Curved Boolean Logic (CBL) Overview

• Curved Boolean Logic is a contextual extension of propositional logic where local truth assignments hold but global consistency fails due to curvature obstructions analogous to coordinate systems in geometry.
• The framework unifies sheaf and exclusivity-hypergraph semantics to capture the transition from locally classical valuations to globally inconsistent structures, enabling new algorithmic operators.
• Its proof theory features a context-aware sequent calculus with overlap rules and flat-limit conservativity, offering insights into satisfiability, complexity, and robustness under noise.

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Curved Boolean Logic (CBL) is a contextual generalization of propositional logic in which local truth assignments need not extend to a single global valuation. It preserves standard Boolean reasoning inside each context, but relaxes the classical assumption that every compatible family of local valuations is induced by one global ledger $g:V\to\{0,1\}$. The resulting failure of global extendability is termed curvature, by analogy with geometry: flat systems admit global coordinates, whereas curved systems exhibit obstructions revealed only when local data are transported around overlaps. In the formulation of "Curved Boolean Logic: A Contextual Generalization of Propositional Logic with Algorithmic Consequences" (Liechtenstein, 6 Oct 2025), this idea is given equivalent sheaf and exclusivity-hypergraph semantics, a context-aware proof calculus, an associated satisfiability problem, and curvature-aware algorithmic operators.

1. Conceptual basis and scope

CBL is proposed as a conservative but strictly more expressive extension of classical propositional logic designed for contextual situations: each local set of propositions can be assigned classical truth values consistently, yet no single global truth assignment is consistent with all of them simultaneously (Liechtenstein, 6 Oct 2025). The motivating phenomenon is familiar from quantum foundations, including Kochen–Specker, Bell, KCBS, and the Mermin square, where outcomes depend on the jointly measured set of observables rather than on a context-independent valuation.

The central contrast is between flat Boolean logic and curved Boolean logic. In the flat case, there exists a single global valuation $g:V\to\{0,1\}$ whose restrictions recover all local assignments. In the curved case, distinct contexts possess local valuations that are compatible on overlaps, but no global valuation exists. The paper explicitly compares this to geometry: Euclidean space admits a global coordinate system, whereas curved space appears locally flat but exhibits an obstruction under transport around loops (Liechtenstein, 6 Oct 2025).

CBL is not presented as a paraconsistent logic. Contradictions are not tolerated locally; rather, they arise as structural obstructions to gluing together locally classical information. This matters for interpretation: the formalism does not abandon classical truth-functional reasoning within a context, but changes the global semantics of how contexts are coordinated.

A recurring misconception is therefore that CBL licenses inconsistent truth values. The formulation rejects that reading. Local reasoning remains strictly classical, and the novelty lies in the possibility that the global Boolean ledger breaks down despite local compatibility (Liechtenstein, 6 Oct 2025).

2. Formal semantics: contexts, global sections, and curvature

A CBL model begins with a finite set of propositional variables $V$ and a finite family of nonempty subsets $\mathcal C=\{C_i\}_{i\in I}$, called contexts, such that each variable appears in at least one context. For a context $C$, a local valuation is a Boolean assignment

$\ell_C:C\to\{0,1\}.$

A family $\{\ell_C\}_{C\in\mathcal C}$ is compatible when

$\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$

for all contexts $C,C'$. A global valuation is a map

$g:V\to\{0,1\}$

whose restriction to each context matches the specified local valuation (Liechtenstein, 6 Oct 2025).

The distinction between flat and curved systems is then semantic. A context system is flat if every compatible family admits a global valuation; otherwise it is curved. The paper abstracts this by a curvature functional

$g:V\to\{0,1\}$0

such that $g:V\to\{0,1\}$1 iff every compatible family admits a global section, and $g:V\to\{0,1\}$2 is monotone under context refinement. A concrete instance is given cohomologically: $g:V\to\{0,1\}$3 with $g:V\to\{0,1\}$4 the context poset and $g:V\to\{0,1\}$5 a presheaf of $g:V\to\{0,1\}$6-valuations under XOR (Liechtenstein, 6 Oct 2025).

