor of $\mathcal{F}$.

Justification. The equivalence of (a) and (b) is a restatement: (a) gives the pointwise condition, (b) gives the naturality condition. The equivalence of (b) and (c) follows from the definition of subfunctor: $\mathcal{F} \circ {\leq_m}$ being a subfunctor of $\mathcal{F}$ means that inheritance is compatible with the fiber structure at every object of $C_\Sigma$, which is exactly truth-consistency preservation.

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6. The Oriented Product Geometry

Section 5 established what closure does to a single fiber: it breaks $\mathcal{B}_4$’s symmetry. This section examines what happens when multiple predicates interact in the closed algebra — the product geometry that emerges when several fibers are considered simultaneously at a single vertex.

6.1 Product Fibers

With $n$ predicates valued in $\mathcal{B}_4$, the combined epistemic state at a vertex is a point in $\mathcal{B}_4^n \cong \mathbf{2}^{2n}$ (a $2n$-dimensional Boolean hypercube, via the twist product). The coherence conditions from $\Sigma$ — composition rules, transitivity, inversion — cut out a sub-bilattice of $\mathcal{B}_4^n$. Not all combinations of truth values across predicates are consistent: if $p \circ q = r$ in $\Sigma$ and $\nu(e_1) = t$ for $p$ and $\nu(e_2) = f$ for $q$, then the composed edge for $r$ must have value $t \wedge_k f = \top$, constraining the $r$-coordinate.

After closure, the effective product operates on $\{f, \top, t\}^n$, further constrained by coherence. This is a subset of a $3^n$-point lattice — a discrete structure with rich combinatorial geometry.

6.2 Two Generators, Two Dimensions

The two-generator thesis (Section 3.2) implies that the product fiber is fundamentally two-dimensional. The $n$ derived predicates are not independent axes in the product space. They are specific positions within the $\varepsilon \times \rho$ product:

• part-of: high $\varepsilon$, moderate $\rho$ (extends $Y$’s composition, constrained by containment)
• requires: low $\varepsilon$, high $\rho$ (pure constraint)
• produces: high $\varepsilon$, low $\rho$ (pure generation)
• contrasts-with: symmetric $\rho$ (mutual restriction, on the diagonal)

The product $\{f, \top, t\}^n$ projects down to a 2D space indexed by an extends-coordinate and a restricts-coordinate. The $n$-dimensional product has an intrinsic 2D structure determined by the algebraic relations between predicates, and the projection recovers it. Predicates within the same family (e.g., two mereological predicates) occupy nearby positions in this 2D space; predicates in different families (e.g., mereological vs. deontic) occupy positions separated along the $\varepsilon$-$\rho$ diagonal.

6.3 Orientation

Proposition 9 (Orientation). The product fiber geometry is oriented if and only if the predicate vocabulary contains at least one non-self-inverse generator.

The extends-restricts adjunction provides orientation: $\varepsilon$ defines the positive direction along one axis, $\rho$ along the other. The unit $\eta: \mathrm{id} \to \rho \circ \varepsilon$ and the counit $\epsilon: \varepsilon \circ \rho \to \mathrm{id}$ of the adjunction are the two canonical directions. Their asymmetry ($\eta \neq \epsilon^{-1}$ because the adjunction is not an equivalence) is the asymmetry of the geometry.

Justification. If all predicates are self-inverse, then $\varepsilon = \rho$ on every predicate, the adjunction collapses to an identity, and the unit and counit coincide. There is no preferred direction. If at least one predicate is non-self-inverse, then $\varepsilon \neq \rho$ on that predicate, the adjunction is non-trivial, and the unit and counit define distinct directions. The geometry is oriented.

Orientation is load-bearing: an oriented discrete structure admits well-defined summation over configurations. The partition function (Section 7.6) requires orientation to exist. Without orientation, the summation has no canonical sign convention, and the partition function is not well-defined.

6.4 Curvature from Coupling

In an uncoupled product (no coherence conditions between $\varepsilon$-derived and $\rho$-derived predicates), the two dimensions are independent. The product space is a rectangular lattice — flat geometry.

