{"_canonicalization":{"envelope_id":"axm_ + sha256(envelope minus {signature, axiom_id, anchors})","envelope_signature":"ed25519(envelope minus {signature, axiom_id})","json":"sort_keys=True, separators=(',',':'), ensure_ascii=False, allow_nan=False, utf-8","leaf_hash":"sha256(0x00 || canonical_json(envelope_full))","seal_signature":"ed25519(seal minus {signature, sig_algorithm})"},"axiom_id":"axm_acab35e9abaa58e6362d55bf8843d642c8c63e328d559c686281ab68ef97bf4d","bitcoin_anchor":{"bitcoin_attestations":[],"calendar_attestations":[],"ots_url":"","stamped_at":"","status":"pending_next_stamp"},"envelope":{"anchors":[{"chain":"crovia.axiom_graph","height":0,"merkle_proof":"spider_vendor_press_v1","root_at_anchor":"spider_vendor_press_v1"}],"axiom_id":"axm_acab35e9abaa58e6362d55bf8843d642c8c63e328d559c686281ab68ef97bf4d","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"eb7d6087ab72836d52ec7d3d812134bb6719fc5aed2c6ab5fc6a3f49c6d67009","published":"Wed, 17 Jun 2026 00:00:00 -0400","receipt_hash":"eb7d6087ab72836d52ec7d3d812134bb6719fc5aed2c6ab5fc6a3f49c6d67009","schema":"spider.news.vendor_press.v1","spider":"vendor_press","spider_record":{"axiom_subtype":"news.vendor_press.v1","category":"news","decision_hint":"POSITIVE","envelope_target":"AX.OBS","fingerprint":"eb7d6087ab72836d52ec7d3d812134bb6719fc5aed2c6ab5fc6a3f49c6d67009","observed_at":"2026-06-17T04:43:17.968423Z","parent_run_hash":"8f56c4deb22b28178ba7974d6ffc5ff17336d44c95dd94de80095705245fa113","published":"Wed, 17 Jun 2026 00:00:00 -0400","runtime_version":"0.1.0","schema":"spider.news.vendor_press.v1","source_status":200,"source_url":"https://export.arxiv.org/rss/cs.AI","spider":"vendor_press","summary_excerpt":"arXiv:2606.17830v1 Announce Type: cross \nAbstract: Neural network parameter spaces are inherently non-injective, as distinct parameter configurations can realize identical functions through functional equivalence. While this symmetry is well understood in classical fully connected and convolutional models, it becomes substantially more intricate in modern attention-based architectures. Existing analyses of multihead attention have largely focused on the vanilla formulation, overlooking positional encodings that fundamentally reshape architectural symmetries. In this work, we provide a formal study of functional equivalence in Transformers with positional encodings. Focusing on the two most widely used variants--sinusoidal and rotary positional encodings (RoPE)--we show that sinusoidal encodings preserve the equivalence structure of vanilla attention, whereas rotary encodings significantly reduce the symmetry group, thereby enhancing expressivity. This offers a principled explanation fo","title":"Functional Equivalence in Attention: A Comprehensive Study with Applications to Linear Mode Connectivity","url":"https://arxiv.org/abs/2606.17830","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.17830v1 Announce Type: cross \nAbstract: Neural network parameter spaces are inherently non-injective, as distinct parameter configurations can realize identical functions through functional equivalence. While this symmetry is well understood in classical fully connected and convolutional models, it becomes substantially more intricate in modern attention-based architectures. Existing analyses of multihead attention have largely focused on the vanilla formulation, overlooking positional encodings that fundamentally reshape architectural symmetries. In this work, we provide a formal study of functional equivalence in Transformers with positional encodings. Focusing on the two most widely used variants--sinusoidal and rotary positional encodings (RoPE)--we show that sinusoidal encodings preserve the equivalence structure of vanilla attention, whereas rotary encodings significantly reduce the symmetry group, thereby enhancing expressivity. This offers a principled explanation fo","title":"Functional Equivalence in Attention: A Comprehensive Study with Applications to Linear Mode Connectivity","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-17T04:43:17Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.17830"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:8cdedf64752f797578a070a753b28fb58b718b41a366f3bf89fd9ec7d8b9930fb7d014e9d23fa185ea84c639bde0a12040a4f02504f61934b6c0e92503833e08","signer":"crovia.substrate","subject":{"observed_at":"2026-06-17T04:43:17Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.17830"},"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":"cd0ebfa00a8eef576a1f805b2416049c68da9571464f9e84c737097846b65e7c","leaf_index":231013,"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":"849d7970a2fd7a93cf18e8ad3f56b6cd5825d239edf95f087c145ada71fdb77c","side":"left"},{"sibling":"e87ba2ad9ed6ddfa4e4e0076d771d2d5204eaff3c23ee8eeca3717e5a35b1976","side":"right"},{"sibling":"601d217123c80d84918d296f2cc686e9a6041601b3a068b29b1f233d328f5597","side":"left"},{"sibling":"597b751b38259cdbe0bd96c26a7629e39064d1eb5f36aa0c382a8249472a1199","side":"right"},{"sibling":"47bfd510ba8997f2908c9e511edda3a2bb8936d69f28f4a47a5cb165f1ddb16b","side":"right"},{"sibling":"5224392f43dd4a302bb61fc34434b5f9c72f8899d53418e9684bcc3cf814e3a4","side":"left"},{"sibling":"8daceae0255c786ffdef12a608da0c1a09b9a2ceedfcc067120bfd86b4ab9952","side":"left"},{"sibling":"ce828166a6a4fc2ff2681c985a56053a1f8b683cc245e0b232fa5939310b3ad9","side":"right"},{"sibling":"ebbec9ce4bf43a3f5f71e2c07df4c91301abefa2da727a4d748099a74e980bc3","side":"right"},{"sibling":"1d74941c32baeab8cad08f8700cb49427d8256f231534f2a225b2bb3e84e4ff8","side":"left"},{"sibling":"d5b9f8b1a2c9f6a46e17982dfbe6ce1f3b5fa4e730220397f2253d114dcc8486","side":"left"},{"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_acab35e9abaa58e6362d55bf8843d642c8c63e328d559c686281ab68ef97bf4d"}}