The sheaf-theoretic semantics follows Abramsky–Brandenburger. Contexts form a poset category $g:V\to\{0,1\}$7 ordered by inclusion, and a presheaf $g:V\to\{0,1\}$8 is defined by $g:V\to\{0,1\}$9, with restriction maps given by function restriction. Compatible families are matching families of sections, and global valuations are global sections in $V$0. Existence of a global section recovers standard Boolean semantics; nonexistence corresponds to contextuality or curvature (Liechtenstein, 6 Oct 2025).

The paper also gives an equivalent exclusivity-hypergraph semantics. Variables become vertices, contexts become hyperedges, and overlaps are encoded by duplicating a variable per context and imposing equality constraints on copies. In that picture, curvature corresponds to cycles whose joint constraints prevent a consistent assignment to all duplicates. The equivalence theorem states that, for finite context systems, sheaf semantics and exclusivity-hypergraph semantics are equivalent: a compatible family admits a global section iff the exclusivity formulation admits an assignment consistent on overlaps (Liechtenstein, 6 Oct 2025).

Minimal witnesses of nonflatness are called curved cores: subfamilies $V$1 with $V$2 such that every proper subfamily has $V$3. The KCBS pentagon and the Mermin square are given as archetypal examples.

3. Proof theory and the flat limit

CBL’s proof theory is a context-aware sequent calculus. Sequents are annotated by a context: $V$4 meaning that within context $V$5, $V$6 is inferred from assumptions $V$7. Inside a fixed context, the calculus is standard propositional sequent calculus with weakening, contraction, cut, and the usual logical rules for $V$8, subject to the requirement that formulas mention only variables in $V$9 (Liechtenstein, 6 Oct 2025).

What distinguishes CBL are the overlap rules. The first is the transport rule: $\mathcal C=\{C_i\}_{i\in I}$0 which expresses that consequences formulated solely on the overlap $\mathcal C=\{C_i\}_{i\in I}$1 may be moved between contexts. The second is the consistency rule: $\mathcal C=\{C_i\}_{i\in I}$2 which fuses complementary contradictions derived in different contexts into a contradiction on the union (Liechtenstein, 6 Oct 2025).

These rules are the proof-theoretic analogue of curvature. Contradictions do not emerge because inconsistent local truth assignments are allowed; they emerge because locally classical derivations, transported across overlaps, fail to admit a globally coherent realization. The soundness theorem states that if $\mathcal C=\{C_i\}_{i\in I}$3 is derivable in CBL, then every local valuation $\mathcal C=\{C_i\}_{i\in I}$4 satisfying $\mathcal C=\{C_i\}_{i\in I}$5 also satisfies $\mathcal C=\{C_i\}_{i\in I}$6, and with (OVL) and (CONS) the system is sound with respect to the sheaf semantics (Liechtenstein, 6 Oct 2025).

A structurally important theorem is flat-limit conservativity: if $\mathcal C=\{C_i\}_{i\in I}$7, then $\mathcal C=\{C_i\}_{i\in I}$8 is derivable in CBL iff the same sequent is derivable in classical propositional logic. In that limit, the overlap machinery is conservative and CBL reduces exactly to standard propositional logic (Liechtenstein, 6 Oct 2025). This theorem sharply delimits the intended role of CBL: it is not a replacement for classical logic in ordinary flat cases, but an extension that becomes nontrivial only when the context system is curved.

The paper identifies completeness in curved settings as an open problem. That omission is consequential: soundness and conservativity are established, but a full proof-theoretic characterization of the curved case remains to be derived (Liechtenstein, 6 Oct 2025).

4. Satisfiability, complexity, and curvature-aware operators

CBL induces an extended satisfiability problem, CBL-SAT. An instance consists of a context system $\mathcal C=\{C_i\}_{i\in I}$9 and a set of clauses $C$0, each attached to a context $C$1. The decision problem asks whether there exists a compatible family of local valuations $C$2 such that every clause $C$3 is satisfied by its designated local valuation $C$4 (Liechtenstein, 6 Oct 2025).

This differs from classical SAT in two precise ways. First, satisfaction is local to contexts, with compatibility constraints on overlaps rather than a single global assignment. Second, the structure of the context family $C$5 carries information beyond the clause set itself. Standard SAT is recovered by taking a single context $C$6 (Liechtenstein, 6 Oct 2025).

