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We introduce \\emph{Bidirectional Provability Fingerprinting} (\\bpf{}), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) \\emph{Counterfactual Probe Generation} (\\cpg{}), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the \\emph{Equivalence Spectrum}, a continuous faithfulness score that replaces brittle binary verdicts; (iii) \\emph{Adaptive Probe Budget Allocation} (\\apba{}), an information-theoretic ","title":"The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements","url":"https://arxiv.org/abs/2606.16541","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.16541v1 Announce Type: new \nAbstract: Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by \\emph{faithfulness}: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended. We introduce \\emph{Bidirectional Provability Fingerprinting} (\\bpf{}), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) \\emph{Counterfactual Probe Generation} (\\cpg{}), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the \\emph{Equivalence Spectrum}, a continuous faithfulness score that replaces brittle binary verdicts; (iii) \\emph{Adaptive Probe Budget Allocation} (\\apba{}), an information-theoretic ","title":"The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-16T04:43:43Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.16541"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:30179196a31edb2f328f115665da53c6d6b76a3d5bedc07efe423f0aaa77cecaa54a9f0dd7fe8c19f02eb16c6c3af900f551146a25d2a0178766b56e4d09a606","signer":"crovia.substrate","subject":{"observed_at":"2026-06-16T04:43:43Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.16541"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"447486e76667a43767d802f7040c3276e5f659b517597800d0f00ed44ca1e955","leaf_index":230277,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"eca0cf0381857d150278c642c06f548438af11e7840f6140dbc41a01aafd305f","side":"left"},{"sibling":"2479ef4a13978298c559990e8c1772e0f33dd518401a17b0988f0ee2e2c1d631","side":"right"},{"sibling":"d6cfdeee30fd604044780d32c661d717564da53954c0619cce9d0c4aee76bfd2","side":"left"},{"sibling":"d5f54d088a056ea15552786bca7044bcbf713ec9f60b18b57f4579ca54c1cc1f","side":"right"},{"sibling":"9332324ee62bebdce79ee50d3daf50a5e6c67b937590e00f082b00bffa58c51d","side":"right"},{"sibling":"f3484bbe9e5a6851aaba461de29ea5ee40da90419525554ab3fc968e0dab7b1e","side":"right"},{"sibling":"1563a36474162fbf76fa1a3ea70f0abdfd1b93b8b7dc614ada2d128d3e0d62e7","side":"right"},{"sibling":"a920a6db38b29333b686b08bcb39326c5a67bb5f4588a0647130fe767a058e04","side":"left"},{"sibling":"03ebe4791d56166247160f6ee7788c15745bdd7e274c62ec21b2438669941587","side":"left"},{"sibling":"74897e850164dddc689c3c65b33f9bae0268ab0bf429867a4e193d9b9b685040","side":"left"},{"sibling":"bde25d7e94e64717e426a97f6fcb4907e92b5c61fc89d92d7e0947a2249c3f6b","side":"right"},{"sibling":"d10d772a4984cae00e65ab24af21d1d260e475bbaae3f17871859eb705bd3999","side":"right"},{"sibling":"e79159853f2f35ddae8e3247d515e433c534277b287d65bbd77ae989aa4992fa","side":"right"},{"sibling":"3054319f1840cce0eaaf0bc4b1ae38e5bf8b6210927d924a750775cc7d77cca6","side":"right"},{"sibling":"0fd8b5059f279c4a4a6688de2472fdbc25543fee183df19b21dacca880354cff","side":"right"},{"sibling":"a04392fb9f2a3a620840e3b3fecd93d12e6d224e481c162389d8ae64e7194599","side":"left"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":232015,"merkle_root":"62bfb7809bb55667ad7eeebdb48267b9b2c1ee89bb808ea1e6d3a814a15aa402","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260617T133701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-17T13:39:59Z","sig_algorithm":"ed25519","signature":"930a3563c643cc7518d048b12a1f5392a96a5edce49533ba0a56f11b6cc69319bb1c800aacc46b52a6941c4d05dae255a45cac757d6093c971b96852c24f140e","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_ed3b42ba8a2c99c7b98ba8e50d1a830b4c9064c852b32754b90e05d9755f1d04"}}