Local Softmax and Global Weights in Non-Boolean Event Structures

Abstract: Softmax and related normalized response functions are widely used in choice theory, machine learning, and cognitive science. In non-Boolean event structures with overlapping contexts, however, local normalization does not automatically yield a global probability weight. We show that imposing single-valuedness on shared atoms -- equivalently, no-disturbance or consistent connectedness -- collapses generalized softmax rules to coordinate parametrizations of the strictly positive part of the admissible-weight polytope. Any strictly positive admissible weight can be represented in this way, while boundary weights arise as limits. Exotic weights that exceed classical or quantum bounds are therefore properties of the event structure and the chosen weight, not of the normalizing link. The resulting hierarchy separates local normalization, cross-context gluing, Cauchy--Gleason linearity, and physical or cognitive realizability.

• The paper establishes a representation theorem linking local softmax normalization with globally consistent admissible weights in pasted non-Boolean event structures.
• It demonstrates that single-valued conditions ensure the softmax parametrization covers classical, quantum, and exotic probability assignments.
• The study outlines practical implications for cognitive science, machine learning, and quantum foundations through analysis of odd-cycle structures.

Local Softmax in Non-Boolean Event Structures: A Technical Synthesis

Overview and Objectives

This paper analytically investigates the behavior and scope of locally normalized softmax (and generalized softmax-like response functions) in finite non-Boolean event structures—specifically, logics constructed via context-gluing that admit overlapping contexts. The central result is a formal equivalence: any strictly positive, single-valued probability assignment (an “admissible weight”) on such a structure is expressible in generalized softmax form, provided local normalization is accompanied by a consistency (“gluing” or single-valuedness/no-disturbance) condition on shared atoms. The study clarifies the interplay between response normalization, global weight admissibility, and the boundary between classical, quantum, and exotic assignments, especially in contextual structures like odd-cycles.

Event Structures and Probability Assignments

The framework operates on a finite pasted event structure $L = (A, \mathcal{M})$—$A$ a set of atoms, $\mathcal{M}$ a family of maximal, mutually exclusive, and jointly exhaustive contexts. Probability weights on $L$ are functions $p: A \to [0,1]$ normalized contextually: $$\sum_{a \in C} p(a) = 1, \quad \forall C \in \mathcal{M}$$ The set of all such $p$ forms a convex polytope $\mathcal{W}(L)$. The subset of convex combinations of classical two-valued weights yields the classical (noncontextual) polytope $\mathcal{C}(L)$, and the subset attainable via Born rule in a suitable Hilbert space comprises the quantum set $\mathcal{Q}(L)$, satisfying the strict inclusion chain $A$0.

A key feature is the presence of “intertwining atoms”—those belonging to multiple overlapping contexts—that drive the distinctions between local normalization and global compatibility.

Generalized Softmax Parametrization

Generalized softmax is presented in this context as

$A$1

where $A$2 is any positive link function and $A$3 are context-dependent scores. If $A$4 or $A$5 is chosen arbitrarily per context, the resulting set of locally normalized distributions may be globally inconsistent: the same atom $A$6 in multiple contexts may receive different probabilities.

The crucial distinction is between:

• Free context-wise normalization: normalized per context, with no cross-context consistency.
• Single-valued/no-disturbance normalization: additional requirement that for every $A$7, $A$8, ensuring global single-valuedness.

Representation Theorem and Implications

A central technical result is the representation theorem: for any strictly positive admissible weight $A$9 and any strictly monotone positive link function $\mathcal{M}$0 with appropriate range, there exist scores $\mathcal{M}$1 such that for all $\mathcal{M}$2 and $\mathcal{M}$3,

$\mathcal{M}$4

Boundary weights (i.e., weights with some zero components), including “exotic” assignments, are obtained in the closure (as limits) of such parametrizations.

A corollary is that softmax normalization does not enlarge or restrict the probabilistic possibilities on $\mathcal{M}$5 once the single-valuedness condition is imposed; the functional form of $\mathcal{M}$6 is largely immaterial from the perspective of which probability assignments are possible.

Gauge Freedom

There is a gauge symmetry: multiplying all $\mathcal{M}$7 by a constant on a connected component does not affect the normalized probabilities. In the exponential link case, this is equivalent to additive translation of scores.

Classical, Quantum, and Exotic Regions

The polytope $\mathcal{M}$8 generally strictly contains both $\mathcal{M}$9 and $L$0. Certain weights in $L$1—such as half-weights on odd-cycles—are normalized and single-valued, yet cannot be represented as convex mixtures of classical deterministic assignments. Similarly, for odd $L$2, the half-weight achieves values exceeding classical ($L$3) and even quantum (Lovász/KCBS) bounds.

Softmax-like parametrizations, under single-valuedness, are capable of representing (as limits) these exotic weights, but do not inherently explain or generate them; their existence is a property of the event structure.

Relation to Cauchy/Gleason Theorems

The global compatibility condition imposed here is distinct from the linearity required in Gleason-type theorems. The current gluing results do not derive the Born rule or linear additive probability assignments, but concern only the compatibility of locally normalized assignments across contexts.

However, the exponential link function can be uniquely singled out if one imposes a compositional axiom: additive increase in score corresponds to multiplicative increase in unnormalized weight—the classic Cauchy functional equation argument, yielding $L$4 under regularity conditions.

Additionally, context-wise entropy maximization (as in Jaynes’ maximum entropy principle) selects the exponential form, but only within contexts; it does not address the gluing/global compatibility problem.

Odd Cycles and Explicit Construction of Exotic Weights

Odd-cycle logics concretely illustrate the distinctions:

• The “half-weight” assignment, $L$5 for each cyclic atom, $L$6 for context-specific atoms, is admissible and normalized.
• This assignment exceeds the classical mixture bound ($L$7) and, for $L$8, the quantum Lovász/KCBS bound.
• Generalized softmax with suppressed context-specific coordinates ($L$9) provides a limiting path to these weights, but not a constructive mechanism within a fixed finite parametrization.

Empirical, Modeling, and Theoretical Implications

For applications in cognitive sciences and machine learning, the mapping of score functions to probabilities must not be conflated with the existence of a global probabilistic structure. Practitioners must:

• First, fit empirical data context-wise.
• Second, test for consistency across shared atoms.
• Third, analyze the location of the reconstructed global weight relative to classical and quantum sets.

Unconstrained softmax or generalized softmax fitting in each context is not explanatory; it has no structural restriction beyond the imposed gluing condition. Only imposing additional structure on the scores (e.g., parametric forms, atom-only dependence, smoothness constraints) imparts meaningful explanatory or generative power.

For foundational quantum theory, the result demonstrates that the nonclassical or beyond-quantum assignments observed in contextual scenarios are rooted not in the choice of normalization function, but in the geometry of the event structure’s weight polytope.

Conclusion

The paper establishes that generalized softmax normalization in non-Boolean event structures, when supplemented with single-valuedness or no-disturbance, is merely an analytic coordinatization of the set of admissible weights. The classical, quantum, or “exotic” status of a probability assignment arises from its position within this polytope, not from the normalization function itself. This clarifies longstanding ambiguities surrounding the probabilistic significance of softmax rules in multi-context logical or physical models, with concrete implications for fields spanning quantum foundations, contextuality, discrete choice, and machine learning.

Future directions include the study of constrained score model classes, extensions to sparse-normalization rules, and the systematic linking of finite event structure analyses to infinite- or effect-algebraic settings relevant for generalized Gleason-type theorems.