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  "slug": "signed-kuramoto-positive-networks",
  "title": "Signed Kuramoto networks inside positive ones: an exact realisation theorem, stable non-manifold equilibria, and Sclosa's closed-manifold conjecture",
  "shortTitle": "Signed Kuramoto networks inside positive ones",
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  "oneLine": "Every ±1 signed Kuramoto network sits exactly inside a network with only positive couplings, with equilibria and stability preserved; this yields a 324-oscillator attractive network whose stable equilibria form a cone over a torus, refuting the closed-manifold and immersed readings of Sclosa's Conjecture 1.3, while its literal arXiv v1 reading is true.",
  "abstract": "Networks of identical phase oscillators with attractive (positive) coupling are the simplest models of synchronisation. This candidate proves that every ±1 signed Kuramoto network is reproduced exactly on an invariant torus of a positive unweighted network: replace each oscillator by a ring of six groups of r clones at phase offsets aπ/3, and realise a negative edge as a matching between groups whose offsets differ by π. For r > 4Δ the complete set of nearby equilibria, the Morse index, the nullity and Lyapunov stability all transfer, because a uniform normal Hessian bound gives an energy inequality even at degenerate equilibria. Applications: a connected 23-regular graph on 324 vertices whose equilibrium set near one stable point is ℝ × C(S¹ × S¹), a cone over a torus, with every nearby equilibrium stable; stable cones ℝ × C(S^k × S^k) for every k ≥ 1; stable crossing tori on 120 vertices; stable smooth continua of every dimension. For Sclosa's Conjecture 1.3, the readings refuted are the published one (a finite union of closed manifolds), its countable version, the version with immersed compact pieces, and pieces closed only relative to the equilibrium or stable set; neither the equilibrium set nor the stable set admits such a cover. The literal arXiv v1 reading (a finite union of manifolds, not necessarily closed) is true for every graph by stratification, as is the reading with boundary allowed. On K_n a closed or immersed compact cover exists exactly when n is odd or n ≤ 4. Everything occurs at exact symmetric resonances and is not robust to generic perturbation.",
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    "Kuramoto model",
    "signed networks",
    "synchronisation",
    "equilibrium set",
    "Lyapunov stability",
    "polygon spaces",
    "stratification",
    "counterexample"
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    "Realisation (Theorem 1): every ±1 signed simple graph with maximum degree Δ is reproduced exactly on an invariant torus of a positive unweighted graph G(H,r), r > 4Δ, with 6nr vertices; complete equilibrium germs, Lyapunov stability, Morse index and nullity transfer.",
    "Stable cone (Theorem 2): G_{6,9}, 324 vertices and 3726 edges, 23-regular, has an equilibrium whose equilibrium germ is ℝ × C(S¹ × S¹), with every nearby equilibrium Lyapunov stable; the Jacobian there has 5 zero and 319 negative eigenvalues (exact certificate).",
    "Sclosa's Conjecture 1.3: refuted in its published closed-manifold wording, its countable version, its immersed-compact version and its relatively-closed version, for both the equilibrium set and the stable set; true in its literal arXiv v1 wording (manifolds not necessarily closed) and with boundary allowed.",
    "Classification (Theorem 3): on K_n a closed (or immersed compact) manifold cover exists exactly when n is odd or n ≤ 4.",
    "Applications: stable cones ℝ × C(S^k × S^k) for every k ≥ 1 (Corollary 13), stable crossing tori on 120 vertices (Corollary 14), stable smooth continua of every dimension (Corollary 15); r = 9 is sufficient, r ≤ 7 is unstable and r = 8 is open (Remark 12)."
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              {
                "field": "uncachedInputTokens",
                "reason": "Not instrumented; unknown is not zero."
              }
            ],
            "calibration": null
          }
        },
        "missingnessReason": null
      }
    ]
  }
}