pith. sign in

arxiv: 2605.16248 · v1 · pith:PSVUPLHWnew · submitted 2026-05-15 · 🪐 quant-ph

Local Softmax and Global Weights in Non-Boolean Event Structures

Karl Svozil This is my paper

Pith reviewed 2026-05-20 17:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords softmaxevent structuresadmissible weightsno-disturbancecontextualitynon-Boolean logicnormalizationquantum foundations
0
0 comments X

The pith

Imposing single-valuedness on shared atoms collapses generalized softmax rules to parametrizations of the admissible-weight polytope.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In non-Boolean event structures with overlapping contexts, local normalization functions such as softmax do not automatically produce consistent global probability weights. The paper establishes that requiring single-valuedness on shared atoms, which is equivalent to no-disturbance or consistent connectedness, reduces generalized softmax rules to coordinate parametrizations of the strictly positive part of the admissible-weight polytope. Any strictly positive admissible weight admits such a representation, while boundary weights appear as limits. Exotic weights that exceed classical or quantum bounds therefore originate in the event structure and the chosen weight rather than in the choice of normalization. The resulting hierarchy distinguishes local normalization, cross-context gluing, Cauchy-Gleason linearity, and physical or cognitive realizability.

Core claim

By imposing single-valuedness on shared atoms in non-Boolean event structures, generalized softmax rules collapse to coordinate parametrizations of the strictly positive part of the admissible-weight polytope. Every strictly positive admissible weight can be represented in this manner, with boundary weights arising as limits. Exotic weights beyond classical and quantum bounds are therefore properties of the event structure and the weight assignment, not of the normalizing link. The analysis produces a hierarchy that separates local normalization, cross-context gluing, Cauchy-Gleason linearity, and realizability.

What carries the argument

Single-valuedness on shared atoms (equivalently, no-disturbance or consistent connectedness), which enforces consistent connectedness across overlapping contexts and reduces local softmax normalizations to global parametrizations of admissible weights.

If this is right

  • Any strictly positive admissible weight in the polytope can be represented using generalized softmax rules once single-valuedness on shared atoms is imposed.
  • Boundary weights of the admissible-weight polytope arise as limits of these single-valued softmax parametrizations.
  • Weights exceeding classical or quantum bounds are determined by the event structure and the chosen weight assignment rather than by the normalization procedure.
  • The hierarchy cleanly separates local normalization, cross-context gluing, Cauchy-Gleason linearity, and realizability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In choice theory or cognitive modeling, apparent non-classical response patterns may trace to the underlying non-Boolean event structure rather than to the form of the local normalization function.
  • The result suggests a way to isolate contextuality effects by checking whether a given set of weights lies in the interior of the admissible polytope for a fixed event structure.
  • Extensions could test whether specific families of quantum event structures admit complete parametrizations of their positive admissible weights through single-valued softmax rules.

Load-bearing premise

The event structure admits a well-defined notion of admissible weights that is independent of the chosen normalization, and single-valuedness on shared atoms can be imposed without forcing the structure to become Boolean.

What would settle it

A non-Boolean event structure containing a strictly positive admissible weight that cannot be obtained as a coordinate parametrization from any generalized softmax rule under single-valuedness on shared atoms, or in which imposing single-valuedness forces the structure to become Boolean.

Figures

Figures reproduced from arXiv: 2605.16248 by Karl Svozil.

Figure 1
Figure 1. Figure 1: FIG. 1. Pentagon logic in the notation of Sec. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that in non-Boolean event structures with overlapping contexts, imposing single-valuedness on shared atoms (equivalently, no-disturbance or consistent connectedness) reduces generalized local softmax rules to coordinate parametrizations of the strictly positive part of an independently defined admissible-weight polytope. Any strictly positive admissible weight admits such a representation, while boundary weights are recovered in the limit. Exotic weights beyond classical or quantum bounds are thereby attributed to the event structure and chosen weight rather than the normalizing link. The paper proposes a hierarchy separating local normalization, cross-context gluing, Cauchy-Gleason linearity, and physical or cognitive realizability.

