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We show how to generalize the above constructions in order to associate, to each point in $(0,\\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(\\alpha)$ for sufficiently small $x$, identifying, in the proc"},"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":"1408.4885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T05:23:41Z","cross_cats_sorted":[],"title_canon_sha256":"ef7f0a03d25baad82070082dd5868e398fd772a1f0e7bea4d4f693f88a2f7dcd","abstract_canon_sha256":"e13aa4381d23a464d17374db8cb07d3620f55b696d81e85de28f5885566042f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:41.428813Z","signature_b64":"maHq8wseVVp6j6+bVo23f6q5XXTTJ6rscdQ4CRlGrcepou6TvLgGCYTiF+2dAs2ljSE6A5JxB6kTpCkK2zLTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d98cac3c2a94d6d17f171d276a23c0e9027ad73b7cd1be2ec01af1f66d6c76f5","last_reissued_at":"2026-05-18T02:44:41.428147Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:41.428147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A collection of metric Mahler measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Charles L. 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