Reproduction logbook (paper-82EJxJzG6r)

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CLAIMS.json

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+{
+  "orid": "82EJxJzG6r",
+  "arxiv_id": "2603.08859v1",
+  "title": "Expressivity-Efficiency Tradeoffs for Hybrid Sequence Models",
+  "claims": [
+    {
+      "index": 1,
+      "text": "Theorem 3.3 proves that any k-layer state-space model solving the function-composition tasks under injectivity conditions must have total log state-space size scaling as Ω(m·log|V| − q·log|Y|), linear in the hidden dimension m (Theorem 3.3)."
+    },
+    {
+      "index": 2,
+      "text": "Theorem 3.7 proves that sliding-window Transformers solving the same tasks under a local-sensitivity condition require total window size scaling with the context-dependency range R (Theorem 3.7)."
+    },
+    {
+      "index": 3,
+      "text": "Theorem 4.3 constructs a two-layer hybrid (Mamba + attention) model that solves the selective copying task using embedding dimension O(max(log|V|, log L)) and working memory Õ(N), versus Ω(L) required by pure Transformers (Theorem 4.3)."
+    },
+    {
+      "index": 4,
+      "text": "Theorem 4.6 constructs a three-layer hybrid model that achieves 99% accuracy on the associative recall task using embedding dimension O(max(log|V|, log L)) and window size Õ(|V|) (Theorem 4.6)."
+    },
+    {
+      "index": 5,
+      "text": "On the selective copying task, the learned hybrid model reaches perfect accuracy with roughly 2,000 parameters while pure Transformer/SSM models need roughly 12,000 parameters to match it, a 6x parameter gap (Figure 4)."
+    },
+    {
+      "index": 6,
+      "text": "On multi-key associative recall, the hybrid model reaches 60% accuracy using 6x fewer parameters than pure Transformers, which plateau near 40% accuracy on single-key associative recall (Figures 5-6)."
+    }
+  ]
+}

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OFFICIAL_VALIDATOR_RUN.json

@@ -0,0 +1,13 @@
+{
+  "validator": "official_validator.py",
+  "authority": "authoritative release gate for this local package only",
+  "not_claimed": [
+    "ICML validator",
+    "OpenReview validator",
+    "paper-author validator",
+    "remote campaign validator"
+  ],
+  "semantic_v4": "SEMANTIC_V4.json",
+  "validation": "VALIDATION.json",
+  "passed": true
+}

README.md

@@ -1,10 +1,36 @@
 ---
-title: Repro Expressivity Efficiency Hybrid Sequence
-emoji: 🐠
-colorFrom: pink
-colorTo: red
+title: Reproduction - Expressivity-Efficiency Hybrid Sequence Models
+emoji: 📐
+colorFrom: blue
+colorTo: green
 sdk: static
 pinned: false
+short_description: Six claims audited, arXiv 2603.08859
+tags:
+  - trackio
+  - open-reproductions
+  - icml2026-repro
+  - paper-82EJxJzG6r
 ---
 
-Check out the configuration reference at https://huggingface.co/docs/hub/spaces-config-reference
+# Local ICML reproduction audit
+
+This package audits the six frozen registered claims for arXiv:2603.08859v1 / OpenReview `82EJxJzG6r`. It is deliberately isolated and local: no Hugging Face Space, campaign ledger, or remote validation target is created or changed.
+
+Run the complete deterministic package with:
+
+```sh
+python3 run_all.py
+```
+
+That runs every evidence route twice, compares the eight result artifacts byte-for-byte, builds all pages, runs `official_validator.py` (the authoritative local-package gate, not an ICML/OpenReview/author validator), writes `SEMANTIC_V4.json` with 12 local semantic checks and `VALIDATION.json`, and finally writes `MANIFEST.sha256`.
+
+The proposed (not created) target slug is `repro-hybrid-seq-82ejxjzg6r`.
+
+Evidence categories are deliberately distinct:
+
+- Claims 1–2: literal theorem audit and direct numerical witnesses.
+- Claims 3–4: independent finite implementations of written constructions, plus direct execution of author notebook cells recorded separately.
+- Claims 5–6: exact authored-source table audit only; learned training is not claimed as rerun because the supplied training path hard-codes CUDA and this host has no CUDA device.
+
+Important limitations are in each claim page. In particular, Claim 1 is falsified at the literal printed scope, Claim 5’s `.999` is not silently rewritten to perfect accuracy, and Claim 6’s Figure 5 and Figure 6 tasks are not conflated.

REPLAY.json

@@ -0,0 +1,64 @@
+{
+  "kind": "paired_deterministic_local_replay",
+  "files": [
+    {
+      "file": "outputs/claim1.json",
+      "replay_a_sha256": "cda7675b2b59deb26050668435bbd6e4d7f5a7fe169e88a851b4a57debd378b6",
+      "replay_b_sha256": "cda7675b2b59deb26050668435bbd6e4d7f5a7fe169e88a851b4a57debd378b6",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim2.json",
+      "replay_a_sha256": "644f05f4af1131b092a8858a3f4ef9931bab04772b9db1097d5079e5c5fb7fa7",
+      "replay_b_sha256": "644f05f4af1131b092a8858a3f4ef9931bab04772b9db1097d5079e5c5fb7fa7",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim3.json",
+      "replay_a_sha256": "c2948406cd663b77c0502c23d6da9b71107b5388540f128f99c64e55d5be0d3b",
+      "replay_b_sha256": "c2948406cd663b77c0502c23d6da9b71107b5388540f128f99c64e55d5be0d3b",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim4.json",
+      "replay_a_sha256": "48c3a0ea60f9e18747607f8ca0b8a40303b32cb6f562407b802dc30c19c0def0",
+      "replay_b_sha256": "48c3a0ea60f9e18747607f8ca0b8a40303b32cb6f562407b802dc30c19c0def0",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim5.json",
+      "replay_a_sha256": "daa15965236e3c890688d7751916192e6502db072eff9a65869f209e05a7f6cf",
+      "replay_b_sha256": "daa15965236e3c890688d7751916192e6502db072eff9a65869f209e05a7f6cf",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim6.json",
+      "replay_a_sha256": "ee900bdcd3c603327736e3533fa296c6337b75f2cdc6a50bbb78cf87c22bddc2",
+      "replay_b_sha256": "ee900bdcd3c603327736e3533fa296c6337b75f2cdc6a50bbb78cf87c22bddc2",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim3_native.json",
+      "replay_a_sha256": "7ba1aee42a132c74b9875edb1a17b43a5f0d9a73cc78ce30cbd407f86ce77d50",
+      "replay_b_sha256": "7ba1aee42a132c74b9875edb1a17b43a5f0d9a73cc78ce30cbd407f86ce77d50",
+      "byte_identical": true
+    },
+    {
+      "file": "outputs/claim4_native.json",
+      "replay_a_sha256": "4f7266b28d80bc0cc7102d5d6cb6d8eafd278d557487299af97fba71515ab33b",
+      "replay_b_sha256": "4f7266b28d80bc0cc7102d5d6cb6d8eafd278d557487299af97fba71515ab33b",
+      "byte_identical": true
+    }
+  ],
+  "byte_identical": true,
+  "python": "3.13.3",
+  "commands": [
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 exp1_ssm_bound.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 exp2_window_bound.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 exp3_selcopy_construction.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 exp4_ar_construction.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 run_native_constructions.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 audit_source_claims.py",
+    "/Library/Frameworks/Python.framework/Versions/3.13/bin/python3 build_logbook.py"
+  ]
+}

SEMANTIC_V4.json

@@ -0,0 +1,68 @@
+{
+  "profile": "semantic-v4-local",
+  "validator": "local_bundle_validator_not_campaign_validator",
+  "checks": [
+    {
+      "name": "frozen_six_claims",
+      "passed": true,
+      "detail": "CLAIMS.json exactly equals the six registered claim strings."
+    },
+    {
+      "name": "complete_eight_routes",
+      "passed": true,
+      "detail": "Index, executive summary, and six claim routes exist."
+    },
+    {
+      "name": "exact_authored_archive",
+      "passed": true,
+      "detail": "arXiv e-print archive matches the retrieval SHA-256."
+    },
+    {
+      "name": "pinned_clean_code_checkout",
+      "passed": true,
+      "detail": "Linked source repository is detached at the recorded clean commit."
+    },
+    {
+      "name": "claim1_literal_falsification",
+      "passed": true,
+      "detail": "Under injectivity the printed RHS is non-positive; one-state 1/2 witnesses are retained."
+    },
+    {
+      "name": "claim2_receptive_field_witnesses",
+      "passed": true,
+      "detail": "Real causal-attention stacks preserve the designed outside-window witness."
+    },
+    {
+      "name": "claim3_construction_and_controls",
+      "passed": true,
+      "detail": "Independent finite construction succeeds while destructive controls fail."
+    },
+    {
+      "name": "claim4_full_vocab_certificate",
+      "passed": true,
+      "detail": "Full-vocabulary construction passes its exact coverage and small-domain gates."
+    },
+    {
+      "name": "claim4_sample_certificate_distinction",
+      "passed": true,
+      "detail": "The one sub-99% finite sample mean is retained separately from the exact coverage certificate."
+    },
+    {
+      "name": "direct_native_notebooks_retained",
+      "passed": true,
+      "detail": "Author notebook cell execution and destructive controls are reported without upgrading scope."
+    },
+    {
+      "name": "claim5_exact_source_scope",
+      "passed": true,
+      "detail": "Figure 4 table is preserved as .999, not silently converted to 1.000."
+    },
+    {
+      "name": "claim6_source_contradiction_marked",
+      "passed": true,
+      "detail": "Figure 6 sixfold row and Figure 5 task distinction are explicitly retained."
+    }
+  ],
+  "passed": true,
+  "check_count": 12
+}

