Emergence and Recovery of (logical) Kochen-Specker Contextuality via Hamilton Extension

• URL: http://arxiv.org/abs/2607.14629v1
• Date: Thu, 16 Jul 2026 06:47:20 GMT
• Title: Emergence and Recovery of (logical) Kochen-Specker Contextuality via Hamilton Extension
• Abstract summary: We show that contextuality can emerge from KS-colorable vector sets through a procedure we call the Hamilton extension.<n>We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions.<n>Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.
• Score: 0.056212519098516274
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.

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