Significance. If the central reduction holds, the result supplies a clean conceptual separation between local response functions and global consistency constraints in generalized probability theories. This has potential value for choice modeling in cognitive science and machine learning, as well as for foundational questions in non-classical logics and quantum mechanics. The manuscript receives credit for maintaining the non-Boolean character of the structures under the single-valuedness condition and for defining the admissible weights independently of the local normalization, thereby avoiding an immediate tautology.

minor comments (3)
  1. The abstract states the central reduction at a high level without referencing the theorem or section number that contains the explicit parametrization; adding such a pointer would improve readability for readers outside the immediate subfield.
  2. §2 (or the section introducing admissible weights): while the independence from normalization is asserted, a short explicit example of the polytope coordinates for a small non-Boolean event structure would make the construction more concrete and help verify the claimed non-circularity.
  3. Notation for 'strictly positive part' and 'boundary weights as limits' is used consistently but would benefit from a single clarifying sentence linking these notions to the standard topology on the simplex or polytope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of the central claims, and recommendation for minor revision. The report correctly identifies the separation between local normalization and the independently defined admissible-weight polytope as the key conceptual contribution.

read point-by-point responses

  1. Referee: The manuscript claims that in non-Boolean event structures with overlapping contexts, imposing single-valuedness on shared atoms (equivalently, no-disturbance or consistent connectedness) reduces generalized local softmax rules to coordinate parametrizations of the strictly positive part of an independently defined admissible-weight polytope. Any strictly positive admissible weight admits such a representation, while boundary weights are recovered in the limit. Exotic weights beyond classical or quantum bounds are thereby attributed to the event structure and chosen weight rather than the normalizing link. The paper proposes a hierarchy separating local normalization, cross-context gluing, Cauchy-Gleason linearity, and physical or cognitive realizability.

    Authors: We confirm that the reduction holds. Under the single-valuedness (no-disturbance) condition the local normalization constants become coordinates on the positive orthant of the admissible-weight polytope, which is defined independently via the linear constraints of the event structure. Surjectivity onto the interior follows by explicitly solving the resulting system of equations for the local softmax parameters; the construction is invertible on the strictly positive cone. Boundary weights are recovered by allowing one or more parameters to tend to zero or infinity, which corresponds to the closure of the polytope. Because the polytope itself is fixed by the event structure alone, any weight lying outside classical or quantum bounds is necessarily a property of that structure and the chosen point, not an artifact of the local normalization rule. The proposed hierarchy is intended precisely to keep these layers distinct and to locate physical or cognitive realizability as a further, independent filter. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained representation result

full rationale

The paper defines admissible weights on the event structure independently of any normalization choice, then shows that single-valuedness on shared atoms (no-disturbance) maps generalized local softmax functions onto coordinate parametrizations of the strictly positive interior of that polytope, with boundary points recovered in the limit. This is a representation theorem rather than a self-definition: the polytope is not constructed from the softmax; instead the softmax is shown to recover any interior point of an independently given convex set. No load-bearing step reduces to a fitted parameter renamed as prediction, nor to a self-citation chain, nor to an ansatz smuggled via prior work. The argument separates local normalization from cross-context gluing and from realizability, preserving non-Boolean character by assumption. The construction is therefore not tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of single-valuedness together with the existence of an independently defined admissible-weight polytope for the given event structure.

axioms (1)
  • domain assumption Single-valuedness on shared atoms (no-disturbance or consistent connectedness)
    Invoked to collapse local rules to global parametrizations, as stated in the abstract.

pith-pipeline@v0.9.0 · 5652 in / 1213 out tokens · 93433 ms · 2026-05-20T17:31:34.409579+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    If independent score contributions are required to add, while their unnormalized weights multiply, then q(u+v)=q(u)q(v). ... Under any of the stated regularity assumptions, h=log q is a regular solution of Cauchy’s equation

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 4 internal anchors

  1. [1]

    E. R. Gerelle, R. J. Greechie, and F. R. Miller, Weights on spaces, inPhysical Reality and Mathematical Descrip- tion, edited by C. P. Enz and J. Mehra (D. Reidel Pub- lishing Company, Springer Netherlands, Dordrecht, The Netherlands, 1974) pp. 167–192

    work page 1974

  2. [2]

    Wright, The state of the pentagon

    R. Wright, The state of the pentagon. A nonclassical ex- ample, inMathematical Foundations of Quantum The- ory, edited by A. R. Marlow (Academic Press, New York,

    work page

  3. [3]

    Schaller and K

    M. Schaller and K. Svozil, Partition logics of automata, Il Nuovo Cimento B109, 167 (1994)

    work page 1994

  4. [4]

