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At the heart of our analytic approach is an \\emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \\emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. 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For the \\emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \\emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context","title":"A Hierarchical Language Model with Predictable Scaling Laws and Provable Benefits of Reasoning","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-14T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.13687"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:03254e8c5cc74fc719288ee853c4c14fb050cd3f25e498639c7c5dbd89aeaa08e5a65f8cf1c0faac027fbd47f5d9c0551325a4af14c92e65a6379afd7df87409","signer":"crovia.substrate","subject":{"observed_at":"2026-05-14T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.13687"},"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":"69e693c5926cf96b361099060a83159d75daf5505bce769b37980df3deba6fb7","leaf_index":132654,"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":"bad5e9e6696689295fddc52c0275d40c76902e651c86ed203cf98051f71b3a7b","side":"right"},{"sibling":"0b72e03ed2534c7e040e861fc1a06d207162299077e5fce5fb22c08803114269","side":"left"},{"sibling":"031b1fd9e3835c3cbff65ae9b2727d353ecaed6ca70f60fd1abb907af178378c","side":"left"},{"sibling":"bfc742c108217953a466742391cc7c327b34a75d752410a8e1de0bac458c2473","side":"left"},{"sibling":"dd638df343785398540bc2001643997aa0c58c84a1d6b143cad5a3db040e7cf0","side":"right"},{"sibling":"bd0857de3ddbbe18a2122a8ad20f9fe94e8837ca5472083a9985cdd38da82647","side":"left"},{"sibling":"990ccbf7dece9bb81548c1b76f72b61af8fe51b5fc9416a0e16fc1cc6d4a4e96","side":"right"},{"sibling":"d5482331e2cc5acdd0d08391a6a4507a80863b313a18e39660a4565d5c9fa7ea","side":"right"},{"sibling":"2ae8cbb1d93652ee36f693c3d63e733765fbafcd7765d6d596692bf393ce0a1d","side":"right"},{"sibling":"a957418f640d5dc3181a7628c2646bb86c6da0ea6888670b451e534693a0c7cb","side":"left"},{"sibling":"c03f0a468f574a08ffe8b17e1a17bd88216e1359f0e33a54447e160cd8675da0","side":"left"},{"sibling":"038ff12da6f55509125ef0d96e1e57dda29a2fe63bf03fba2af4cf7cbcd88b36","side":"right"},{"sibling":"b0419206fe62ef216df3900ca93cffd44df267435a4643dda354c1310079cf91","side":"right"},{"sibling":"0d4a9c03674f9d0ce64df15c15e9f656a54b41f93c428aac8e615fda26291956","side":"right"},{"sibling":"7856d920f3f1f1d2194c1ed7351bf0d674440df3cb423a6911f89cb3e9578c0b","side":"right"},{"sibling":"6ac6396bdd2e9df315427a46155531476e7f9012b4bd962e0d2d6d1209b11723","side":"right"},{"sibling":"7c8dc85cbfe43e19ac759ad176cfa11dba2467ae17927c471d5c55663c4f490d","side":"right"},{"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":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_a9b79ef27da8440fbe615f27976f83fa2f1db2acf01cc08dca70f63a431f1384"}}