SOURCE_PROVENANCE.json

@@ -0,0 +1,53 @@
+{
+  "arxiv_id": "2603.08859v1",
+  "arxiv_eprint_url": "https://export.arxiv.org/e-print/2603.08859v1",
+  "archive": "source/2603.08859v1.tar.gz",
+  "archive_sha256": "e8d22bfd259aaa60385841d8643109ecb66f7eb1081dd76429f5215f05a032e8",
+  "archive_sha256_expected_from_retrieval": "e8d22bfd259aaa60385841d8643109ecb66f7eb1081dd76429f5215f05a032e8",
+  "authored_tree_file_sha256": {
+    "source/authored/00README.json": "d1f6b1a71c9accb812ec5121d1fff7910c025ca7bb201948bacdb40941bc14e5",
+    "source/authored/appendix/app_prelim.tex": "58305508fa935a85634e9a69ee3a919648ea7e3cf25061120fb0cd1eae723999",
+    "source/authored/appendix/construction_conventions.tex": "f76427efb23af284fd0852afedf188310d3fad58cda11414930b87429155178a",
+    "source/authored/appendix/constructions.tex": "4fbf21f1fa7544ff602fcd896a72a3babd50986fc2c5711f2d70b9844ba1a026",
+    "source/authored/appendix/experiment_details.tex": "4caa42ab4df1d828ea44fb9167e017827477b66265d70f24c1fad42104d02b77",
+    "source/authored/appendix/missing_proof_lb.tex": "7e729f2198b6c5b7190355a7985b56096822f9c7c462ac1d42e6d62a03487406",
+    "source/authored/fig/construction_structure.pdf": "bd8c0384a10f6baf22cc4652d8c12e4b176fe439f90c41ad3c2f4aa8df200a2a",
+    "source/authored/fig/example.png": "f304cbf59f3c06bfa5a9e1df347701b4197cda26bc4ebd67e2471a104a84e50a",
+    "source/authored/fig/exps/assoc_recall_mk/layers_dim.png": "b6bbbf7f68754086361ab6641322b7acf24f933b56fcb044c779a8b8f862212c",
+    "source/authored/fig/exps/assoc_recall_mk/layers_num_heads.png": "047a87d7c7b5e81595113875f27955cf30ce33056894921b6532998d7ff6b01f",
+    "source/authored/fig/exps/assoc_recall_mk/layers_state_dim.png": "c12d55000a42d3899f51599619e90ba7a24455c87aa113c0252a96ac1216e612",
+    "source/authored/fig/exps/assoc_recall_mk/layers_window.png": "c5355c62564a80012078b25777a9fdbaedc96359ccd3081e2e09d4bfb30f933f",
+    "source/authored/fig/exps/binary_recall/layers_dim_3.png": "b6dd8cf4936994649a6d74ca2a1210c057571e19f0518584f6f15447a4d7272e",
+    "source/authored/fig/exps/constructions/decode_recall_input.pdf": "76ec4177346017cf7233b3a2cabadb2a517d7ca43ca60a0e238bcd866b8b8783",
+    "source/authored/fig/exps/constructions/decode_recall_output.pdf": "5b66865ef2ef1491cae85f7faa370e6d17657b583481076709b56dbda1f1cc92",
+    "source/authored/fig/exps/constructions/selective_copy_input.pdf": "392d7bd0ed19eddc8fd084c48d8367618af273c1fbf25fe00b00a66fa6af77b8",
+    "source/authored/fig/exps/constructions/selective_copy_output.pdf": "563189f4e77fc950e0482d6432c2293c2298f9e6539cfd918fbf0ea8c333c7a7",
+    "source/authored/fig/exps/mini_decode_recall.png": "3200afa19cd9ebfd1ea5f6b1f53d02ad3683dcc3741ae00c67f8e34a89d6affc",
+    "source/authored/fig/exps/needle/layers_dim.png": "b03026887e3d4046aafb90f766b6a5154a2b742222ee71c53a007c6a1ca57c91",
+    "source/authored/fig/exps/var_copy/layers_dim.png": "d40b41b76dc484233b2c83d6a59f613393fdca8d5f19d413727bcafc23660bc5",
+    "source/authored/fig/exps/var_copy/layers_dim_1000.png": "07c2d5ccc723acbc5a302b0b386f50244d8010f42c82aa616200076796067c23",
+    "source/authored/fig/exps/var_copy/layers_dim_200.png": "3a6dfd95e7aa31a031f42f9db1b99456cb2230096033e652a57b76d7516eb380",
+    "source/authored/fig/exps/var_copy/layers_dim_adversary.png": "c49b8ed16b17f782392821672415a68b3b6878981bece5637c78729b0176ae47",
+    "source/authored/fig/exps/var_copy/layers_num_heads.png": "a64ad32988841e8cfe2537c7df96442711dd8e5c6c81a05ec638f7ed5f25554d",
+    "source/authored/fig/exps/var_copy/layers_state_dim.png": "d8b8da4a094e7ac4cd878cfcb7ca6fc98913c8c6fd28ab7995045720e881dec8",
+    "source/authored/fig/exps/var_copy/layers_window.png": "f154fdb8104c5d0c3492019f22846c1e69cad5b1a110f24881523ea7d6be07b9",
+    "source/authored/fig/selective_copy_construction_structure.pdf": "b70c660739eba172c0dc3959139c7f931657c4887a8354448e87e13eb593988a",
+    "source/authored/main.tex": "92a399acca0a701a2fa2c1ed038192ac84dd6ab2430932081cc7283416699b8f",
+    "source/authored/mathcommands.tex": "3a7753a5a76463a9093767c57462ed42bd99fbe7d9594248b40c4fdd91f9a2e2",
+    "source/authored/reference.bib": "78b0d784946e5b987b8d5d62755f4463d122d50e49c6312f9989f28a7b16dc43",
+    "source/authored/sections/conclusion.tex": "c5db291da441eef14b0288d1a3f082fbddb0430b3ec3299d6f83f21982cb7fb8",
+    "source/authored/sections/experiments.tex": "fcd88332cbe90aff62337c3add333f71fc542736b3e459015a28f0b08ae3c55b",
+    "source/authored/sections/func_comp_and_construct.tex": "bc06eda70de577b8bfe7290e9edd6237bfa7d1feaa793de30306e49dd3d6f226",
+    "source/authored/sections/introduction.tex": "cc0766f21b73d5251d78cded3e7573a0a6a4b6cabe454fd06a686ba7f8bcae17",
+    "source/authored/sections/prelim_and_notation.tex": "63836f14cc9a172d631ea42ae31e4cb7ca8c09660923ddaaedbb17543d24d559",
+    "source/authored/sections/related.tex": "8e80ed9742331f8c5d3954ee88e7b71d2db50f6e2050638625010a703c8b551f",
+    "source/authored/sections/tasks.tex": "80ba12f976423975af8377f1e80fb4b4b2622dd8dcca33fe6bbe398a28406408",
+    "source/authored/sections/upper_bound.tex": "de2fba5f19810c4de6cc82a404726376a353b8910414c472363a95102041ee3f"
+  },
+  "paper_linked_code": {
+    "remote": "https://github.com/SprocketLab/hybrid-expressivity",
+    "checkout": "6be8f8fbc2169290af6f4ba5e4bd53a5c6485f7b",
+    "status_porcelain": []
+  },
+  "code_version_selection": "The arXiv source links to the repository without a commit hash. The detached 6be8f8f checkout is the latest repository commit dated before the 2026-03-09 arXiv v1 submission; it is not asserted to be an author-pinned archival snapshot."
+}

VALIDATION.json

@@ -0,0 +1,78 @@
+{
+  "validator": "validate_logbook.py (local bundle validator)",
+  "semantic_v4": {
+    "profile": "semantic-v4-local",
+    "validator": "local_bundle_validator_not_campaign_validator",
+    "checks": [
+      {
+        "name": "frozen_six_claims",
+        "passed": true,
+        "detail": "CLAIMS.json exactly equals the six registered claim strings."
+      },
+      {
+        "name": "complete_eight_routes",
+        "passed": true,
+        "detail": "Index, executive summary, and six claim routes exist."
+      },
+      {
+        "name": "exact_authored_archive",
+        "passed": true,
+        "detail": "arXiv e-print archive matches the retrieval SHA-256."
+      },
+      {
+        "name": "pinned_clean_code_checkout",
+        "passed": true,
+        "detail": "Linked source repository is detached at the recorded clean commit."
+      },
+      {
+        "name": "claim1_literal_falsification",
+        "passed": true,
+        "detail": "Under injectivity the printed RHS is non-positive; one-state 1/2 witnesses are retained."
+      },
+      {
+        "name": "claim2_receptive_field_witnesses",
+        "passed": true,
+        "detail": "Real causal-attention stacks preserve the designed outside-window witness."
+      },
+      {
+        "name": "claim3_construction_and_controls",
+        "passed": true,
+        "detail": "Independent finite construction succeeds while destructive controls fail."
+      },
+      {
+        "name": "claim4_full_vocab_certificate",
+        "passed": true,
+        "detail": "Full-vocabulary construction passes its exact coverage and small-domain gates."
+      },
+      {
+        "name": "claim4_sample_certificate_distinction",
+        "passed": true,
+        "detail": "The one sub-99% finite sample mean is retained separately from the exact coverage certificate."
+      },
+      {
+        "name": "direct_native_notebooks_retained",
+        "passed": true,
+        "detail": "Author notebook cell execution and destructive controls are reported without upgrading scope."
+      },
+      {
+        "name": "claim5_exact_source_scope",
+        "passed": true,
+        "detail": "Figure 4 table is preserved as .999, not silently converted to 1.000."
+      },
+      {
+        "name": "claim6_source_contradiction_marked",
+        "passed": true,
+        "detail": "Figure 6 sixfold row and Figure 5 task distinction are explicitly retained."
+      }
+    ],
+    "passed": true,
+    "check_count": 12
+  },
+  "operational": {
+    "replay_required": true,
+    "byte_identical_replay": true,
+    "no_remote_publication": true,
+    "official_campaign_validator": "not run: no campaign target was created or modified"
+  },
+  "passed": true
+}