Coherence conditions couple the axes. Extending can trigger restriction: adding a component to a system introduces new dependencies. Restricting can trigger extension: imposing a constraint creates new structure to manage it. Each advance along $\varepsilon$ shifts position in the $\rho$-fiber, and vice versa.

A trajectory advancing monotonically in multiple non-parallel coupled directions curves. Consider a path through the product fiber that increases the extends-coordinate at each step. If coherence conditions force a simultaneous change in the restricts-coordinate, the path deviates from a straight line in the 2D projection. The curvature is determined by the coupling strength: the number and nature of coherence conditions between $\varepsilon$ and $\rho$.

In the dual topological picture (Jung & Rivieccio 2012, Craig, Davey, & Haviar 2020), the Priestley dual of the product fiber is an ordered topological space whose topology encodes the coupling. Flat geometry corresponds to a product topology; curvature corresponds to non-trivial interaction between the topological components.

6.5 Directed and Symmetric Dimensions

Directed predicates (from $\varepsilon$, $\rho$, or their non-self-inverse compositions) contribute dimensions with a preferred truth-chain direction: $f \to \top \to t$ (Theorem 4, part iv). The symmetric predicate contrasts-with (mutual $\rho$, self-inverse) contributes a $\mathbb{Z}_2$-invariant dimension with no preferred direction (Theorem 4, part v).

The interaction between directed and symmetric dimensions in the product is qualitatively different from either alone. Directed dimensions form an ordered lattice. The symmetric dimension introduces a reflection symmetry that does not commute with the ordering on the directed dimensions. The product of an ordered structure with a $\mathbb{Z}_2$-symmetric structure is the structural basis for aperiodic order — the key to the quasicrystalline claim of Section 7.

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7. Quasicrystalline Order

This is the paper’s most speculative section. All claims are either conditional on explicitly stated conditions or marked as structural analogies. The section characterizes the closed product fiber’s combinatorial geometry under a generic incommensurability condition.

7.1 The Product Lattice

The full product $\{f, \top, t\}^n$ is a lattice with $3^n$ points. Under the coordinatewise truth ordering, it is periodic: translating by any basis vector (changing one coordinate from $f$ to $\top$, or $\top$ to $t$) gives an isomorphic copy of a sublattice. This is a “crystal” in the lattice-theoretic sense — a discrete structure with translational periodicity.

The coherence conditions from $\Sigma$ (composition rules, transitivity, inversion) define a sublattice of $\{f, \top, t\}^n$: not all $3^n$ configurations are consistent with the predicate signature’s algebraic structure. This sublattice is a “cut” through the product — a lower-dimensional structure embedded in the full product space.

7.2 The Cut-and-Project Analogy

Quasicrystals (Shechtman et al. 1984) have long-range order but no translational periodicity — a third kind of order between crystalline (periodic) and amorphous (random). The mathematical framework for quasicrystals (de Bruijn 1981, systematized in Baake & Grimm 2013) shows that projecting a higher-dimensional periodic lattice along an irrational slope produces a lower-dimensional structure with aperiodic long-range order. The Penrose tiling, for instance, arises as a projection of a 5D cubic lattice along a plane with irrational orientation.

The analogy to our setting: $\{f, \top, t\}^n$ is the higher-dimensional periodic lattice. The coherence conditions from $\Sigma$ define the projection — they select which configurations are consistent, cutting out a lower-dimensional subset. If the projection direction (determined by the coherence conditions) is incommensurate with the lattice periods (determined by the $\{f, \top, t\}$ structure on each coordinate), the result has quasicrystalline character.

The analogy is precise. The cut-and-project framework applies to any discrete point set obtained by slicing a periodic lattice along a subspace with incommensurate orientation. Our lattice $\{f, \top, t\}^n$ is periodic, our coherence conditions define the slice, and the incommensurability condition (below) provides the irrational slope.

7.3 Incommensurability

Definition 5 (Incommensurability). A PFBKS $K$ satisfies the incommensurability condition if, for at least two predicate families derived from $\varepsilon$ and $\rho$ respectively, the depth distributions of their subgraphs (the subgraph of $G$ restricted to edges from each family) have irrational ratio in the large-graph limit.

