Upper bounds on ultrametric OGPs at levels 1 and 2 for symmetric binary perceptrons are approximately 1.6578 and 1.6219, closely matching the 3rd and 4th lifting-level parametric RDT estimates, supporting conjectures that the algorithmic threshold equals the infinite-level limits of both frameworks.
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Strengthened Dirac-type minimum degree conditions guarantee that the k-switch reconfiguration graphs on perfect matchings are connected and expanders, with matching lower-bound constructions showing exponential numbers of components below certain degree thresholds.
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Ultrametric OGP - parametric RDT \emph{symmetric} binary perceptron connection
Upper bounds on ultrametric OGPs at levels 1 and 2 for symmetric binary perceptrons are approximately 1.6578 and 1.6219, closely matching the 3rd and 4th lifting-level parametric RDT estimates, supporting conjectures that the algorithmic threshold equals the infinite-level limits of both frameworks.
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Dirac's theorem and the switch geometry of perfect matchings
Strengthened Dirac-type minimum degree conditions guarantee that the k-switch reconfiguration graphs on perfect matchings are connected and expanders, with matching lower-bound constructions showing exponential numbers of components below certain degree thresholds.