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However, the theorem does not specify when this mechanism becomes empirically observable, nor whether it leaves a measurable spectral signature in the parameter updates. We test this directly on Tian's modular addition setup ($ M = 71 $, $ K = 2048 $, MSE loss) and observe a clear structure-mechanism dissociation. The predicted sign rule holds robustly on the top-200 most-similar feature pairs across activations (empirical sign-match rising from 0.865 to 0.985 on $ \\sigma = x^2 $ across 5 seeds, and saturating at 1.000 on $ \\sigma = \\operatorname{ReLU} $). However, the spectral signature in the parameter updates is strongly activ","title":"Feature Repulsion and Spectral Lock-in: An Empirical Study of Two-Layer Network Grokking","url":"https://arxiv.org/abs/2605.08119","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.08119v1 Announce Type: cross \nAbstract: Tian (2025) proves a repulsion theorem (Theorem 6) for the matrix $ B = (\\widetilde{F}^\\top \\widetilde{F} + \\eta I)^{-1} $ during the interactive feature-learning stage of grokking: similar features have negative off-diagonal entries $ B_{j\\ell} $, producing an effective repulsive force that drives them apart. However, the theorem does not specify when this mechanism becomes empirically observable, nor whether it leaves a measurable spectral signature in the parameter updates. We test this directly on Tian's modular addition setup ($ M = 71 $, $ K = 2048 $, MSE loss) and observe a clear structure-mechanism dissociation. The predicted sign rule holds robustly on the top-200 most-similar feature pairs across activations (empirical sign-match rising from 0.865 to 0.985 on $ \\sigma = x^2 $ across 5 seeds, and saturating at 1.000 on $ \\sigma = \\operatorname{ReLU} $). However, the spectral signature in the parameter updates is strongly activ","title":"Feature Repulsion and Spectral Lock-in: An Empirical Study of Two-Layer Network Grokking","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-12T04:43:42Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.08119"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:3123fe86ff7f6719cee15854d66e47d61cfc7fab2d92c310b02632ac4ec7e4ec762faf6316fb9b3eb3f3bb7be96d96e5c220616b9c1da166fd78479fa7b5fd09","signer":"crovia.substrate","subject":{"observed_at":"2026-05-12T04:43:42Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.08119"},"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":"f0d3406f8a9c7ffac891dfe3668866c0c62c6545ace7bf91b16d8dc6e6622b73","leaf_index":128460,"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":"95bc3a01f9586753c3e1377710b14f11f53200d25e78fe4c424f05935d201d9a","side":"right"},{"sibling":"9657a75a52bc497541c4d6d73fad43fbb2dcbc1930b4417431777b257c49ed8b","side":"right"},{"sibling":"901e9c283d9a04554b901f3f153c4c319e904cf5bd75f91e74623944813b8a64","side":"left"},{"sibling":"402b3d3650d2ebd902122afe9fdcea28f77dac78dd91d7e4815146ace3968ca0","side":"left"},{"sibling":"0ef4692e6c5429c9870b5aa075883dc4357d7bbb9e28ccd8ffe6fa1d39e1e149","side":"right"},{"sibling":"584f7351ea8abce1efad8787266629d24824a20b95d0b0d8ae4a978c78e050b3","side":"right"},{"sibling":"80a5f0118f1cab257d86ade2f02746f5ce1158951c70a9e1f7025391472dc0ef","side":"left"},{"sibling":"0fac4565968e7e49cfc1d10e28cfc4966c72b4dd0deb14bce20d28acb7a0d65a","side":"left"},{"sibling":"aa3ddc60ad25fcdd4bf93520d8baeca26d807789d80fb0deffbb19c4f8882286","side":"left"},{"sibling":"077005bbaed7b42537914de0b512ed52d33243215eab288181aba807d37bf04e","side":"right"},{"sibling":"c6eaf7a4fcab2db96e9e9423acb6922c80f64882d0f3f50d09e53a4807d23084","side":"left"},{"sibling":"df12eaabc0a370aff5d0488478f48d2b3f90d025c903643f98ea804b017688ec","side":"right"},{"sibling":"ffc4d51379293bc3e1910c7d612f409dc610fd9acf8241793fb89f82e1bad4ef","side":"left"},{"sibling":"62ac6554017807bd83187f5a3e5f4f72d6c482616429c2780e9fff1f4845fa04","side":"left"},{"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_87d97edd6175b1b845870cec74b63f5a13f1d75d1a7d4bb19c0a5bfc0aa9f292"}}