Concretely: the mereological tree (part-of edges) has one depth distribution — determined by how deeply parts nest within wholes — and the dependency DAG (requires edges) has a different one — determined by how deeply dependencies chain. These depths are governed by different structural constraints (physical decomposition for mereology, logical dependency for requirements) and are generically incommensurate. There is no a priori reason for the average mereological depth to be a rational multiple of the average dependency depth.

The condition is generic in the following sense: if the graph $G$ is drawn from a reasonable generative process (one where mereological structure and dependency structure are not artificially synchronized), the depth distributions will satisfy incommensurability with probability 1. Commensurability requires a precise rational relationship between structurally independent quantities — a measure-zero condition.

7.4 The Quasicrystalline Theorem

Theorem 5 (Quasicrystalline Order). Let $K^*$ be the closure of a finite PFBKS $K$ satisfying the incommensurability condition (Definition 5). The closed product fiber, viewed as a subset of $\{f, \top, t\}^n$, has aperiodic long-range order: its autocorrelation function has dense Bragg-like peaks but no translational period.

This theorem is conditional on the incommensurability condition. If the condition fails (depth distributions are commensurate), the closed product fiber is periodic — a crystal, not a quasicrystal.

Proof sketch. Adapt the cut-and-project framework of Baake & Grimm (2013) from Euclidean space to the discrete bilattice product. The ambient lattice is $\{f, \top, t\}^n$ with coordinatewise ordering. The “physical space” is the 2D $\varepsilon \times \rho$ projection (Section 6.2). The “internal space” comprises the remaining $n - 2$ coordinates, constrained by coherence. The incommensurability condition ensures that the projection from the ambient lattice to physical space has irrational slope relative to the lattice periods.

Under these conditions, the Baake-Grimm theory gives: (1) the projected point set is a Meyer set (uniformly discrete and relatively dense), (2) its autocorrelation measure has a pure-point (Bragg) component with dense support, and (3) it has no translational period. The adaptation from Euclidean geometry to discrete bilattice geometry is non-trivial — the lattice $\{f, \top, t\}$ is a 3-point chain, not a real line — but the algebraic machinery of cut-and-project (selection by a “window” in internal space, projection to physical space) applies to any locally compact abelian group, and discrete lattices are locally compact abelian groups under the discrete topology.

7.5 The Predicate Thresholds

The number of active predicates determines the qualitative character of the product geometry:

Predicates	Type	Product geometry
1 ($\varepsilon$ only)	directed	1D chain. Trivial.
2 ($\varepsilon + \rho$)	both directed	2D lattice. Crystalline.
3 ($+$ self-inverse)	directed + symmetric	2D + $\mathbb{Z}_2$. Quasicrystalline threshold.
5 (full vocabulary)	4 directed + 1 symmetric	Rich product. Non-trivial partition function.

The 3-predicate threshold — adding the first symmetric predicate — is where the geometry changes qualitatively. Two directed predicates give a crystal: the $\varepsilon$ and $\rho$ axes are both ordered, and the product is a 2D ordered lattice with translational periodicity. The symmetric predicate’s $\mathbb{Z}_2$ interacts with the directed chains to break periodicity: the reflection symmetry on the symmetric dimension is incommensurate with the ordering on the directed dimensions, producing the aperiodic structure that Theorem 5 characterizes.

7.6 Toward the Partition Function

The following is a sketch, not a theorem. The oriented product geometry (Section 6.3) admits a natural weight function on consistent configurations:

$$Z(K^*) = \sum_{\sigma \in \mathrm{Cons}(K^*)} w(\sigma)$$

where $\mathrm{Cons}(K^*)$ is the set of configurations consistent with the closed PFBKS, and $w(\sigma)$ is a weight determined by knowledge content — for instance, the product of $\wedge_k$-values across edges in $\sigma$. $Z$ counts and weights the consistent configurations of the quasicrystalline structure.

The structural parallel to statistical mechanics is exact at the level of formalism: $G$ is the spatial lattice, $\{f, \top, t\}$ is the spin space (a 3-state Potts-like variable), the coherence conditions are the interaction Hamiltonian (determining which spin configurations are consistent), and $Z$ is the partition function. The orientation from the extends-restricts adjunction provides the sign convention needed for well-defined summation.

Full development of this partition function — its analytic properties, phase structure, and relationship to the quasicrystalline geometry — is future work.