audit_source_claims.py

@@ -0,0 +1,157 @@
+"""Exact authored-source audit for the two learned-model claims.
+
+No local GPU training is represented as a rerun: the official training code is
+CUDA-only on this host and the repository contains learning-rate metadata and
+figures, but no per-run scalar result artifacts.  The appropriate evidence is
+therefore the SHA-pinned authored TeX table/caption itself, including any
+internal numeric conflict.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+import re
+from pathlib import Path
+
+import torch
+
+
+ROOT = Path(__file__).resolve().parent
+OUT = ROOT / "outputs"
+TEX = ROOT / "source" / "authored" / "sections" / "experiments.tex"
+TRAIN = ROOT / "source" / "official-code" / "micro_hf" / "train_utils.py"
+
+
+def sha(path: Path) -> str:
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def line_of(text: str, needle: str) -> int:
+    return text[: text.index(needle)].count("\n") + 1
+
+
+def table_after(text: str, heading: str) -> tuple[str, list[dict[str, float]]]:
+    start = text.index(heading)
+    end = text.index("\\end{figure}", start)
+    block = text[start:end]
+    rows = []
+    pat = re.compile(
+        r"\$\\sim(\d+)\$\s*&\s*([0-9.]+)\s*&\s*([0-9.]+)"
+        r"\s*&\s*([0-9.]+)\s*&\s*([0-9.]+)"
+    )
+    for m in pat.finditer(block):
+        rows.append(
+            {
+                "parameters_approx": float(m.group(1)),
+                "pure_tf": float(m.group(2)),
+                "pure_ssm": float(m.group(3)),
+                "tf_to_ssm": float(m.group(4)),
+                "ssm_to_tf": float(m.group(5)),
+            }
+        )
+    if len(rows) != 4:
+        raise RuntimeError(f"expected four rows after {heading!r}, found {len(rows)}")
+    return block, rows
+
+
+def training_route() -> dict[str, object]:
+    text = TRAIN.read_text()
+    return {
+        "official_train_file": str(TRAIN.relative_to(ROOT)),
+        "official_train_file_sha256": sha(TRAIN),
+        "hard_coded_cuda_calls": text.count(".to('cuda')"),
+        "torch_cuda_available_on_this_host": torch.cuda.is_available(),
+        "result_artifacts_present": False,
+        "reason_no_local_training_rerun": (
+            "The official micro_hf entry point hard-codes CUDA tensors; this "
+            "host has no CUDA device. The checkout contains figures and lrs.json "
+            "metadata but no per-run evaluation JSON/CSV/checkpoint results."
+        ),
+    }
+
+
+def main() -> None:
+    OUT.mkdir(exist_ok=True)
+    text = TEX.read_text()
+    var_block, var_rows = table_after(text, "\\noindent \\textbf{Selective Copy.}")
+    mk_block, mk_rows = table_after(text, "\\noindent \\textbf{Multi-Key Associative Recall.}")
+    route = training_route()
+
+    var_2k = next(r for r in var_rows if r["parameters_approx"] == 2000)
+    var_12k = next(r for r in var_rows if r["parameters_approx"] == 12000)
+    claim5 = {
+        "kind": "exact_authored_source_audit",
+        "source": str(TEX.relative_to(ROOT)),
+        "source_sha256": sha(TEX),
+        "table_heading_line": line_of(text, "\\noindent \\textbf{Selective Copy.}"),
+        "caption_line": line_of(text, "At 2000 parameters, hybrid models consistently"),
+        "table_rows": var_rows,
+        "literal_table_comparison": {
+            "hybrid_ssm_to_tf_at_approximately_2000": var_2k["ssm_to_tf"],
+            "pure_tf_at_approximately_12000": var_12k["pure_tf"],
+            "pure_ssm_at_approximately_12000": var_12k["pure_ssm"],
+            "parameter_ratio_12000_over_2000": 12000 / 2000,
+            "strict_table_value_is_exactly_one": var_2k["ssm_to_tf"] == 1.0,
+        },
+        "source_caption_says_perfect": "consistently attain perfect accuracy" in var_block,
+        "source_results_says_roughly_six_times": "factor of $6 \\times$" in text,
+        "assessment": (
+            "The authored table supports the approximate 6x/near-perfect comparison "
+            "(.999 versus .923/.931), but its printed .999 is not literally 1.000 "
+            "while the caption calls it perfect. This is source evidence, not an "
+            "independent learned-model reproduction."
+        ),
+        "training_route": route,
+    }
+
+    mk_2k = next(r for r in mk_rows if r["parameters_approx"] == 2000)
+    mk_6k = next(r for r in mk_rows if r["parameters_approx"] == 6000)
+    mk_12k = next(r for r in mk_rows if r["parameters_approx"] == 12000)
+    single_key_sentence = "none of the pure models achieved greater than 40\\% accuracy"
+    claim6 = {
+        "kind": "exact_authored_source_audit",
+        "source": str(TEX.relative_to(ROOT)),
+        "source_sha256": sha(TEX),
+        "mkar_table_heading_line": line_of(text, "\\noindent \\textbf{Multi-Key Associative Recall.}"),
+        "mkar_caption_line": line_of(text, "hybrid models could perform the task to 60\\%"),
+        "single_key_figure_result_line": line_of(text, single_key_sentence),
+        "mkar_table_rows": mk_rows,
+        "literal_table_checks": {
+            "hybrid_ssm_to_tf_at_approximately_2000": mk_2k["ssm_to_tf"],
+            "hybrid_ssm_to_tf_at_approximately_6000": mk_6k["ssm_to_tf"],
+            "pure_tf_at_approximately_12000": mk_12k["pure_tf"],
+            "sixfold_parameter_ratio_from_2000_to_12000": 12000 / 2000,
+            "hybrid_reaches_0_60_at_approximately_2000": mk_2k["ssm_to_tf"] >= 0.60,
+            "hybrid_reaches_0_60_at_approximately_6000": mk_6k["ssm_to_tf"] >= 0.60,
+            "ratio_for_6000_vs_12000": 12000 / 6000,
+        },
+        "source_caption_claims_60pct_and_six_times": (
+            "60\\% accuracy with $6 \\times$ fewer" in mk_block
+        ),
+        "single_key_statement_is_a_different_task": True,
+        "assessment": (
+            "The authored MKAR table is internally insufficient for the literal "
+            "60%-at-6x conjunction: at the 6x row it reports .512 (<.60), while "
+            "at .990 the nearest shown pure-TF row is only 2x larger. The caption "
+            "asserts the headline, but raw points needed to locate an unshown "
+            "60% crossing were not released. The <=40% statement is explicitly "
+            "about associative recall with decoding (Figure 5), not MKAR (Figure 6)."
+        ),
+        "training_route": route,
+    }
+
+    (OUT / "claim5.json").write_text(json.dumps(claim5, indent=2) + "\n")
+    (OUT / "claim6.json").write_text(json.dumps(claim6, indent=2) + "\n")
+    print(
+        "claim5 source: hybrid@2k=%.3f, pure@12k=(%.3f, %.3f), ratio=%.1f"
+        % (var_2k["ssm_to_tf"], var_12k["pure_tf"], var_12k["pure_ssm"], 6.0)
+    )
+    print(
+        "claim6 source: hybrid@2k=%.3f, hybrid@6k=%.3f, pureTF@12k=%.3f"
+        % (mk_2k["ssm_to_tf"], mk_6k["ssm_to_tf"], mk_12k["pure_tf"])
+    )
+
+
+if __name__ == "__main__":
+    main()

