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  "slug": "boundary-crossing-paths-as-a-local-computable-bound-on-network-emergence",
  "title": "Boundary-Crossing Paths as a Local, Computable Bound on Network Emergence",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Emergent features of a networked system—behaviors visible in the coupled whole but absent from any isolated part—are usually measured globally, which obscures where in the network they originate. Building on the recent proposal that every emergent feature of a network system is produced by a route crossing the interface between parts, and that the number of such boundary-crossing paths bounds the total emergence discrepancy [1], [2], we develop a fully explicit walk-counting formalism for this claim. We model observations built route by route as walk-counting functionals W_L on the system's graph, define the emergence discrepancy D_L as the excess of whole-system walk mass over the sum of part-level walk masses, and prove that D_L equals exactly the number of walks that touch the inter-part boundary at least once. We verify the identity by exhaustive enumeration on a four-node reference network, computing per-length discrepancies D^(1) = 4, D^(2) = 12, D^(3) = 28, and cumulative D_≤ 3 = 44, matching the closed-form projection D_≤ L = 2^L+3 - 8 - 4L. [...]",
  "abstract_markdown": "Emergent features of a networked system—behaviors visible in the coupled whole but absent from any isolated part—are usually measured globally, which obscures where in the network they originate. Building on the recent proposal that every emergent feature of a network system is produced by a route crossing the interface between parts, and that the number of such boundary-crossing paths bounds the total emergence discrepancy [1], [2], we develop a fully explicit walk-counting formalism for this claim. We model observations built route by route as walk-counting functionals $W_L$ on the system's graph, define the emergence discrepancy $D_L$ as the excess of whole-system walk mass over the sum of part-level walk masses, and prove that $D_L$ equals exactly the number of walks that touch the inter-part boundary at least once. We verify the identity by exhaustive enumeration on a four-node reference network, computing per-length discrepancies $D^{(1)} = 4$, $D^{(2)} = 12$, $D^{(3)} = 28$, and cumulative $D_{\\le 3} = 44$, matching the closed-form projection $D_{\\le L} = 2^{L+3} - 8 - 4L$. [...]",
  "date": "2026-10-06",
  "first_published": "2026-10-06",
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  "url": "https://papers.qnfo.org/papers/boundary-crossing-paths-as-a-local-computable-bound-on-network-emergence",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Boundary-Crossing Paths as a Local, Computable Bound on Network Emergence"
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        "description": "Emergent features of a networked system—behaviors visible in the coupled whole but absent from any isolated part—are usually measured globally, which obscures where in the network they originate. Building on the recent proposal that every emergent feature of a network system is produced by a route crossing the interface between parts, and that the number of such boundary-crossing paths bounds the total emergence discrepancy [1], [2], we develop a fully explicit walk-counting formalism for this claim. We model observations built route by route as walk-counting functionals W_L on the system's graph, define the emergence discrepancy D_L as the excess of whole-system walk mass over the sum of part-level walk masses, and prove that D_L equals exactly the number of walks that touch the inter-part boundary at least once. We verify the identity by exhaustive enumeration on a four-node reference network, computing per-length discrepancies D^(1) = 4, D^(2) = 12, D^(3) = 28, and cumulative D_≤ 3 = 44, matching the closed-form projection D_≤ L = 2^L+3 - 8 - 4L. [...]",
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