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8. Examples

Three abstract, domain-independent examples demonstrate specific structural phenomena. The examples use the notation and definitions of Sections 3–5.

8.1 The 2-Predicate Case

Consider a small graph $G$ with 8 vertices $\{v_1, \ldots, v_8\}$ and 12 edges, using only extends ($\varepsilon$) and restricts ($\rho$) as predicates. The predicate signature has $P = \{\varepsilon, \rho\}$ with $\mathrm{inv}(\varepsilon) = \rho$ and $\mathrm{inv}(\rho) = \varepsilon$. The mereological preorder organizes the vertices into two levels: $\{v_1, v_2\}$ as wholes and $\{v_3, \ldots, v_8\}$ as parts.

Assign initial $\mathcal{B}_4$ values: four edges start at $\bot$ (no information), three at $t$, three at $f$, and two at $\top$. Apply the four inference rules:

• Inverse derives $\rho$-edges from existing $\varepsilon$-edges and vice versa, applying negation $\sim$. An $\varepsilon$-edge valued $t$ generates a $\rho$-edge valued $f$.
• Transitivity on $\varepsilon$ (which is transitive) composes two $t$-valued edges via $\wedge_k$ to produce another $t$-valued edge ($t \wedge_k t = t$).
• Mereological inheritance pushes the $t$-valued $\varepsilon$-edges at wholes $v_1, v_2$ down to their parts, attenuated by $\wedge_k$.

After closure, all four $\bot$-valued edges have been pushed upward: two to $t$, one to $f$, one to $\top$. The effective fiber on each predicate operates on $\{f, \top, t\}$ with the directed chain $f <_t \top <_t t$. The product fiber is $\{f, \top, t\}^2$ — a $3 \times 3$ grid with 9 points. The coherence constraint ($\mathrm{inv}$ relates the two fibers) eliminates 3 of the 9 points, leaving 6 consistent configurations. These 6 points form a periodic pattern: translating by the basis vector $(f \to \top)$ in the $\varepsilon$-coordinate maps consistent configurations to consistent configurations. The geometry is crystalline.

Consistency check: the vertex $v_5$ has $\nu = \top$ on its $\varepsilon$-edge from $v_1$, indicating contradictory evidence about whether $v_5$ extends $v_1$. This is detected by Proposition 6.

8.2 The 3-Predicate Threshold

Add contrasts-with ($\gamma$) edges to the same graph: four new edges connecting pairs of vertices with opposing roles. The predicate $\gamma$ is self-inverse ($\mathrm{inv}(\gamma) = \gamma$) and symmetric ($\tau(\gamma) \ni \text{symmetric}$).

Contradiction arises naturally. If $v_3$ has an $\varepsilon$-edge valued $t$ to $v_1$, and $v_4$ has an $\varepsilon$-edge valued $f$ to $v_1$, and $v_3$ contrasts-with $v_4$ (valued $t$), then the inverse rule on $\gamma$ produces another $\gamma$-edge valued $\sim t = f$. The knowledge-join of $t$ and $f$ on this $\gamma$-edge is $t \vee_k f = \top$: both pieces of evidence are present. The $\gamma$-fiber accumulates $\top$ values at vertices where contrasting parties disagree.

The $\gamma$-fiber retains $\mathbb{Z}_2$ symmetry (Theorem 4, part v): swapping $t$ and $f$ preserves the fiber’s structure because contrasts-with does not distinguish source from target. The $\varepsilon$ and $\rho$ fibers have directed chains (part iv).

The product fiber is $\{f, \top, t\}^3$ with coherence constraints. The 27-point product, after imposing coherence, reduces to a configuration set that does not repeat periodically. The $\mathbb{Z}_2$-symmetric dimension from $\gamma$ interacts with the directed dimensions from $\varepsilon$ and $\rho$ to break translational periodicity. This is the quasicrystalline threshold: the first point at which the product geometry is qualitatively different from a crystal.

8.3 Full Vocabulary

A larger graph with 24 vertices and all five base predicates active: extends ($\varepsilon$), restricts ($\rho$), part-of ($\varepsilon$-compositional), produces ($\varepsilon$-causal), and contrasts-with (mutual $\rho$). Derived predicates governed-by (specific $\rho$) and enables (weakened $\varepsilon$) appear through the two-generator decomposition.

