Contextual Probability and Operational Minimality in Tug-of-War Decision Making

Song-Ju Kim

quant-phJan 15, 2026

Contextual Probability and Operational Minimality in Tug-of-War Decision Making

Authors: Song-Ju Kim

Organizations: SOBIN Institute LLC, 3-38-7 Keyakizaka, Kawanishi, Hyogo, 666-0145, Japan.

Abstract

What connects minimal decision-making to contextual probability? We address this question through the Tug-of-War (TOW) principle, a resource-coupled mechanism for multi-armed bandit decision-making. Starting from classical probability, we show that when reward-learning functionality and specified information requirements are fixed, minimizing the necessary relaxation of common event semantics can lead to nonclassical probability. We derive the exact minimum semantic repair and identify the conditions under which information efficiency, calibration, and compositional consistency select contextual statistics. We then construct a single quantum Tug-of-War (QTOW) instrument in which the same decision--reward process both improves stationary decision performance and drives the unconditional learned state beyond the Klyachko--Can--Binicioğlu--Shumovsky (KCBS) noncontextuality bound. A finite diagnostic based on the instrument's recovery operations also yields a corrected contextuality witness. These results connect minimal decision-making, operational economy, and the emergence of contextual probability.

Figures & tables

Panel of Figure 1 — Figure 1: The classical TOW principle and one bandit trial. (a) A finite volume is divided among two competing arms and a reserve; moving the resource toward one arm changes the balance without creating volume. The heights are schematic rather than measured physical dimensions. (b) At trial $t$ , the current state and a fluctuation produce a deterministic comparison of the two sides; the selected external slot machine then generates a stochastic reward; this observation updates the shared preference for trial $t+1$ . The diagram distinguishes randomness in the physical comparator from randomness in the external reward source.

Panel of Figure 2 — Figure 2: The five-cycle exclusivity graph $C_{5}$ used in the diagnostic service. Each vertex denotes a shared Yes/No event $e_{i}$ ; each edge denotes a jointly measured exclusive pair $c_{i}=(e_{i},e_{i+1})$ (indices modulo five). The five contexts use the same event labels, rather than five unrelated pairs of outcomes.

Panel of Figure 3 — Figure 3: Additional operational conditions imposed on the $C_{5}$ diagnostic of Fig. 2 . Two independently prepared copies supply product events (left); the highlighted exclusive clique is part of the GE2 constraint $K^{2}\leq 5$ . An independently specified efficient calibration has a quadratic probability budget (right; schematic, not to scale). Neither condition is entailed by native TOW reward learning.

Panel of Figure 4 — Figure 4: Learning and diagnosis in one QTOW instrument. The upper loop shows its repeated binary decision, external reward, and reward-conditioned update. Recovery $W_{k}$ preserves the logical preference plane; the update retains the state with weight $1-\gamma$ or resets it to $\tau_{k,r}$ with weight $\gamma$ . The lower, dashed path represents an independent fixed $C_{5}$ test on fresh copies of the unselected learned ensemble $\bar{\rho}_{t}$ . The diagnostic is an additional service, not a step in ordinary reward learning.

Figure 5: Same-instrument, unconditional learning trajectory. The solid curve is the ideal sharp-probe KCBS sum of the ensemble after

t

decision–reward trials; the dashed line is the noncontextual bound. The curve crosses the bound at

t=13

and approaches

K_{*}=2.065248\ldots

. The inset gives the expected reward, which grows from the uniform-decision baseline

0.5

0.648824\ldots

. Every trial branch is included.

Completed trials $t$	Expected reward $R_{t}$	Ideal KCBS $K_{t}$	Finite-gate corrected gap
$0$	$0.500000$	$1.809017$	$-0.195950$
$13$	$0.610995$	$2.000117$	$-0.006320$
$14$	$0.614777$	$2.006630$	$+0.000143$
$20$	$0.630730$	$2.034096$	$+0.027397$
$30$	$0.642515$	$2.054386$	$+0.047531$
Stationary	$\mathbf{0.648824}$	$\mathbf{2.065248}$	$\mathbf{+0.058309}$

Table 1: Decision and diagnostic results for one physical recovery–reset instrument, averaged over all reward histories.

Model or service	Established result	Required capabilities
Two-state TOW learner	L1–L4 and stationary reward advantage	Coupled preference, choice and reward update
Five-context information service	Exact minimal semantic repair and fixed-information boundary; conditional information-optimal selection	Shared events, neutral calibration, information requirement and GE2
Two-copy calibrated diagnostic	Quantum-compatible maximal $C_{5}$ vector under GE2 and efficiency	Independent copies and adjustable response calibration
Recovery–reset QTOW	Stationary gain and unconditional ideal KCBS violation; finite corrected CbD witness	One native learning instrument; independent diagnostic reuse of recovery gates

Table 2: Successive operational tasks considered in the TOW framework. Each row adds a question about the same decision-making functionality rather than introducing a competing learning algorithm.