build_logbook.py

@@ -0,0 +1,276 @@
+"""Build static claim pages and provenance from deterministic result artifacts."""
+
+from __future__ import annotations
+
+import hashlib
+import json
+import subprocess
+from pathlib import Path
+
+
+ROOT = Path(__file__).resolve().parent
+OUT = ROOT / "outputs"
+PAGES = ROOT / "pages"
+AUTHORED_ARCHIVE = ROOT / "source" / "2603.08859v1.tar.gz"
+AUTHORED = ROOT / "source" / "authored"
+CODE = ROOT / "source" / "official-code"
+
+
+CLAIM_SLUGS = [
+    "claim-1-theorem-3-3-literal-bound",
+    "claim-2-theorem-3-7-window-bound",
+    "claim-3-theorem-4-3-selective-copy",
+    "claim-4-theorem-4-6-associative-recall",
+    "claim-5-figure-4-selective-copy-learning",
+    "claim-6-figures-5-6-associative-recall-learning",
+]
+
+
+def sha(path: Path) -> str:
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def load(name: str) -> dict:
+    return json.loads((OUT / name).read_text())
+
+
+def code_info() -> dict:
+    return {
+        "remote": "https://github.com/SprocketLab/hybrid-expressivity",
+        "checkout": subprocess.check_output(
+            ["git", "-C", str(CODE), "rev-parse", "HEAD"], text=True
+        ).strip(),
+        "status_porcelain": subprocess.check_output(
+            ["git", "-C", str(CODE), "status", "--porcelain"], text=True
+        ).splitlines(),
+    }
+
+
+def source_provenance() -> dict:
+    files = {}
+    for path in sorted(AUTHORED.rglob("*")):
+        if path.is_file():
+            files[str(path.relative_to(ROOT))] = sha(path)
+    return {
+        "arxiv_id": "2603.08859v1",
+        "arxiv_eprint_url": "https://export.arxiv.org/e-print/2603.08859v1",
+        "archive": str(AUTHORED_ARCHIVE.relative_to(ROOT)),
+        "archive_sha256": sha(AUTHORED_ARCHIVE),
+        "archive_sha256_expected_from_retrieval": "e8d22bfd259aaa60385841d8643109ecb66f7eb1081dd76429f5215f05a032e8",
+        "authored_tree_file_sha256": files,
+        "paper_linked_code": code_info(),
+        "code_version_selection": (
+            "The arXiv source links to the repository without a commit hash. "
+            "The detached 6be8f8f checkout is the latest repository commit dated "
+            "before the 2026-03-09 arXiv v1 submission; it is not asserted to be "
+            "an author-pinned archival snapshot."
+        ),
+    }
+
+
+def claim_page(title: str, status: str, body: str, evidence: list[str], limitations: str) -> str:
+    bullets = "\n".join(f"- {item}" for item in evidence)
+    return (
+        f"# {title}\n\n"
+        f"**Assessment: {status}.**\n\n"
+        f"{body}\n\n"
+        "## Evidence\n\n"
+        f"{bullets}\n\n"
+        "## Scope and limitations\n\n"
+        f"{limitations}\n"
+    )
+
+
+def write_pages() -> dict:
+    claims = json.loads((ROOT / "CLAIMS.json").read_text())["claims"]
+    c1, c2, c3, c4, c5, c6 = [load(f"claim{i}.json") for i in range(1, 7)]
+    n3, n4 = load("claim3_native.json"), load("claim4_native.json")
+    PAGES.mkdir(exist_ok=True)
+
+    pages = []
+    bodies = [
+        claim_page(
+            claims[0]["text"],
+            "falsified as a literal positive linear lower-bound claim",
+            "Assumption 3.2 makes G: V^m → Y^q injective, so cardinality gives "
+            "m·log₂|V| ≤ q·log₂|Y|. The printed right-hand side "
+            "m·log|V| − q·log|Y| is therefore never positive under its own "
+            "assumption. The appendix instead derives a different Fano bound under "
+            "error < 1/8; that is not the printed probability-1/2 theorem.",
+            [
+                "Exact authored statement: source/authored/sections/func_comp_and_construct.tex:26–28.",
+                f"1,296 admissible cardinality configurations: maximum printed RHS = {c1['printed_bound_audit']['max_literal_bound_over_admissible_grid']:.1f}.",
+                "Exhaustive binary selective-copy partitions: one state attains exactly 1/2 success for m=2 and m=3, matching the theorem's printed threshold.",
+                "The appendix source records its stronger, different prerequisite as err < 1/8 (source/authored/appendix/missing_proof_lb.tex).",
+            ],
+            "This is a literal-statement audit, not a claim that no corrected lower "
+            "bound can be proved. The exact selective-copy enumeration is finite; the "
+            "cardinality implication itself is general.",
+        ),
+        claim_page(
+            claims[1]["text"],
+            "supported by explicit paper-task witnesses and exact receptive-field checks",
+            "For the selective-copy witnesses, changing only a token outside the "
+            "causal receptive field leaves the terminal logits bit-identical. On the "
+            "two-point witness distribution this forces accuracy 1/2, below 2/3. "
+            "A full-window control separates the same pair.",
+            [
+                f"{c2['receptive_field_sweep']['configs_tested']} real float64 causal-attention stacks; outside-RF violations = {c2['receptive_field_sweep']['violations_outside_receptive_field']}.",
+                f"Maximum terminal-logit change outside RF = {c2['max_abs_delta_when_sumW_below_R']:.1f} for the hard witnesses.",
+                f"Full-window control separates {c2['control_full_window_separates_frac']:.0%} of tested witness configurations.",
+                "The source proof explicitly identifies the terminal dependency as the last sum_i W_i tokens (appendix/missing_proof_lb.tex).",
+            ],
+            "Finite numerical checks cannot prove the universal theorem. They directly "
+            "exercise its stated quantities (window, R, indistinguishability and "
+            "success threshold) on the paper's selective-copy task rather than a proxy.",
+        ),
+        claim_page(
+            claims[2]["text"],
+            "supported for the literal construction; official notebook does not establish its universal scope",
+            "A direct transcription of Appendix D.2/D.5 achieves exact final-position "
+            "selective copying across exhaustive and large deterministic sweeps, with a "
+            "window-minus-one and no-SSM destructive controls. The separately executed "
+            "author notebook is included unchanged as provenance and scores only .825 on "
+            "the notebook generator's 1,024 final-position examples, so it is not used "
+            "as evidence of the theorem's 'every input' quantifier.",
+            [
+                f"Independent construction: {c3['total_inputs_tested']:,} valid inputs; minimum accuracy = {c3['min_accuracy_over_all_configs']:.6f}.",
+                "Includes L=100, |V|=32 and longer L=1,024/4,096 sweeps; these are labelled construction checks, not learned-model runs.",
+                f"Native notebook baseline/control: {n3['baseline']['last_position_accuracy']:.6f} / {n3['destructive_control']['last_position_accuracy']:.6f} on {n3['baseline']['eligible_last_position_examples']} eligible examples.",
+                "The destructive construction controls remove one attention position, zero the query, or disable selective state update.",
+            ],
+            "The independent checker is a faithful finite implementation of the written "
+            "construction, not a mechanized proof. The native notebook's aggregate failure "
+            "is retained rather than overwritten or relabelled as a success.",
+        ),
+        claim_page(
+            claims[3]["text"],
+            "supported by a full-vocabulary construction and exact coverage certificate; native notebook is only a limited check",
+            "The Appendix D.4 construction was implemented with the stated full-vocabulary "
+            "binary code, last-match positional bias, and a finite softmax separation. "
+            "Its exact iid coverage formula is at least .99 for each measured scale; the "
+            "5×20,000-instance point estimates track that certificate (one mean, .989820, "
+            "is below .99 and is not relabelled as an empirical pass). The implementation "
+            "also exhausts three small full-window domains.",
+            [
+                f"All analytic coverage certificates meet 99%: {c4['all_meet_99pct']}.",
+                f"Minimum five-seed empirical mean = {c4['min_success_at_theorem_window']:.6f}; maximum gap from exact coverage = {c4['max_gap_vs_analytic']:.6f}.",
+                "Exhaustive full-window domains (|M|, L)=(4,8),(4,9),(8,8) all score 1.0.",
+                f"Native authored notebook baseline/control: {n4['baseline']['last_position_accuracy']:.6f} / {n4['destructive_control']['last_position_accuracy']:.6f}; it has no windowed 99%-coverage test.",
+            ],
+            "The native notebook is a direct one-construction execution, not the paper's "
+            "probabilistic window experiment. The 99% support comes from the written "
+            "construction plus an exact iid coverage calculation and finite tests; it is "
+            "not a claimed learned-model training rerun.",
+        ),
+        claim_page(
+            claims[4]["text"],
+            "authored-source supported, but not independently reproduced",
+            "The exact authored Figure 4 table gives SSM→TF .999 at approximately 2,000 "
+            "parameters and pure TF/SSM .923/.931 at approximately 12,000 parameters, a "
+            "sixfold nominal parameter ratio. The caption calls .999 'perfect', so the "
+            "strict word perfect and printed value are internally inconsistent at the "
+            "shown precision.",
+            [
+                f"Pinned source table: hybrid@2k={c5['literal_table_comparison']['hybrid_ssm_to_tf_at_approximately_2000']:.3f}; pure TF/SSM@12k={c5['literal_table_comparison']['pure_tf_at_approximately_12000']:.3f}/{c5['literal_table_comparison']['pure_ssm_at_approximately_12000']:.3f}.",
+                "The TeX table, caption, and source-file SHA are in outputs/claim5.json.",
+                "The official micro_hf training path contains hard-coded CUDA transfers and this machine has no CUDA device; no local run is presented as a reproduction.",
+            ],
+            "The repository did not contain per-run scalar results, checkpoints, or an "
+            "author-pinned code commit in the arXiv archive. This page reports the paper's "
+            "own exact table, not an independent empirical confirmation.",
+        ),
+        claim_page(
+            claims[5]["text"],
+            "not independently established; the authored MKAR table conflicts with the literal numeric conjunction",
+            "The Figure 6 table's sixfold row is approximately 2,000 versus 12,000 "
+            "parameters, where SSM→TF is .512—not 60%. At the first shown hybrid result "
+            "above 60% (.990 at approximately 6,000), the nearest shown pure-TF row is "
+            "approximately 12,000, only 2× larger. The caption asserts 60%-at-6×, but the "
+            "raw points needed to locate a different crossing were not released.",
+            [
+                f"Pinned MKAR values: hybrid@2k={c6['literal_table_checks']['hybrid_ssm_to_tf_at_approximately_2000']:.3f}, hybrid@6k={c6['literal_table_checks']['hybrid_ssm_to_tf_at_approximately_6000']:.3f}, pure TF@12k={c6['literal_table_checks']['pure_tf_at_approximately_12000']:.3f}.",
+                "The <=40% sentence is from Figure 5's associative recall with decoding, a different task from Figure 6's MKAR; it is not treated as an MKAR plateau measurement.",
+                "The exact TeX/table SHA and CUDA-only non-rerun route are recorded in outputs/claim6.json.",
+            ],
+            "This is an authored-source/data audit. It does not infer a curve between "
+            "unreleased points or substitute the Figure 5 task for MKAR. A GPU rerun would "
+            "require compatible CUDA hardware and a declared protocol; neither is claimed here.",
+        ),
+    ]
+
+    for slug, page in zip(CLAIM_SLUGS, bodies):
+        dest = PAGES / slug
+        dest.mkdir(exist_ok=True)
+        (dest / "page.md").write_text(page)
+        pages.append(str((dest / "page.md").relative_to(ROOT)))
+
+    rows = [
+        ("1", "Theorem 3.3 literal lower bound", "falsified", "printed RHS non-positive under injectivity"),
+        ("2", "Theorem 3.7 window lower bound", "supported", "paper-task witnesses and real attention stacks"),
+        ("3", "Theorem 4.3 selective copy", "supported", "independent construction; native notebook limitation retained"),
+        ("4", "Theorem 4.6 associative recall", "supported", "full-vocabulary construction and exact coverage"),
+        ("5", "Figure 4 learned selective copy", "source-supported", "reported table only; .999/perfect rounding conflict"),
+        ("6", "Figures 5–6 learned recall", "not established", "MKAR table conflicts with 60%-at-6x conjunction"),
+    ]
+    table = "\n".join(f"| {a} | {b} | {c} | {d} |" for a, b, c, d in rows)
+    executive = (
+        "# Executive summary\n\n"
+        "This is a six-claim audit of arXiv:2603.08859v1 / OpenReview `82EJxJzG6r`. "
+        "Claim text is frozen in `CLAIMS.json`; evidence does not rewrite scope.\n\n"
+        "| claim | subject | assessment | headline |\n| --- | --- | --- | --- |\n"
+        + table
+        + "\n\nThe package distinguishes (1) native author-notebook execution, "
+        "(2) an independent implementation of the written constructions, and (3) "
+        "exact authored-table evidence. GPU-only learned-model training was not rerun "
+        "on this non-CUDA host.\n"
+    )
+    (PAGES / "executive-summary").mkdir(exist_ok=True)
+    (PAGES / "executive-summary" / "page.md").write_text(executive)
+    index = (
+        "# Reproduction audit: Expressivity-Efficiency Tradeoffs for Hybrid Sequence Models\n\n"
+        "Paper: arXiv:2603.08859v1 · OpenReview `82EJxJzG6r`\n\n"
+        "Six registered claims are preserved verbatim in `CLAIMS.json`. Run `python3 run_all.py` "
+        "for paired deterministic replays, page generation, validation, and a recursive manifest.\n"
+    )
+    (PAGES / "index.md").write_text(index)
+    return {"claim_pages": pages}
+
+
+def write_logbook(route_info: dict) -> None:
+    claims = json.loads((ROOT / "CLAIMS.json").read_text())["claims"]
+    children = [
+        {"slug": "executive-summary", "title": "Executive summary", "file": "pages/executive-summary/page.md", "children": []}
+    ]
+    for claim, slug in zip(claims, CLAIM_SLUGS):
+        children.append(
+            {
+                "slug": slug,
+                "title": f"Claim {claim['index']}: {claim['text']}",
+                "file": f"pages/{slug}/page.md",
+                "children": [],
+            }
+        )
+    logbook = {
+        "schema_version": 1,
+        "title": "Reproduction audit: Hybrid Sequence Models",
+        "emoji": "🔬",
+        "proposed_target_slug": "repro-hybrid-seq-82ejxjzg6r",
+        "publication_status": "local-only; no Space created or modified",
+        "paper": {"arxiv_id": "2603.08859v1", "openreview_id": "82EJxJzG6r"},
+        "updated_at": "2026-07-28T00:00:00+00:00",
+        "root": {"slug": "index", "title": "Reproduction audit", "file": "pages/index.md", "children": children},
+        "routes_built": route_info,
+    }
+    (ROOT / "logbook.json").write_text(json.dumps(logbook, indent=2) + "\n")
+
+
+def main() -> None:
+    route_info = write_pages()
+    write_logbook(route_info)
+    (ROOT / "SOURCE_PROVENANCE.json").write_text(json.dumps(source_provenance(), indent=2) + "\n")
+    print("built six claim pages, logbook routes, and source provenance")
+
+
+if __name__ == "__main__":
+    main()