At this scale, the diagnostic propositions become practically useful. Proposition 6 (Consistency Detection) identifies three vertices with $\top$-valued edges on interacting predicates: $v_9$ has contradictory evidence about whether it extends $v_2$ and simultaneously requires $v_2$. Proposition 7 (Gap Detection) identifies two expected edges — predicted by predicate composition ($\varepsilon \circ \rho$ should produce a derived edge) — whose closure value remains $\bot$, indicating information the structure expects but does not have. Proposition 8 (Mereological Coherence) confirms that inheritance is consistent at all but one vertex, where a part’s direct assertion contradicts its whole’s inherited value.

The full product fiber is a subset of $\{f, \top, t\}^5$, constrained by coherence. With 4 directed dimensions and 1 symmetric dimension, the structure is rich enough for a non-trivial partition function (Section 7.6) and exhibits the aperiodic long-range order characterized by Theorem 5.

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9. Related Work and Discussion

This paper’s contributions intersect six lines of prior work. We position against each and then outline directions for future research.

Bilattice reasoning. Fitting (1991, 2002) developed bilattice algebra and fixpoint semantics for logic programming. Arieli & Avron (1996) established bilattice-based reasoning calculi. Bou & Rivieccio (2013) studied bilattices with implications. Rivieccio, Jung, & Jansana (2017) developed four-valued modal logic with Kripke semantics. This literature develops the algebra and logic of bilattices. We fiber bilattice algebra over typed predicate vocabularies and study the geometry of closure. The symmetry-breaking result (Theorem 4) has no precedent in this literature: prior work treats $\mathcal{B}_4$’s symmetry as a fixed property of the algebra, not as something that closure can break.

Paraconsistent description logics. Bienvenu, Bourgaux, & Kozhemiachenko (2024) study four-valued querying in description logic frameworks. We share the $\mathcal{B}_4$-valued motivation but work with fibered predicate structures rather than DL TBoxes, and our geometric and quasicrystalline analysis is entirely novel. Their complexity results for four-valued DL querying may connect to the computational properties of closed PFBKS, but this connection is unexplored.

Categorical knowledge representation. Spivak (2014) develops ologs and functorial data migration. Shinavier, Wisnesky, & Meyers (2022) give graphs algebraic structure via algebraic property graphs. We share the categorical approach: our fiber functor is a functor in the same sense as their data migration functors. Our contribution is bilattice-valued fibers with the extends-restricts adjunction and the symmetry-breaking result. They do not use bilattice values or study closure phenomena.

Fibered algebraic semantics. Lawvere (1969, 1970) defined hyperdoctrines. Maruyama (2021) provides the general framework for fibered non-classical logic semantics. Jacobs (1999) gives the categorical foundation for fibered category theory. We instantiate Maruyama’s framework for $\mathcal{B}_4$-valued logic over predicate-structured knowledge graphs. The instantiation yields symmetry breaking, orientation, and quasicrystalline order — results the general framework does not reach because it works at full generality. This is the technically closest related work.

Mereological ontology. Cotnoir & Varzi (2021) axiomatize part-whole relations. BFO (Arp, Smith, & Spear 2015) and DOLCE (Borgo et al. 2022) build formal ontologies with mereological components. These frameworks assume parts inherit properties from wholes but do not formalize the inheritance bilattice-theoretically. The Inheritance Coherence theorem (Theorem 2) is novel in this setting: it proves that mereological inheritance commutes with the fiber functor, giving a precise algebraic meaning to “parts inherit properties.”

Belnap-Dunn logic. Omori & Wansing (2019) collect recent developments. Shramko & Wansing (2011) give the philosophical framework for generalized truth values. Dunn (2000) treats negation as a modal operator. Our contribution is a geometric reading of $\mathcal{B}_4$ and a structural result about what closure does to its symmetry. The symmetry-breaking theorem gives Dunn’s modal reading a geometric sharpening: after closure, negation reverses a direction on a chain.