The basic complexity classification is classical in form but structurally informative: CBL-SAT is in NP, and SAT reduces to it by taking one context, so the problem is NP-complete. The paper is explicit that CBL does not yield a polynomial-time satisfiability algorithm in general; any algorithmic advantage is structural, arising from earlier pruning on some curved instances rather than from improved worst-case complexity (Liechtenstein, 6 Oct 2025). It also notes tractable subclasses, for example when the context hypergraph has bounded treewidth and each clause is fully contained in some bag, in which case standard dynamic programming on tree decompositions gives polynomial-time algorithms.

Two operational operators are central. CBL-AC (Curved Arc-Consistency) generalizes arc-consistency so that a value $C$7 is removed if every local assignment involving $C$8 within some minimal curved face leads to an overlap violation. CBL-CONS (Curved Overlap Consistency) operationalizes the proof rule (CONS): if local proof engines in overlapping contexts derive contradiction from $C$9 in one context and from $\ell_C:C\to\{0,1\}.$0 in the other, the solver emits a global cut on $\ell_C:C\to\{0,1\}.$1 with a dual witness (Liechtenstein, 6 Oct 2025).

The paper proves two qualitative properties. On flat instances, the operators are conservative: CBL-AC collapses to standard arc-consistency or yields no extra cuts. On curved instances, they are stronger on curved cores: if a minimal curved face exists, then CBL-AC eliminates at least one domain value classical AC would not, or CBL-CONS generates a new global cut (Liechtenstein, 6 Oct 2025). The sketched CBL-Solve procedure extends CDCL-like search with context-aware propagation, early overlap checks, curvature cuts, and learning of overlap-derived clauses.

The KCBS pentagon is the canonical toy example. Under the CBL-Solve sketch, selecting $\ell_C:C\to\{0,1\}.$2 propagates around the pentagon until overlap constraints force $\ell_C:C\to\{0,1\}.$3, so a global contradiction is detected early, before exploring many unrelated branches (Liechtenstein, 6 Oct 2025). A plausible implication is that CBL’s algorithmic utility lies where clause-local syntax alone obscures global obstruction patterns induced by overlaps.

5. Canonical examples, robustness, and statistical testing

Two contextual structures organize the intuition of CBL. In the KCBS pentagon, variables $\ell_C:C\to\{0,1\}.$4 are arranged in a 5-cycle with contexts $\ell_C:C\to\{0,1\}.$5. Each context enforces exclusivity and “exactly one” by the clauses

$\ell_C:C\to\{0,1\}.$6

Each context is locally satisfiable, but on an odd cycle the requirement “exactly one true per edge” is globally impossible. Hence every context admits a consistent local valuation while no global valuation exists, and $\ell_C:C\to\{0,1\}.$7 (Liechtenstein, 6 Oct 2025).

The Mermin square presents the same phenomenon with nine variables arranged in a $\ell_C:C\to\{0,1\}.$8 grid, contexts taken as rows and columns, and parity constraints that are locally satisfiable but globally contradictory when row and column products are compared. In both cases, CBL treats contextuality as logical curvature rather than as an external anomaly (Liechtenstein, 6 Oct 2025).

The framework also formalizes robustness. Axiom A2 introduces $\ell_C:C\to\{0,1\}.$9-bounded perturbations, bounding how much perturbations can change empirical statistics on any face per unit mass. Three noise regimes are studied: i.i.d. flips with rate $\{\ell_C\}_{C\in\mathcal C}$0, AR(1)-correlated noise with effective flip rate $\{\ell_C\}_{C\in\mathcal C}$1, and adversarially bounded perturbations controlling the $\{\ell_C\}_{C\in\mathcal C}$2-norm of a folded count perturbation (Liechtenstein, 6 Oct 2025). The stated bounds imply that legal inferences are Lipschitz-stable under the stipulated perturbation budgets.

