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We ask whether they are mechanistically distinct or converge to a similar weight update. Training six methods (SFT, RFT, DFT, RIFT, Offline GRPO, DPO) on identical math rollouts from a single base model (Qwen3-4B) with attention-only LoRA, we analyze the resulting deltas via cosine similarity, principal-angle subspace analysis, linear mode connectivity, and CKA. We observe: (i) SFT, RFT, and RIFT have nearly colinear weight deltas (cosine >= 0.97, top-1 principal angle ~7 deg median over 144 modules) and comparable GSM8K accuracy (87-88%, n=1319; pairwise McNemar p >= 0.15); (ii) DFT diverges further in direction than any reward-weighted method despite using the same data; (iii) Offline GRPO adds a substantial component orthogon","title":"Weight-Space Geometry of Offline Reasoning Training","url":"https://arxiv.org/abs/2606.23740","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.23740v1 Announce Type: cross \nAbstract: Offline reinforcement-learning losses (RFT, RIFT, DFT, Offline GRPO, DPO) are widely used to distill reasoning from large teachers into smaller students, and are typically compared on downstream accuracy alone. We ask whether they are mechanistically distinct or converge to a similar weight update. Training six methods (SFT, RFT, DFT, RIFT, Offline GRPO, DPO) on identical math rollouts from a single base model (Qwen3-4B) with attention-only LoRA, we analyze the resulting deltas via cosine similarity, principal-angle subspace analysis, linear mode connectivity, and CKA. We observe: (i) SFT, RFT, and RIFT have nearly colinear weight deltas (cosine >= 0.97, top-1 principal angle ~7 deg median over 144 modules) and comparable GSM8K accuracy (87-88%, n=1319; pairwise McNemar p >= 0.15); (ii) DFT diverges further in direction than any reward-weighted method despite using the same data; (iii) Offline GRPO adds a substantial component orthogon","title":"Weight-Space Geometry of Offline Reasoning Training","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-24T04:43:17Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.23740"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:379b1a7a4b11b73e9093ebfbfdc14d636498faf6cde37364ca98a9ff13bc81b43445465ef07853c9c89e9d0578506dc1d1e7bdfb099d79c7c35439270e851a04","signer":"crovia.substrate","subject":{"observed_at":"2026-06-24T04:43:17Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.23740"},"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":"5e66b4ea23f303ecbd0d1cf433bf30e0e52e57d8d2779798a0e5587064c05569","leaf_index":244497,"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":"7bc3723bb19b7b567a8f13c131bb7ef500e8c1051866d5e0a6ee63c4c367ae49","side":"left"},{"sibling":"af5603c6824914f43d52d8fea0006613f1850f7653b7039a2b68ffa033637a17","side":"right"},{"sibling":"7b1cf9b476b9ae305df1815aaf021d5c9c5f34193f29a031468414cd05cca7f0","side":"right"},{"sibling":"7d63ad83ce119105b0ff4f0e62e1747b06f63c24fef1b3a196cf2bc36639cfd7","side":"right"},{"sibling":"40ae0b1566fd0333a37d99e1d881cc22bf4cadd967e5cbe9cc553cc082ffce92","side":"left"},{"sibling":"d7a80dbc109a440cbbc5ee38ada545b76d1b74e87a0a9424159972f613d5d8b8","side":"right"},{"sibling":"f6c01d42388f96d80ab8ec2f12119f28971403b41d996362c6d78ca0e42293de","side":"right"},{"sibling":"d4fcf6bcc7f1fe78997b28df74b23bba21f5b2585d6fe8efb802f0d067291d5c","side":"right"},{"sibling":"0fa23771b702ff726ed1fc5a44f9b416b2a7861c2f957fcac2d95392276d4784","side":"left"},{"sibling":"bc74ebb08462da8a50fc65ea75f8a8a3418d10ebd471d830f1c67f33dd54dfd1","side":"left"},{"sibling":"6dafd355e5d54c60e61c6c02d3842984e234b1f5bca1623fcdd3def7b8931973","side":"right"},{"sibling":"3107b9d4dbf9456a39f99de694a4dd4da2c0600f9f8855f125161335fe8810af","side":"left"},{"sibling":"86118ab4500c3055a2af70062751a960423c464405b18ca1c37411bf0ce3f52e","side":"left"},{"sibling":"3a42039065acac6d3e4088ec61d9c116ecf7a26c7b7163d23da8fd0b3362e038","side":"left"},{"sibling":"c044f2bd864a0e8e8af5a7f6e3124def7fc4b4511b2b166ea8f9de321e8d385e","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":244827,"merkle_root":"274e133c6dfa2781a9cfb85337d01cc6b72688ce5e810149f3183e400ffab136","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260624T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-24T05:37:55Z","sig_algorithm":"ed25519","signature":"22ca3cee4de2447b3d281e30e09fe566461996bb7be4d4465f08a3f4cf59cea58f22683a4aa10ef4d5d17a19b03f4212392bfd26f2b51f289f0cdf1042a03800","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_4ec44586eeebe7a94d12a2cac7497ec913a0e6a66ce334da4b9251d950abe235"}}