exp1_ssm_bound.py

@@ -0,0 +1,409 @@
+"""Claim 1 - Theorem 3.3: SSM state-space lower bound for function composition.
+
+The paper (Definition 3.1) considers F(u, v) with u in V^m, v in V^n, and
+Assumption 3.2: there is a query set Q = {v^(1),...,v^(q)} such that
+    G(u) := (F(u,v^(1)), ..., F(u,v^(q)))
+is an injection.  Theorem 3.3 then asserts that any k-layer SSM computing F
+needs  sum_i log|S_i| >= Omega(m log|V| - q log|Y|).
+
+We instantiate the *canonical* member of the family, the one the paper itself
+uses in Theorem 4.2 / D.1 for selective copying:
+
+    F(u, v) = u_v ,   u in V^m,  v in [m],   Y = V,   Q = {1,...,m},  q = m.
+
+G(u) = u is an injection, so Assumption 3.2 holds by construction.
+
+What we compute (all exact, no sampling):
+
+  A. Fooling set.  All |V|^m prefixes are pairwise separated by some query.
+     Checked over every unordered pair.
+  B. Minimum state count.  A single-layer SSM's behaviour on this task is
+     determined by (i) the partition of prefixes induced by the state after
+     reading u and (ii) an arbitrary decoder r(state, query).  We grant the
+     decoder unlimited power, which only *weakens* the lower bound we verify.
+     Exhaustive enumeration over ALL set partitions gives the exact optimal
+     accuracy A*(s) for every state budget s.  A*(s) = 1 iff s >= |V|^m.
+  C. Negative control with a closed form.  With exactly |V|^m - 1 states the
+     best possible accuracy is exactly 1 - 1/(m |V|^m): one pair of prefixes
+     must be merged, the cheapest pair is at Hamming distance 1, and merging
+     costs exactly one wrong (prefix, query) cell.  Verified against brute
+     force over all C(N,2) merges.
+  D. Lemma 3.5.  k stacked SSM layers are simulated by one layer whose state
+     space is the product; verified behaviourally on all short inputs, and the
+     product bound is shown tight on a witness.
+  E. Audit of the two constants in the printed statement of Theorem 3.3.
+
+Run:  python3 exp1_ssm_bound.py
+"""
+
+import itertools
+import json
+import math
+import os
+import random
+
+import numpy as np
+
+OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "outputs")
+os.makedirs(OUT, exist_ok=True)
+
+
+# --------------------------------------------------------------------------
+# A. fooling set: every pair of prefixes is separated by some query in Q
+# --------------------------------------------------------------------------
+def fooling_set(m, V):
+    """Return (n_prefixes, n_pairs, n_separated, min_separating_queries)."""
+    prefixes = list(itertools.product(range(V), repeat=m))
+    n = len(prefixes)
+    sep = 0
+    min_sep_q = m + 1
+    for a in range(n):
+        ua = prefixes[a]
+        for b in range(a + 1, n):
+            ub = prefixes[b]
+            # queries j in [m] on which the required answers differ
+            k = sum(1 for j in range(m) if ua[j] != ub[j])
+            if k > 0:
+                sep += 1
+            min_sep_q = min(min_sep_q, k)
+    return n, n * (n - 1) // 2, sep, min_sep_q
+
+
+# --------------------------------------------------------------------------
+# B. exact optimal accuracy for a given state budget, by exhaustive partition
+# --------------------------------------------------------------------------
+def _partitions(collection):
+    """All set partitions of a list (Knuth / standard recursive generator)."""
+    collection = list(collection)
+    if len(collection) == 1:
+        yield [collection]
+        return
+    first = collection[0]
+    for smaller in _partitions(collection[1:]):
+        for i, subset in enumerate(smaller):
+            yield smaller[:i] + [[first] + subset] + smaller[i + 1:]
+        yield [[first]] + smaller
+
+
+def block_correct(block, m):
+    """Number of correct (prefix, query) cells achievable by the best decoder."""
+    tot = 0
+    for j in range(m):
+        counts = {}
+        for u in block:
+            counts[u[j]] = counts.get(u[j], 0) + 1
+        tot += max(counts.values())
+    return tot
+
+
+def exact_accuracy_curve(m, V):
+    """A*(s) for every s = 1..|V|^m, by exhaustive enumeration of partitions."""
+    prefixes = list(itertools.product(range(V), repeat=m))
+    n = len(prefixes)
+    best = {s: 0 for s in range(1, n + 1)}
+    total_cells = n * m
+    for part in _partitions(prefixes):
+        s = len(part)
+        c = sum(block_correct(b, m) for b in part)
+        if c > best[s]:
+            best[s] = c
+    # A*(s) is monotone non-decreasing in s (more states never hurt)
+    run = 0
+    curve = {}
+    for s in range(1, n + 1):
+        run = max(run, best[s])
+        curve[s] = run / total_cells
+    return curve
+
+
+def min_states_for(curve, p):
+    for s in sorted(curve):
+        if curve[s] >= p - 1e-12:
+            return s
+    return None
+
+
+# --------------------------------------------------------------------------
+# C. negative control: |V|^m - 1 states, exact closed form
+# --------------------------------------------------------------------------
+def best_accuracy_one_merge(m, V):
+    """Brute force over all C(N,2) merges; the rest of the prefixes stay alone."""
+    prefixes = list(itertools.product(range(V), repeat=m))
+    n = len(prefixes)
+    total_cells = n * m
+    best = -1
+    witness = None
+    for a in range(n):
+        for b in range(a + 1, n):
+            ua, ub = prefixes[a], prefixes[b]
+            # everything outside the merged pair is correct on all m queries
+            c = (n - 2) * m + block_correct([ua, ub], m)
+            if c > best:
+                best = c
+                witness = (ua, ub)
+    return best / total_cells, witness
+
+
+# --------------------------------------------------------------------------
+# D. Lemma 3.5: k SSM layers collapse to one with product state space
+# --------------------------------------------------------------------------
+def lemma_35_check(rng, k, sizes, alphabet, max_len):
+    """Build k random SSM layers, simulate them stacked and as a product
+    automaton, and compare outputs on every input string up to max_len."""
+    layers = []
+    alpha_in = alphabet
+    for j in range(k):
+        s = sizes[j]
+        u = rng.integers(0, s, size=(s, alpha_in))       # update rule
+        r = rng.integers(0, alphabet, size=s)            # output map
+        layers.append((u, r, s))
+        alpha_in = alphabet
+
+    def run_stacked(seq):
+        cur = list(seq)
+        for (u, r, s) in layers:
+            st = 0
+            out = []
+            for tok in cur:
+                st = int(u[st, tok])
+                out.append(int(r[st]))
+            cur = out
+        return tuple(cur)
+
+    # single-layer product simulation
+    prod_states = {}
+
+    def run_product(seq):
+        st = tuple(0 for _ in layers)
+        out = []
+        for tok in seq:
+            new = []
+            sym = tok
+            for (u, r, s), cs in zip(layers, st):
+                ns = int(u[cs, sym])
+                new.append(ns)
+                sym = int(r[ns])
+            st = tuple(new)
+            prod_states[st] = True
+            out.append(sym)
+        return tuple(out)
+
+    mismatches = 0
+    tested = 0
+    for L in range(1, max_len + 1):
+        for seq in itertools.product(range(alphabet), repeat=L):
+            tested += 1
+            if run_stacked(seq) != run_product(seq):
+                mismatches += 1
+    return {
+        "k": k,
+        "sizes": list(sizes),
+        "product_bound": int(np.prod(sizes)),
+        "reachable_product_states": len(prod_states),
+        "sequences_tested": tested,
+        "behaviour_mismatches": mismatches,
+        "bound_respected": bool(len(prod_states) <= int(np.prod(sizes))),
+    }
+
+
+# --------------------------------------------------------------------------
+# E. audit of the printed bound
+# --------------------------------------------------------------------------
+def h2(p):
+    if p <= 0 or p >= 1:
+        return 0.0
+    return -p * math.log2(p) - (1 - p) * math.log2(1 - p)
+
+
+def audit_printed_bound():
+    """Two independent audits of the literal statement of Theorem 3.3.
+
+    (i) The body writes  Omega(m log|V| - q log|Y|).  Assumption 3.2 requires
+        G: V^m -> Y^q to be injective, hence |V|^m <= |Y|^q, hence
+        m log|V| - q log|Y| <= 0 for EVERY admissible (m,|V|,q,|Y|).
+    (ii) The appendix (proof of Theorem 3.3, eq. before the Q.E.D.) actually
+         derives  log|S| >= m log|V| - q (H2(1/8) + log|Y|/8)  under err < 1/8.
+         That form is strictly positive and linear in m.
+    """
+    rows = []
+    worst_literal = -math.inf
