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We define the \\emph{energy field}, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs.\n  Two classes of invariants emerge. \\emph{Mechanism-level} invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. \\emph{Model-level} regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call \\emph{key incoherence}.\n  These invariants have practical c","title":"On the Invariants of Softmax Attention","url":"https://arxiv.org/abs/2605.02907","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.02907v1 Announce Type: cross \nAbstract: Softmax attention maps every query--key interaction into a probability distribution, but the underlying structure remains largely unexplored. We define the \\emph{energy field}, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs.\n  Two classes of invariants emerge. \\emph{Mechanism-level} invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. \\emph{Model-level} regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call \\emph{key incoherence}.\n  These invariants have practical c","title":"On the Invariants of Softmax Attention","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-07T04:43:30Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.02907"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:69cd74d66abf946a81ad5dcf2ea17170e4bca15cfa6ee051095ba21493742c730d3d870eaa0291a50a3657bb6dafb08c503d80af159f11fab2b7be0a5e853503","signer":"crovia.substrate","subject":{"observed_at":"2026-05-07T04:43:30Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.02907"},"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":"0cf882403d0d09ede71da0a685e7e25adece3a5c9fc8787e461b7904ba302552","leaf_index":118212,"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":"7a9eec4d3ff82de0f26b7a4deb737412033a3d65a2b1cc12b853417b595082ce","side":"right"},{"sibling":"3c8ebb32ca101a8acc642a325753632a83a1ed2cc3757e896fd3f52ab26f1c90","side":"right"},{"sibling":"b772f35ff79b4a3ebf025337972b98ba5ca393664fd27e47467ad7d0d8791398","side":"left"},{"sibling":"c99c8fd7c44280b6b8083060dbbc01a905fa2f5db7d5ae63a6ac6551e5161027","side":"right"},{"sibling":"a6ef598ab40441a8ed079d84074d06b5febea003a04fb72b309560a7d89025f5","side":"right"},{"sibling":"c063c39ae694b656fe885810193365d271b3b69afdf3da9ee1836f170f43c681","side":"right"},{"sibling":"1b5d9d099a141dbd9b65685d9424ef07b508b27c0e01ef2b925d1dd5d95de03a","side":"left"},{"sibling":"c25c853cb3dd8c7f1b99fc80fd602c044932de34c1b34bd69ab1d5b8c94e42b1","side":"left"},{"sibling":"3d3124224341dac361f0e93f24f70795342ba216dc1c0065e7fa7592d943b488","side":"left"},{"sibling":"438bef084db8613e8143527769e3c677e12a59f8a6e2dea114f88afab0446a43","side":"right"},{"sibling":"8e758a4477dfc0838d2e8ea2bdbd583bc3192683e75fd9cde217ee6a3b2f3ac8","side":"left"},{"sibling":"4219f746e463e594ccd6447debdf736a57e77319b12bcb74d8ece2b5860064c6","side":"left"},{"sibling":"05f89b32c00462e60adf95c1fe4579cdc2791b36e8b17573d8f3b5fd5da95a0b","side":"right"},{"sibling":"8ccd9937a2c0d5c04044d07d1557791b7d07bb31eac41a39a675608d44b38f23","side":"right"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_8388671ad817ea1d0e160497daeee9618315a4b666c11f92689d43c25c930932"}}