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This gives a matrix-theoretic and graph-theoretic version of the so called Berger construction: it extends the simply laced affine Dynkin stars while remaining accessible through elementary linear algebra. For a star with arm lengths $r_1,\\ldots,r_m$ we compute the determinant, the inertia, the positive-definite and affine regimes, and the primitiv"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.23011","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-21T20:32:41Z","cross_cats_sorted":[],"title_canon_sha256":"cbb0fdd5615050d9db83867d5a6a21558fef0b1de343099a63093a877f5d1e2e","abstract_canon_sha256":"e37ee6c22179016dce950af2fc8b4379cc873c66185ac98c54eb07e590bdfab9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:34.284128Z","signature_b64":"KVJ3ijaEipE0RE11EONruqRXeFNVKbhGU5FtiKs7aXJCuttHzlgB+uwFC6Hy3Ng6iqCEgxxCIDo3mOTHJp//BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5865035f7ff163d6e7bb9d64bba89c7e0c5f4520699fe0a951218aa529c8d6dd","last_reissued_at":"2026-05-25T02:01:34.283410Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:34.283410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Star-Shaped Integral Cartan-Type Matrices and an Egyptian-Fraction Classification of Affine Weighted Trees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emilio Torrente-Lujan","submitted_at":"2026-05-21T20:32:41Z","abstract_excerpt":"We study a concrete family of symmetric integral $Z$-matrices attached to weighted star trees. 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