+    for m in range(1, 13):
+        for Vs in (2, 3, 4, 8, 16, 32):
+            for Ys in (2, 3, 4, 8, 16, 32):
+                # smallest q for which injectivity V^m -> Y^q is possible
+                qmin = math.ceil(m * math.log2(Vs) / math.log2(Ys))
+                for q in (qmin, qmin + 1, qmin + 3):
+                    lit = m * math.log2(Vs) - q * math.log2(Ys)
+                    app = m * math.log2(Vs) - q * (h2(1 / 8) + math.log2(Ys) / 8)
+                    worst_literal = max(worst_literal, lit)
+                    rows.append({"m": m, "V": Vs, "Y": Ys, "q": q,
+                                 "literal": lit, "appendix": app})
+    # the selective-copying instantiation used in Theorem 4.2: m = q = N, Y = V
+    inst = []
+    for Ms in (2, 4, 8, 16, 26, 32):
+        for N in range(2, 13):
+            lit = N * math.log2(Ms) - N * math.log2(Ms)
+            app = N * math.log2(Ms) - N * (h2(1 / 8) + math.log2(Ms) / 8)
+            inst.append({"N": N, "M": Ms, "literal": lit, "appendix": app})
+    # slope of the appendix form in N, for each M
+    slopes = {}
+    for Ms in (2, 4, 8, 16, 26, 32):
+        xs = np.array([r["N"] for r in inst if r["M"] == Ms], float)
+        ys = np.array([r["appendix"] for r in inst if r["M"] == Ms], float)
+        A = np.vstack([xs, np.ones_like(xs)]).T
+        coef, *_ = np.linalg.lstsq(A, ys, rcond=None)
+        pred = math.log2(Ms) - (h2(1 / 8) + math.log2(Ms) / 8)
+        slopes[str(Ms)] = {"fitted_slope": float(coef[0]),
+                           "closed_form_slope": pred,
+                           "abs_err": abs(float(coef[0]) - pred)}
+    return {
+        "n_admissible_configs": len(rows),
+        "max_literal_bound_over_admissible_grid": worst_literal,
+        "literal_bound_ever_positive": bool(worst_literal > 0),
+        "min_appendix_bound_on_instantiation": min(r["appendix"] for r in inst),
+        "appendix_slope_in_m": slopes,
+        "note": ("literal = m*log2|V| - q*log2|Y| (Thm 3.3 as printed); "
+                 "appendix = m*log2|V| - q*(H2(1/8)+log2|Y|/8) (proof, err<1/8)"),
+    }
+
+
+def trivial_guessing_accuracy(Ys):
+    """A 1-state (constant) SSM's success probability under uniform Y."""
+    return 1.0 / Ys
+
+
+# --------------------------------------------------------------------------
+def main():
+    res = {"task": "F(u,v) = u_v with Q = [m]; G(u)=u is injective (Asm 3.2)"}
+
+    # ---- A + B + C on a grid ------------------------------------------
+    grid = []
+    for V in (2, 3, 4):
+        for m in range(2, 11):
+            n = V ** m
+            if n > 4096:
+                continue
+            row = {"m": m, "V": V, "n_prefixes": n}
+            row["log2_states_required"] = math.log2(n)
+            row["theory_m_log2_V"] = m * math.log2(V)
+            row["abs_residual"] = abs(row["log2_states_required"] - row["theory_m_log2_V"])
+            if n <= 512:
+                np_, pairs, sep, minq = fooling_set(m, V)
+                row["pairs_checked"] = pairs
+                row["pairs_separated"] = sep
+                row["all_pairs_separated"] = bool(sep == pairs)
+                row["min_separating_queries"] = minq
+            # closed-form negative control
+            row["control_states"] = n - 1
+            row["control_acc_closed_form"] = 1.0 - 1.0 / (m * n)
+            if n <= 256:
+                acc, wit = best_accuracy_one_merge(m, V)
+                row["control_acc_bruteforce"] = acc
+                row["control_acc_gap"] = abs(acc - row["control_acc_closed_form"])
+                row["control_witness_pair"] = [list(wit[0]), list(wit[1])]
+            grid.append(row)
+    res["grid"] = grid
+    res["max_residual_log2_states"] = max(r["abs_residual"] for r in grid)
+    res["max_control_gap"] = max(r["control_acc_gap"] for r in grid
+                                 if "control_acc_gap" in r)
+    res["all_pairs_separated_everywhere"] = all(
+        r.get("all_pairs_separated", True) for r in grid)
+
+    # ---- linear-in-m fit ------------------------------------------------
+    fits = {}
+    for V in (2, 3, 4):
+        xs = np.array([r["m"] for r in grid if r["V"] == V], float)
+        ys = np.array([r["log2_states_required"] for r in grid if r["V"] == V], float)
+        A = np.vstack([xs, np.ones_like(xs)]).T
+        coef, *_ = np.linalg.lstsq(A, ys, rcond=None)
+        pred = A @ coef
+        ss_res = float(np.sum((ys - pred) ** 2))
+        ss_tot = float(np.sum((ys - ys.mean()) ** 2))
+        fits[str(V)] = {
+            "fitted_slope": float(coef[0]),
+            "theory_slope_log2_V": math.log2(V),
+            "slope_abs_err": abs(float(coef[0]) - math.log2(V)),
+            "intercept": float(coef[1]),
+            "R2": 1.0 - ss_res / ss_tot,
+            "max_abs_residual": float(np.max(np.abs(ys - pred))),
+        }
+    res["linear_in_m_fit"] = fits
+
+    # ---- exhaustive optimal-accuracy curves ------------------------------
+    curves = []
+    for (m, V) in [(2, 2), (3, 2), (2, 3)]:
+        c = exact_accuracy_curve(m, V)
+        n = V ** m
+        curves.append({
+            "m": m, "V": V, "n_prefixes": n,
+            "partitions_enumerated": "Bell(%d)" % n,
+            "A_star": {str(s): c[s] for s in sorted(c)},
+            "min_states_acc_1.0": min_states_for(c, 1.0),
+            "min_states_acc_0.875": min_states_for(c, 0.875),
+            "min_states_acc_0.9": min_states_for(c, 0.9),
+            "min_states_acc_0.5": min_states_for(c, 0.5),
+            "A_star_at_n_minus_1": c[n - 1],
+            "closed_form_at_n_minus_1": 1.0 - 1.0 / (m * n),
+        })
+    res["exhaustive_accuracy_curves"] = curves
+    res["exhaustive_min_states_equals_Vm"] = all(
+        c["min_states_acc_1.0"] == c["n_prefixes"] for c in curves)
+    res["max_curve_control_gap"] = max(
+        abs(c["A_star_at_n_minus_1"] - c["closed_form_at_n_minus_1"]) for c in curves)
+
+    # ---- D. Lemma 3.5 ----------------------------------------------------
+    rng = np.random.default_rng(0)
+    lem = []
+    for trial, (k, sizes, alpha, ml) in enumerate([
+        (2, (2, 3), 2, 8), (3, (2, 2, 2), 2, 7), (2, (3, 4), 3, 5),
+        (4, (2, 2, 2, 2), 2, 6), (3, (2, 3, 2), 3, 4)]):
+        lem.append(lemma_35_check(rng, k, sizes, alpha, ml))
+    res["lemma_3_5"] = lem
+    res["lemma_3_5_total_mismatches"] = sum(x["behaviour_mismatches"] for x in lem)
+    res["lemma_3_5_bound_always_respected"] = all(x["bound_respected"] for x in lem)
+
+    # tightness: search random layer stacks for one whose reachable product
+    # state count meets the product bound with equality
+    tight = []
+    for (k, sizes, alpha, ml) in [(2, (2, 3), 3, 6), (2, (3, 3), 3, 6),
+                                  (3, (2, 2, 2), 3, 6)]:
+        best = None
+        for _ in range(4000):
+            r = lemma_35_check(np.random.default_rng(rng.integers(1 << 30)),
+                               k, sizes, alpha, ml)
+            if best is None or r["reachable_product_states"] > best["reachable_product_states"]:
+                best = r
+            if r["reachable_product_states"] == r["product_bound"]:
+                break
+        best["bound_is_tight"] = bool(
+            best["reachable_product_states"] == best["product_bound"])
+        tight.append(best)
+    res["lemma_3_5_tightness"] = tight
+    res["lemma_3_5_tight_witnesses"] = sum(1 for t in tight if t["bound_is_tight"])
+
+    # ---- E. audit of the printed bound ------------------------------------
+    res["printed_bound_audit"] = audit_printed_bound()
+    res["one_state_guessing_accuracy"] = {
+        str(y): trivial_guessing_accuracy(y) for y in (2, 3, 4, 8, 26, 32)}
+
+    with open(os.path.join(OUT, "claim1.json"), "w") as f:
+        json.dump(res, f, indent=1)
+
+    print("max residual log2(states) vs m*log2|V| :",
+          res["max_residual_log2_states"])
+    print("all fooling pairs separated            :",
+          res["all_pairs_separated_everywhere"])
+    print("exhaustive min-states == |V|^m         :",
+          res["exhaustive_min_states_equals_Vm"])
+    print("max |bruteforce - closed form| control :", res["max_control_gap"])
+    print("Lemma 3.5 mismatches                   :",
+          res["lemma_3_5_total_mismatches"])
+    print("max of literal bound over grid         :",
+          res["printed_bound_audit"]["max_literal_bound_over_admissible_grid"])
+    for V, f in fits.items():
+        print("  |V|=%s  slope=%.16f (theory %.16f)  R2=%.16f"
+              % (V, f["fitted_slope"], f["theory_slope_log2_V"], f["R2"]))
+
+
+if __name__ == "__main__":
+    main()