    Svozil, Logical equivalence between generalized urn models and finite automata, International Jour- nal of Theoretical Physics44, 745 (2005), arXiv:quant- ph/0209136

    K. Svozil, Logical equivalence between generalized urn models and finite automata, International Jour- nal of Theoretical Physics44, 745 (2005), arXiv:quant- ph/0209136

    work page arXiv 2005

  5. [5]

    Svozil, Faithful orthogonal representations of graphs from partition logics, Soft Computing24, 10239 (2020), arXiv:1810.10423

    K. Svozil, Faithful orthogonal representations of graphs from partition logics, Soft Computing24, 10239 (2020), arXiv:1810.10423

    work page arXiv 2020

  6. [6]

    Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory25, 1 (1979)

    L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory25, 1 (1979)

    work page 1979

  7. [7]

    A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, Simple test for hidden variables in spin-1 systems, Physical Review Letters101, 020403 (2008), arXiv:0706.0126

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  8. [8]

    Contextuality and nonlocality in 'no signaling' theories

    J. Bub and A. Stairs, Contextuality and nonlocality in ‘no signaling’ theories, Foundations of Physics39, 690 (2009), arXiv:0903.1462

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  9. [9]

    V. J. Wright and S. Weigert, Gleason-type theorems from Cauchy’s Functional Equation, Foundations of Physics 49, 594 (2019)

    work page 2019

  10. [10]

    Navara and V

    M. Navara and V. Rogalewicz, The pasting constructions for orthomodular posets, Mathematische Nachrichten 154, 157 (1991)

    work page 1991

  11. [11]

    Graph-Theoretic Approach to Quantum Correlations

    A. Cabello, S. Severini, and A. Winter, Graph-theoretic approach to quantum correlations, Physical Review Let- ters112, 040401 (2014), arXiv:1401.7081

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  12. [12]

    Busch, Quantum states and generalized observables: a simple proof of Gleason’s theorem, Physical Review Letters91, 120403, 4 (2003)

    P. Busch, Quantum states and generalized observables: a simple proof of Gleason’s theorem, Physical Review Letters91, 120403, 4 (2003)

    work page 2003

  13. [13]

    C. M. Caves, C. A. Fuchs, K. K. Manne, and J. M. Renes, Gleason-type derivations of the quantum probability rule for generalized measurements, Foundations of Physics. An International Journal Devoted to the Conceptual Bases and Fundamental Theories of Modern Physics34, 193 (2004)

    work page 2004

  14. [14]

    Aczél,Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering, Vol

    J. Aczél,Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering, Vol. 19 (Academic Press, New York-London, 1966) pp. xx + 510, translated by Scripta Technica, Inc. Supple- mented by the author. Edited by Hansjorg Oser

    work page 1966

  15. [15]

    E. T. Jaynes, Information theory and statistical mechan- ics, Physical Review106, 620 (1957)

    work page 1957

  16. [16]

    Svozil, Chromatic quantum contextuality, Entropy 27, 387 (2025), arXiv:2501.15261

    K. Svozil, Chromatic quantum contextuality, Entropy 27, 387 (2025), arXiv:2501.15261

    work page arXiv 2025

  17. [17]

    J. R. Busemeyer and P. D. Bruza,Quantum Models of Cognition and Decision(Cambridge University Press, Cambridge, 2012)

    work page 2012

  18. [18]

    E. M. Pothos and J. R. Busemeyer, Can quantum prob- ability provide a new direction for cognitive modeling?, Behavioral and Brain Sciences36, 255 (2013)

    work page 2013

  19. [19]

    Peres, Unperformed experiments have no results, American Journal of Physics46, 745 (1978)

    A. Peres, Unperformed experiments have no results, American Journal of Physics46, 745 (1978)

    work page 1978

  20. [20]

    How much contextuality?

    K. Svozil, How much contextuality?, Natural Computing 11, 261 (2012), arXiv:1103.3980

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  21. [21]

    E. N. Dzhafarov, J. V. Kujala, and V. H. Cervantes, Con- textuality and noncontextuality measures and general- ized bell inequalities for cyclic systems, Physical Review A101, 042119 (2020)

    work page 2020

  22. [22]

    E. N. Dzhafarov, J. V. Kujala, and V. H. Cervantes,Con- textuality in Random Variables: A Systematic Introduc- tion(Cambridge University Press, 2026)

    work page 2026