Quantity	Value
$K_{\mathrm{av}}$	$2.0676577898997393$
$E$	$0.0003768918497872960$
$2+3E/(2\alpha)$ , $\alpha=1/3$	$2.0016960133240428$
Excess over preparation-noncontextual bound	$0.0659617765756965$
Smallest conditional correction weight	$0.00233121530637$
Smallest six-word source weight	$0.0341460296793$

Table S1: Finite positive-word calibrated preparation-noncontextuality witness for the unselected learned source at

m=0.6

. The error

E

is defined in Eq. ( S3.3 ).

Panel of Figure 9 — Figure S1: Conceptual structure of QTOW. In the ordinary learning trajectory, the qutrit undergoes a two-outcome decision instrument $\{K_{A},K_{B}\}$ . For finite sharpness, the instrument perturbs the state while preserving part of the pre-decision state information. The selected option $d_{t}$ and reward $r_{t}$ then determine a norm-preserving feedback operation $U_{d_{t},r_{t}}$ . The third level is an auxiliary mode of the present construction and supports the KCBS-type projective contextuality diagnostic used below.

Figure S2: Fixed-probe learning witness. The auxiliary-axis pentagon does not violate on the ordinary learning subspace. The pentagon oriented along

|A\rangle

has

K<2

on the initial state and after the A readout, but

K>2

following an A decision, external reward, and feedback. Angles refer to state amplitudes.

Figure S3: Unconditional learning dynamics for eight finite trials. Weighted sums over all

4^{n}

histories; the pure-state clipping convention and the globally fixed unitary feedback are shown separately. Both have

K>2

in the finite-time ensemble by trial six. The fixed-angle version is a linear quantum channel; the clipped version is a controlled pure-state trajectory.

History after one complete trial	Probability	$K$ after feedback
$A$ , win	0.400	2.0541913
$A$ , loss	0.100	1.8986734
$B$ , win	0.100	1.5638427
$B$ , loss	0.400	1.7193606

Table S2: Complete decision–reward branches for the neutral initial state.

	C: history-controlled clipping		U: fixed rotation
$n$	$\Pr(K>2)$	$\mathbb{E}[K]$	$\Pr(K>2)$	$\mathbb{E}[K]$
0	0	1.809017	0	1.809017
1	0.400000	1.855672	0.400000	1.855672
2	0.289288	1.896650	0.289288	1.896650
4	0.540011	1.961852	0.540011	1.961616
6	0.641963	2.009084	0.641963	2.004802

Table S3: Finite-time probability of a KCBS violation and the unconditional KCBS sum for the clipped (C) and fixed-angle (U) protocols.

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Multi-Armed Bandits study how to choose among uncertain actions over time while balancing immediate rewards with learning.

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May 1, 2026J/K move · Enter open · S save

Spacetime Formation under Requirements: Contextual Realization and Form-Dependent Probability

Song-Ju Kim

SOBIN Institute LLC, Kawanishi, Hyogo, Japan

Cognitive Modeling

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Quantum cognition often explains order effects, contextuality, and violations of the law of total probability by replacing classical probability with quantum probability on a fixed event structure. This paper proposes a different interpretation: quantum probability is the fixed-spacetime projection of contextual spacetime formation under finite-state requirements. The framework begins not with time, space, objects, or probabilities, but with requirements such as finite representational capacity, single-state semantic stability, context-sensitive intervention, avoidance of explicit context labels, coherent world-formation, and intersubjective transformability. When these requirements cannot be realized within a single global Boolean event structure, the mismatch appears, under fixed-spacetime projection, as noncommutativity, interference, and quantum-like probability. Building on prior single-state approaches to contextuality, we reinterpret classical contextual bookkeeping cost as the fixed-spacetime shadow of contextual spacetime formation. Auxiliary memory or context labels in a classical representation correspond, in this account, to holonomy-like mismatch among locally Boolean logic-worlds. The interference term is the cross term generated when locally classical realization contributions are nontrivially glued and projected back into a fixed classical spacetime form. The result is a transcendental-operational realist account: objecthood, eventhood, probability, and spacetime are treated as forms of realization under requirements, while objectivity is defined by invariants preserved across observer- and history-dependent spacetime formations.

Abstract

Multi-Armed Bandits

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