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Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if $\\psi'(G)>\\psi'(D_8) = \\frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. 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Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if $\\psi'(G)>\\psi'(D_8) = \\frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if ψ'(G)>ψ'(D_8) = 19/43, then G belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The critical threshold is exactly ψ'(D8), which is assumed to be the largest value attained by groups outside the modular-lattice class; this relies on exhaustive case analysis or classification results for small-order groups that are not detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finite groups with ψ'(G) > 19/43 have modular subgroup lattices; all groups with ψ'(G) > 31/77 are classified, completing the supersolubility criterion.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If the normalized sum of element orders in a finite group exceeds 19/43, the value for the dihedral group of order 8, then the group has a modular subgroup lattice.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5130e7e6d96d8f34c9dfc68164de3b982d851f3296a3d305f901c8de3af974cf"},"source":{"id":"2601.11253","kind":"arxiv","version":2},"verdict":{"id":"063f679e-4281-42c1-af78-691f701123d2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:57:24.133761Z","strongest_claim":"if ψ'(G)>ψ'(D_8) = 19/43, then G belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound","one_line_summary":"Finite groups with ψ'(G) > 19/43 have modular subgroup lattices; 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