exp2_window_bound.py

@@ -0,0 +1,353 @@
+"""Claim 2 - Theorem 3.7: sliding-window Transformers need total window >= R.
+
+Assumption 3.6 (R-local sensitivity): there exist x, x' with
+    x[L-R+1:L] == x'[L-R+1:L]   but   F(x) != F(x').
+Theorem 3.7: any stack of k Transformer layers with window sizes W_1..W_k that
+computes F with probability 2/3 must satisfy sum_i W_i >= R.
+
+Three independent pieces of evidence, all exact:
+
+  A. RECEPTIVE FIELD.  Run *real* float64 sliding-window causal Transformer
+     stacks (softmax attention + MLP + residual + layernorm) with random
+     weights.  Perturb input position p, read the output at position L.
+     Prediction: the output changes only for p > L - (1 + sum_i (W_i - 1)).
+     Since 1 + sum(W_i - 1) <= sum W_i, sum_i W_i < R implies the two
+     local-sensitivity witnesses are indistinguishable.  We record the maximum
+     |delta| in the output logits outside the receptive field; the theorem
+     requires it to be exactly 0.
+
+  B. WITNESSES.  Explicit local-sensitivity witness pairs for the paper's own
+     selective-copying task (Definition 4.1).  We search for the largest R
+     admitting a witness and compare to the paper's claim (R = L/2 in the
+     proof of Theorem 4.2, giving Omega(L)).
+
+  C. THE ACTUAL FAILURE PROBABILITY.  Feed the witness pair through a
+     window-limited stack; if sum W_i < R the two outputs are bit-identical,
+     so under the uniform distribution on {x, x'} the model is correct with
+     probability exactly 1/2 < 2/3.  Measured, not assumed.
+
+  NEGATIVE CONTROL.  With sum_i W_i >= R we *construct* a stack that separates
+  the same witness pair (outputs differ), so the bound is tight rather than
+  vacuous.
+
+Run:  python3 exp2_window_bound.py
+"""
+
+import json
+import math
+import os
+
+import numpy as np
+
+OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "outputs")
+os.makedirs(OUT, exist_ok=True)
+
+
+# Apple Accelerate raises spurious divide/overflow RuntimeWarnings inside
+# matmul even on finite inputs.  numerical_gate() below re-derives the same
+# products with einsum and asserts finiteness before we silence them.
+np.seterr(all="ignore")
+
+
+def numerical_gate():
+    rng = np.random.default_rng(0)
+    worst = 0.0
+    for n in (8, 16, 64, 256, 512):
+        A = rng.normal(size=(n, 8))
+        B = rng.normal(size=(8, 16))
+        worst = max(worst, float(np.max(np.abs(A @ B - np.einsum("ij,jk->ik", A, B)))))
+    finite = True
+    mx = 0.0
+    for L, k in ((64, 4), (128, 3), (256, 4), (512, 4)):
+        m = WindowedTransformer(L, 8, [max(2, (L - 2) // k)] * k, seed=L + k)
+        y = m.forward(np.random.default_rng(L).integers(0, 8, size=L))
+        finite = finite and bool(np.all(np.isfinite(y)))
+        mx = max(mx, float(np.max(np.abs(y))))
+    return {"max_abs_matmul_minus_einsum": worst,
+            "all_forward_outputs_finite": finite,
+            "max_abs_logit": mx,
+            "note": "Accelerate RuntimeWarnings are spurious; verified here"}
+
+
+def jsonable(o):
+    if isinstance(o, (np.integer,)):
+        return int(o)
+    if isinstance(o, (np.floating,)):
+        return float(o)
+    if isinstance(o, np.ndarray):
+        return o.tolist()
+    raise TypeError(str(type(o)))
+
+
+# --------------------------------------------------------------------------
+# a real sliding-window causal Transformer stack (float64, numpy)
+# --------------------------------------------------------------------------
+def layernorm(X, eps=1e-5):
+    mu = X.mean(axis=1, keepdims=True)
+    sd = X.std(axis=1, keepdims=True)
+    return (X - mu) / (sd + eps)
+
+
+def softmax_rows(S):
+    S = S - S.max(axis=1, keepdims=True)
+    E = np.exp(S)
+    return E / E.sum(axis=1, keepdims=True)
+
+
+class WindowedTransformer:
+    """k layers, layer i has causal sliding window W_i (attends to the W_i most
+    recent positions, itself included)."""
+
+    def __init__(self, L, d, windows, seed):
+        rng = np.random.default_rng(seed)
+        self.L, self.d, self.windows = L, d, list(windows)
+        self.emb = rng.normal(size=(64, d))          # token -> R^d
+        self.pos = rng.normal(size=(L, d)) * 0.1
+        self.layers = []
+        for W in windows:
+            self.layers.append({
+                "Wq": rng.normal(size=(d, d)) / math.sqrt(d),
+                "Wk": rng.normal(size=(d, d)) / math.sqrt(d),
+                "Wv": rng.normal(size=(d, d)) / math.sqrt(d),
+                "Wo": rng.normal(size=(d, d)) / math.sqrt(d),
+                "U1": rng.normal(size=(d, 2 * d)) / math.sqrt(d),
+                "U2": rng.normal(size=(2 * d, d)) / math.sqrt(2 * d),
+                "W": W,
+            })
+        self.head = rng.normal(size=(d, 8)) / math.sqrt(d)
+        # causal sliding-window masks
+        idx = np.arange(L)
+        self.masks = []
+        for W in windows:
+            m = (idx[:, None] >= idx[None, :]) & (idx[:, None] - idx[None, :] < W)
+            self.masks.append(m)
+
+    def forward(self, tokens):
+        X = self.emb[np.asarray(tokens)] + self.pos
+        for lay, mask in zip(self.layers, self.masks):
+            Q, K, V = X @ lay["Wq"], X @ lay["Wk"], X @ lay["Wv"]
+            S = (Q @ K.T) / math.sqrt(self.d)
+            S = np.where(mask, S, -np.inf)
+            A = softmax_rows(S)
+            X = X + (A @ V) @ lay["Wo"]
+            X = layernorm(X)
+            H = X @ lay["U1"]
+            X = X + np.maximum(H, 0) @ lay["U2"]
+            X = layernorm(X)
+        return X[-1] @ self.head           # logits at the last position
+
+
+def receptive_field(windows):
+    """Positions L-rf+1..L can influence output L; rf = 1 + sum(W_i - 1)."""
+    return 1 + sum(w - 1 for w in windows)
+
+
+# --------------------------------------------------------------------------
+# A. receptive-field sweep with real float attention
+# --------------------------------------------------------------------------
+def receptive_field_sweep(Ls, seeds, d=8, vocab=8):
+    rows = []
+    worst_outside = 0.0
+    worst_inside_zero = 0
+    violations = 0
+    n_cfg = 0
+    rng = np.random.default_rng(12345)
+    for L in Ls:
+        for k in (1, 2, 3, 4):
+            for s in range(seeds):
+                cap = max(2, (L - 2) // k)
+                windows = [int(rng.integers(1, cap + 1)) for _ in range(k)]
+                rf = receptive_field(windows)
+                if rf >= L:
+                    continue
+                n_cfg += 1
+                model = WindowedTransformer(L, d, windows, seed=1000 * L + 10 * k + s)
+                base = np.array(rng.integers(0, vocab, size=L))
+                y0 = model.forward(base)
+                # perturb one position strictly outside the receptive field
+                p_out = L - rf - 1                       # 0-indexed
+                alt = base.copy()
+                alt[p_out] = (alt[p_out] + 1) % vocab
+                d_out = float(np.max(np.abs(model.forward(alt) - y0)))
+                # perturb the position just inside the receptive field boundary
+                p_in = L - rf
+                alt2 = base.copy()
+                alt2[p_in] = (alt2[p_in] + 1) % vocab
+                d_in = float(np.max(np.abs(model.forward(alt2) - y0)))
+                worst_outside = max(worst_outside, d_out)
+                if d_out != 0.0:
+                    violations += 1
+                if d_in == 0.0:
+                    worst_inside_zero += 1
+                if len(rows) < 40:
+                    rows.append({"L": L, "k": k, "windows": windows,
+                                 "sum_W": int(sum(windows)), "rf": rf,
+                                 "max_abs_delta_outside_rf": d_out,
+                                 "max_abs_delta_inside_rf": d_in})
+    return {
+        "configs_tested": n_cfg,
+        "violations_outside_receptive_field": violations,
+        "max_abs_delta_outside_receptive_field": worst_outside,
+        "configs_with_no_effect_just_inside_rf": worst_inside_zero,
+        "sample_rows": rows,
+    }
+
+
+# --------------------------------------------------------------------------
+# B. local-sensitivity witnesses for selective copying (Definition 4.1)
+# --------------------------------------------------------------------------
+def selcopy_F(x, N):
+    """x is a list of ints; tokens 1..N are number tokens with value = token,
+    other tokens are >= N+1.  F(x) = x[L+1-n] (1-indexed) with n the value of
+    the LAST number token.  Returns None if no number token."""
+    L = len(x)
+    n = None
+    for i in range(L):
+        if 1 <= x[i] <= N:
+            n = x[i]
+    if n is None or n > L:
+        return None
+    return x[L - n]                      # 0-indexed: position L+1-n
+
+
+def largest_R_witness(L, N, M, rng, tries=20000):
+    """Largest R for which we can exhibit x, x' agreeing on the last R
+    positions but with F(x) != F(x')."""
+    V = N + M
+    best = None
+    for _ in range(tries):
+        x = list(rng.integers(N + 1, V + 1, size=L))     # all non-number
+        # put the only number token at position 0 (1-indexed position 1)
+        n1 = int(rng.integers(1, N + 1))
+        n2 = int(rng.integers(1, N + 1))
+        if n1 == n2:
+            continue
+        x1 = x.copy(); x1[0] = n1
+        x2 = x.copy(); x2[0] = n2
+        f1, f2 = selcopy_F(x1, N), selcopy_F(x2, N)
+        if f1 is None or f2 is None or f1 == f2:
+            continue
+        # they agree on positions 2..L  ->  R = L - 1
+        R = L - 1
+        assert x1[1:] == x2[1:]
+        if best is None or R > best["R"]:
+            best = {"R": R, "x": x1, "x_prime": x2, "F_x": f1, "F_xprime": f2}
+        if best["R"] == L - 1:
+            break
+    return best
+
+
+# --------------------------------------------------------------------------
+# C/control. does a window-limited stack actually confuse the witness pair?
+# --------------------------------------------------------------------------
+def witness_indistinguishability(L, N, M, seeds=5):
+    rng = np.random.default_rng(7)
+    rows = []
+    max_delta_when_short = 0.0
+    n_short = n_long = 0
+    n_long_separated = 0
+    for s in range(seeds):
+        w = largest_R_witness(L, N, M, np.random.default_rng(100 + s))
+        R = w["R"]
+        for k in (1, 2, 3):
+            # (i) sum W_i < R  ->  must be indistinguishable
+            budget = R - 1
+            windows = [max(1, budget // k)] * k
+            while receptive_field(windows) > R - 1 and windows[0] > 1:
+                windows = [x - 1 for x in windows]
+            model = WindowedTransformer(L, 8, windows, seed=42 + 7 * s + k)
+            dshort = float(np.max(np.abs(model.forward(w["x"])
+                                         - model.forward(w["x_prime"]))))
+            max_delta_when_short = max(max_delta_when_short, dshort)
+            n_short += 1
+            # (ii) sum W_i >= R  ->  the stack CAN separate them (control)
+            windows2 = [L] * k
+            model2 = WindowedTransformer(L, 8, windows2, seed=99 + 7 * s + k)
+            dlong = float(np.max(np.abs(model2.forward(w["x"])
+                                        - model2.forward(w["x_prime"]))))
+            n_long += 1
+            if dlong > 0:
+                n_long_separated += 1
+            rows.append({"L": L, "R": R, "k": k, "windows_short": windows,
+                         "sum_W_short": int(sum(windows)),
+                         "rf_short": receptive_field(windows),
+                         "max_abs_delta_short": dshort,
+                         "windows_long": windows2,
+                         "sum_W_long": int(sum(windows2)),
+                         "max_abs_delta_long": dlong})
+    return {
+        "pairs": rows,
+        "n_short_window_tests": n_short,
+        "max_abs_delta_when_sumW_below_R": max_delta_when_short,
+        "n_long_window_tests": n_long,
+        "n_long_window_separated": n_long_separated,
+    }
+
+
+def main():
+    res = {}
+    res["numerical_gate"] = numerical_gate()
+
+    # ---- A ---------------------------------------------------------------
+    res["receptive_field_sweep"] = receptive_field_sweep(
+        Ls=[16, 32, 64, 128, 256, 512], seeds=40)
+
+    # ---- B ---------------------------------------------------------------
+    wit = []
+    for L in (8, 16, 32, 64, 128, 256):
+        for s in range(5):
+            rng = np.random.default_rng(1000 * L + s)
+            w = largest_R_witness(L, N=6, M=26, rng=rng)
+            same_last_R = w["x"][L - w["R"]:] == w["x_prime"][L - w["R"]:]
+            wit.append({
+                "L": L, "seed": s,
+                "R_measured": w["R"],
+                "R_theory_max": L - 1,
+                "R_claimed_in_Thm_4_2": L // 2,
+                "same_in_every_window_below_R": bool(same_last_R),
+                "F_x": w["F_x"], "F_xprime": w["F_xprime"],
+                "outputs_differ": bool(w["F_x"] != w["F_xprime"]),
+            })
+    res["local_sensitivity_witnesses"] = wit
+    res["all_witnesses_valid"] = all(
+        r["same_in_every_window_below_R"] and r["outputs_differ"] and
+        r["R_measured"] == r["R_theory_max"] for r in wit)
+    res["R_measured_equals_L_minus_1_everywhere"] = all(
+        r["R_measured"] == r["L"] - 1 for r in wit)
+
+    # ---- C + negative control -------------------------------------------
+    ind = {}
+    for L in (16, 32, 64):
+        ind[str(L)] = witness_indistinguishability(L, N=6, M=26, seeds=3)
+    res["witness_indistinguishability"] = ind
+    res["max_abs_delta_when_sumW_below_R"] = max(
+        v["max_abs_delta_when_sumW_below_R"] for v in ind.values())
+    res["control_full_window_separates_frac"] = (
+        sum(v["n_long_window_separated"] for v in ind.values())
+        / sum(v["n_long_window_tests"] for v in ind.values()))
+    # success probability under D = uniform{x, x'} when the model cannot see R
+    res["success_prob_when_sumW_below_R"] = 0.5
+    res["threshold_in_theorem"] = 2.0 / 3.0
+    res["fails_theorem_threshold"] = bool(0.5 < 2.0 / 3.0)
+
+    with open(os.path.join(OUT, "claim2.json"), "w") as f:
+        json.dump(res, f, indent=1, default=jsonable)
+
+    print("numerical gate |matmul-einsum|          :",
+          res["numerical_gate"]["max_abs_matmul_minus_einsum"])
+    rf = res["receptive_field_sweep"]
+    print("configs tested (real float attention)   :", rf["configs_tested"])
+    print("violations outside receptive field      :",
+          rf["violations_outside_receptive_field"])
+    print("max |delta| outside receptive field     :",
+          rf["max_abs_delta_outside_receptive_field"])
+    print("R_measured == L-1 for every witness     :",
+          res["R_measured_equals_L_minus_1_everywhere"])
+    print("max |delta| on witness when sum W < R   :",
+          res["max_abs_delta_when_sumW_below_R"])
+    print("control: full window separates fraction :",
+          res["control_full_window_separates_frac"])
+
+
+if __name__ == "__main__":
+    main()

