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We resolve this paradox for vacancy-doped\n  monolayer $1T$-\\ptis~by demonstrating that the insulator-to-half-metal transition is governed by universal geometric percolation\n  of the defect network, extending the percolation framework established for three-dimensional diluted magnetic semiconductors into\n  the 2D vacancy-doped regime. 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We resolve this paradox for vacancy-doped\n  monolayer $1T$-\\ptis~by demonstrating that the insulator-to-half-metal transition is governed by universal geometric percolation\n  of the defect network, extending the percolation framework established for three-dimensional diluted magnetic semiconductors into\n  the 2D vacancy-doped regime. Half-metallicity emerges via a two-step mechanism: crys"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the insulator-to-half-metal transition is governed by universal geometric percolation of the defect network... At critical vacancy concentration xc ≈ 12.5%, a percolation transition drives the majority-spin impurity band from flat, localized levels (W < 0.1 eV) to a dispersive 1.5 eV-wide band with 100% spin polarization and a minority-spin gap of 1.0 eV.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the supercell-size dependence (2×2 cells showing antiferromagnetism while 4×4 cells show ferromagnetism) arises solely from the presence or absence of a spanning percolation cluster rather than from finite-size artifacts, boundary conditions, or other details of the underlying electronic-structure calculations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Geometric percolation of Ti vacancies in 1T-TiS2 monolayers triggers half-metallic ferromagnetism at xc ≈ 12.5%, creating a functional window of 11% < x < 15% with 100% spin-polarized transport.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Percolation of sulfur vacancies at roughly 12.5 percent concentration switches vacancy-doped titanium disulfide from an insulator to a half-metal.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b47a85b2b8da10cbb552eac68211a09767fa2d8671d3613a092006b7bb06c5fb"},"source":{"id":"2605.01754","kind":"arxiv","version":1},"verdict":{"id":"530764ac-f676-4507-97a2-edca7c326073","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:40:47.081050Z","strongest_claim":"the insulator-to-half-metal transition is governed by universal geometric percolation of the defect network... 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