For testing contextuality under noise, the paper specifies a permutation-based protocol. For each window $\{\ell_C\}_{C\in\mathcal C}$3, a context-sensitive statistic $\{\ell_C\}_{C\in\mathcal C}$4 is computed; an empirical null is generated by permutation, producing

$\{\ell_C\}_{C\in\mathcal C}$5

and the resulting family of $\{\ell_C\}_{C\in\mathcal C}$6-values is corrected using Benjamini–Hochberg FDR at level $\{\ell_C\}_{C\in\mathcal C}$7 (Liechtenstein, 6 Oct 2025). The prescribed figure standard is explicit: observed curve, permutation null band $\{\ell_C\}_{C\in\mathcal C}$8, BH-corrected $\{\ell_C\}_{C\in\mathcal C}$9-values, significance markers, and captioned parameters $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$0 and seed.

These statistical components broaden the scope of CBL beyond pure proof theory. The formalism is intended not only to represent contextual obstruction semantically, but also to support reproducible claims about curvature in noisy data (Liechtenstein, 6 Oct 2025).

6. Related frameworks and broader geometric interpretations

CBL is positioned relative to contextuality theory, SAT/CSP, and more geometric treatments of Boolean structure. The paper states that KCBS, Mermin, and related contextuality scenarios appear in CBL as specific curved cores, while Abramsky–Brandenburger sheaf semantics are directly embedded in its semantic layer. Cabello–Severini–Winter exclusivity graphs correspond to the exclusivity-hypergraph semantics, and Table 1 reportedly embeds CSW-type exclusivity into a geometric parameter $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$1 describing deviation from flat logic (Liechtenstein, 6 Oct 2025). What is added, on the paper’s own account, is a full proof calculus, a SAT/CSP layer, and explicit treatment of noise and robustness.

From the SAT/CSP perspective, CBL introduces an axis of structure orthogonal to the usual emphasis on constraint language and polymorphisms. Two clause sets may share the same local syntax while differing in context overlap structure, curved cores, and solver behavior. A plausible implication is that CBL is less a new clause formalism than a reorganization of satisfiability around overlap topology (Liechtenstein, 6 Oct 2025).

Two earlier arXiv lines of work illuminate this broader geometric orientation. "The Clifford algebra of $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$2 and the Boolean Satisfiability Problem" represents Boolean formulas as idempotents in $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$3, identifies primitive idempotents with Boolean atoms, and reformulates unsatisfiability as a covering condition on the orthogonal group $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$4 (Budinich, 2021). That paper explicitly presents this as a geometric or “curved” formulation of Boolean logic and SAT, with logical states realized as regions on a curved manifold rather than isolated vertices. Although its construction differs from the contextual-sheaf semantics of CBL, it supplies a related program: embedding Boolean reasoning into continuous geometric structures without claiming an escape from NP-completeness.

A different antecedent is "Quantum geometry of Boolean algebras and de Morgan duality," which equips Boolean algebras over $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$5 with differential calculi, metrics, bimodule connections, curvature, and Ricci tensors, treating propositions as subsets, 1-forms as arrow sets of graphs, and $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$6 as the arrows crossing the boundary of $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$7 (Majid, 2019). In that setting, small graphs exhibit flat, curved, and Ricci-flat geometries; for example, the line graph $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$8 is non-flat but Ricci flat. This suggests a family resemblance rather than an identity of frameworks: both CBL and the $\ell_C|_{C\cap C'}=\ell_{C'}|_{C\cap C'}$9 quantum-geometric approach treat Boolean structure as capable of curvature, but they do so with different mathematical primitives.

The 2025 CBL paper also sketches links to robustness and adapter stability in LLMs. Prompt contexts are compared to measurement settings, local reasoning steps to contexts, and contradictory instruction traces to globally curved proof trees; layerwise curvature surrogates are proposed for compression operators such as LoRA and quantization (Liechtenstein, 6 Oct 2025). The paper itself characterizes these as outlines and speculative connections rather than fully developed algorithms. That restraint is important: the formal core of CBL concerns contextual propositional logic, while the AI applications remain prospective.

Open directions stated in the paper include deriving the per-face phase $C,C'$0 axiomatically, characterizing geometric tail decay in holonomy, connecting the CBL invariant $C,C'$1 to spectral graph invariants, proving completeness of the context-sequent calculus in curved settings, and extending the framework to probabilistic and weighted logics (Liechtenstein, 6 Oct 2025). Taken together, these indicate that CBL is intended as a logically precise framework for contextual obstruction, with algorithmic and robustness consequences, rather than as a finished universal theory of nonclassical inference.