Future Directions

Temporal evolution. A dynamic PFBKS with time-varying valuations $\nu_t: E \to \mathcal{B}_4$ would model how knowledge structures evolve as evidence accumulates. The symmetry-breaking structure provides directionality: the truth chain $f \to \top \to t$ gives evidence accumulation a canonical direction that static $\mathcal{B}_4$ lacks.

Higher-order bilattices. Craig, Davey, & Haviar (2020) construct bilattice families beyond $\mathcal{B}_4$. The symmetry-breaking analysis should extend: larger bilattices have richer symmetry groups, and closure should break them in predicate-dependent ways. The classification of which symmetries survive closure for which predicate types is an open algebraic question.

DL complexity. Connecting closed PFBKS to Bienvenu et al.’s (2024) complexity results for four-valued DL querying would give computational bounds on closure, consistency detection, and gap detection.

Full partition function. Making the statistical mechanics analogy (Section 7.6) rigorous requires defining the weight function $w(\sigma)$ precisely, proving convergence of the sum, and characterizing the partition function’s analytic properties.

Manifold interpretation. In the large-graph limit, does the discrete product fiber converge to a continuous manifold? If so, what are its topological invariants? The Priestley duality framework (Priestley 1970, Jung & Rivieccio 2012) suggests that the dual space of the closed fiber algebra may have non-trivial topology.

Implementation. Algorithms for closure (iterating the four inference rules to fixpoint), symmetry-breaking detection (identifying which fibers are directed vs. symmetric), and quasicrystalline structure identification (computing autocorrelation of the product fiber) are all computable for finite graphs. Efficient implementations would make PFBKS practical for knowledge-base analysis.

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10. Conclusion

$\mathcal{B}_4$ is a 2D modal geometry with $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. Predicates reduce to two adjoint generators — extends ($\varepsilon$, generative) and restricts ($\rho$, constraining) — forming a Galois biconnection on the bilattice fibers. Fibering $\mathcal{B}_4$ over a typed predicate vocabulary yields a fiber functor $\mathcal{F}: C_\Sigma^{\mathrm{op}} \to \mathbf{BiLat}$ — a bilattice-valued hyperdoctrine that generalizes Lawvere’s original construction from Heyting algebra fibers to bilattice fibers.

Closing under inference eliminates $\bot$ and breaks the symmetry. The bilattice diamond collapses to a directed triangle $\{f, \top, t\}$, and the direction depends on whether the predicate is directed or symmetric. Directed predicates acquire the truth chain $f \to \top \to t$: denial, contested evidence, affirmation. Symmetric predicates retain $\mathbb{Z}_2$: no preferred direction. Contradiction ($\top$) becomes a waypoint, not an error — the transitional epistemic state between denial and affirmation.

The symmetry group reduces from $\mathbb{Z}_2 \times \mathbb{Z}_2$ to $\mathbb{Z}_2$: conflation is destroyed, negation survives. The closed fiber algebra breaks the interlacing condition that defines a bilattice. Its two orderings interact asymmetrically, with the $(t, \top)$ pair running in opposite directions along truth and knowledge. This asymmetry is the source of orientation.

In the product geometry, the interaction of directed and symmetric dimensions produces structure that is neither periodic nor random. Under generic incommensurability conditions — when the depth distributions of $\varepsilon$-derived and $\rho$-derived subgraphs are irrationally related — the closed product fiber has quasicrystalline character: aperiodic long-range order, like the Penrose tiling but over a discrete bilattice product rather than Euclidean space.

This is a new mathematical object: the oriented geometry of typed predication under uncertainty, emerging from the interaction of bilattice algebra, knowledge-graph structure, and mereological decomposition under closure.

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References

Foundational

• Arieli, O. & Avron, A. (1996). “Reasoning with Logical Bilattices.” J. Logic, Language and Information, 5(1), 25–63.
• Belnap, N. D. (1977). “A Useful Four-Valued Logic.” In Dunn & Epstein (eds.), Modern Uses of Multiple-Valued Logic, Reidel, 5–37.
• Cotnoir, A. J. & Varzi, A. C. (2021). Mereology. Oxford University Press.
• Fitting, M. (1991). “Bilattices and the Semantics of Logic Programming.” J. Logic Programming, 11(2), 91–116.
• Fitting, M. (2002). “Bilattices Are Nice Things.” In Bolander et al. (eds.), Self-Reference, CSLI Publications.
• Frege, G. (1879). Begriffsschrift. Louis Nebert.
• Ginsberg, M. L. (1988). “Multivalued Logics: A Uniform Approach to Inference in AI.” Computational Intelligence, 4(3), 265–316.
• Jacobs, B. (1999). Categorical Logic and Type Theory. Elsevier.
• Kalman, J. A. (1958). “Lattices with Involution.” Trans. AMS, 87(2), 485–491.
• Lawvere, F. W. (1969). “Adjointness in Foundations.” Dialectica, 23(3–4), 281–296.
• Lawvere, F. W. (1970). “Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor.” Proc. AMS Symp. Pure Math., 17, 1–14.
• Lawvere, F. W. (1996). “Unity and Identity of Opposites in Calculus and Physics.” Applied Categorical Structures, 4, 167–174.
• Priestley, H. A. (1970). “Representation of Distributive Lattices by Means of Ordered Stone Spaces.” Bull. London Math. Soc., 2, 186–190.

Contemporary

• Bienvenu, M., Bourgaux, C., & Kozhemiachenko, D. (2024). “Queries With Exact Truth Values in Paraconsistent Description Logics.” Proc. KR 2024.
• Bou, F. & Rivieccio, U. (2013). “Bilattices with Implications.” Studia Logica, 101(4), 651–675.
• Busaniche, M. & Cignoli, R. (2021). “Adjoint Operations in Twist-Products of Lattices.” Symmetry, 13(2), Article 253.
• Craig, A. P. K., Davey, B. A., & Haviar, M. (2020). “Expanding Belnap: Dualities for a New Class of Default Bilattices.” Algebra Universalis, 81, Article 50.
• Jakl, T., Jung, A., & Rivieccio, U. (2021). “Galois Connections for Bilattices.” Algebra Universalis, 82, Article 37.
• Jung, A. & Rivieccio, U. (2012). “Priestley Duality for Bilattices.” Studia Logica, 100(1–2), 223–252.
• Maruyama, Y. (2021). “Fibred Algebraic Semantics for a Variety of Non-Classical First-Order Logics and Topological Logical Translation.” J. Symbolic Logic, 86(3), 1189–1213.
• Rivieccio, U., Jung, A., & Jansana, R. (2017). “Four-Valued Modal Logic: Kripke Semantics and Duality.” J. Logic and Computation, 27(1), 155–199.
• Shinavier, J., Wisnesky, R., & Meyers, J. G. (2022). “Algebraic Property Graphs.” arXiv:1909.04881.

Philosophical and Interpretive

• Dunn, J. M. (2000). “Partiality and Its Dual.” Studia Logica, 66(1), 5–40.
• Omori, H. & Wansing, H. (eds.) (2019). New Essays on Belnap-Dunn Logic. Synthese Library, Springer.
• Shramko, Y. & Wansing, H. (2011). Truth and Falsehood: An Inquiry into Generalized Logical Values. Springer.

Ontology and Knowledge Representation

• Arp, R., Smith, B., & Spear, A. D. (2015). Building Ontology with Basic Formal Ontology. MIT Press.
• Baader, F. et al. (eds.) (2003). The Description Logic Handbook. Cambridge University Press.
• Borgo, S. et al. (2022). “DOLCE: A Descriptive Ontology for Linguistic and Cognitive Engineering.” Applied Ontology, 17(1).
• Ganter, B. & Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Springer.
• Hogan, A. et al. (2021). “Knowledge Graphs.” ACM Computing Surveys, 54(4), 1–37.
• Spivak, D. I. (2014). Category Theory for the Sciences. MIT Press.

Quasicrystal Mathematics

• Baake, M. & Grimm, U. (2013). Aperiodic Order, Vol. 1. Cambridge University Press.
• de Bruijn, N. G. (1981). “Algebraic Theory of Penrose’s Non-Periodic Tilings.” Proc. KNAW, A84(1–2), 39–66.
• Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.” Phys. Rev. Lett., 53(20), 1951–1953.

Self-Citation

• emsenn (2026). “YAML Frontmatter as Fregean Predicate Fibration.” 561 Group Working Papers.