exp3_selcopy_construction.py

@@ -0,0 +1,362 @@
+"""Claim 3 - Theorem 4.3: two-layer hybrid (Mamba + attention) solves selective
+copying with embedding dim O(max(log|V|, log L)) and working memory ~O(N).
+
+The construction is transcribed literally from Appendix D.2:
+
+  Embedding (blocks of width c = max(bits(|V|), bits(L)) each):
+      b0 = psi'(x_i)                     binary +-1 code of the token
+      b1 = 1{x_i in N} psi'(x_i)         gated copy (the paper's second row)
+      b2 = 0                             SSM output slot
+      b3 = 0                             attention output slot
+      b4 = phi(i) = psi'(L + 1 - i)      position code, as in D.2
+      g  = 1{x_i in N}                   one extra scalar, the Mamba gate
+
+  Mamba layer (W_A = I, Delta(x) = 1{x in N}, W_B picks b1, W_C writes b2):
+      H_t = (1 - Delta_t) H_{t-1} + Delta_t (W_B Phi_t)
+  so H_t = psi'(n_t) with n_t the value of the most recent number token.
+
+  Attention layer, causal sliding window of size N:
+      q_i = M * b2(i) = M psi'(n_i),  k_j = b4(j) = psi'(L+1-j),  v_j = psi'(x_j)
+  The score M <psi'(n_i), psi'(L+1-j)> is maximal (= M c) exactly at
+  j* = L + 1 - n_i, so the readout is psi'(x_{L+1-n_L}) = F(x).
+
+  As the paper states in D.5, the construction is only required to be correct
+  at the LAST position, which is where we score it.
+
+Evidence: exhaustive enumeration of every valid input at small scale, large
+random sweeps at paper scale, and three negative controls that must break.
+
+Run:  python3 exp3_selcopy_construction.py
+"""
+
+import itertools
+import json
+import math
+import os
+
+import numpy as np
+
+np.seterr(all="ignore")          # Accelerate spurious warnings; gated below
+OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "outputs")
+os.makedirs(OUT, exist_ok=True)
+
+BIG = 30.0                       # softmax inverse temperature M of the proof
+
+
+def jsonable(o):
+    if isinstance(o, np.integer):
+        return int(o)
+    if isinstance(o, np.floating):
+        return float(o)
+    if isinstance(o, np.ndarray):
+        return o.tolist()
+    raise TypeError(str(type(o)))
+
+
+# --------------------------------------------------------------------------
+class SelCopy:
+    """Definition 4.1.  Number tokens carry ids equal to their offset value."""
+
+    def __init__(self, offsets, M, L):
+        self.offsets = sorted(offsets)
+        self.M = M
+        self.L = L
+        self.max_off = max(self.offsets)
+        # token ids: number tokens = the offsets themselves; others follow
+        self.other = list(range(self.max_off + 1, self.max_off + 1 + M))
+        self.vocab = self.offsets + self.other
+        self.V = len(self.vocab)
+        self.is_num = np.zeros(max(self.vocab) + 1, bool)
+        self.is_num[self.offsets] = True
+        self.c = max((max(self.vocab)).bit_length(), L.bit_length())
+        self.window = self.max_off
+
+    # ---- reference implementation of the task ---------------------------
+    def target(self, X):
+        """X: (B, L) token ids.  Returns (target, n_last, valid)."""
+        B, L = X.shape
+        num = self.is_num[X]
+        n_last = np.where(num, X, 0)
+        # running max index of a number token -> value of the LAST number token
+        last = np.zeros(B, int)
+        out = np.zeros((B, L), int)
+        for t in range(L):
+            last = np.where(num[:, t], X[:, t], last)
+            out[:, t] = last
+        nL = out[:, -1]
+        valid = (nL > 0) & (nL <= L)
+        jstar = L - nL                       # 0-indexed position L+1-n
+        tgt = X[np.arange(B), np.clip(jstar, 0, L - 1)]
+        return tgt, nL, valid, jstar, out
+
+    # ---- codes ----------------------------------------------------------
+    def code(self, ints):
+        """+-1 binary code of a non-negative integer array, width c."""
+        a = np.asarray(ints)
+        bits = ((a[..., None] >> np.arange(self.c)) & 1).astype(np.float64)
+        return 2.0 * bits - 1.0
+
+    def embed(self, X):
+        B, L = X.shape
+        c = self.c
+        num = self.is_num[X].astype(np.float64)
+        E = np.zeros((B, L, 5 * c + 1))
+        E[:, :, 0:c] = self.code(X)
+        E[:, :, c:2 * c] = num[..., None] * self.code(X)
+        # b2, b3 stay zero
+        pos = np.broadcast_to(np.arange(1, L + 1), (B, L))
+        E[:, :, 4 * c:5 * c] = self.code(L + 1 - pos)
+        E[:, :, 5 * c] = num
+        return E
+
+    # ---- layers ---------------------------------------------------------
+    def mamba(self, E, selective=True):
+        """H_t = (1-D_t) H_{t-1} + D_t W_B Phi_t ; writes H into block b2."""
+        B, L, d = E.shape
+        c = self.c
+        WB = np.zeros((c, d)); WB[:, c:2 * c] = np.eye(c)        # picks b1
+        WC = np.zeros((d, c)); WC[2 * c:3 * c, :] = np.eye(c)     # writes b2
+        gate_sel = np.zeros(d); gate_sel[5 * c] = 1.0             # reads g
+        H = np.zeros((B, c))
+        out = E.copy()
+        for t in range(L):
+            phi = E[:, t, :]
+            D = (phi @ gate_sel)[:, None] if selective else np.ones((B, 1))
+            H = (1.0 - D) * H + D * (phi @ WB.T)
+            out[:, t, :] += H @ WC.T
+        return out
+
+    def attention_last(self, Z, window, big=BIG, blind_query=False):
+        """Causal sliding-window attention evaluated at the last position."""
+        B, L, d = Z.shape
+        c = self.c
+        Wq = np.zeros((c, d)); Wq[:, 2 * c:3 * c] = big * np.eye(c)   # b2
+        Wk = np.zeros((c, d)); Wk[:, 4 * c:5 * c] = np.eye(c)         # b4
+        Wv = np.zeros((c, d)); Wv[:, 0:c] = np.eye(c)                 # b0
+        lo = max(0, L - window)
+        q = Z[:, -1, :] @ Wq.T
+        if blind_query:
+            q = np.zeros_like(q)
+        K = Z[:, lo:L, :] @ Wk.T
+        V = Z[:, lo:L, :] @ Wv.T
+        S = np.einsum("bc,btc->bt", q, K)
+        S = S - S.max(axis=1, keepdims=True)
+        A = np.exp(S)
+        A /= A.sum(axis=1, keepdims=True)
+        read = np.einsum("bt,btc->bc", A, V)
+        return read, A, lo
+
+    # ---- decoding -------------------------------------------------------
+    def decode(self, read):
+        """Nearest +-1 codeword among the vocabulary."""
+        cw = self.code(np.array(self.vocab))               # (V, c)
+        sims = read @ cw.T
+        idx = np.argmax(sims, axis=1)
+        top = np.sort(sims, axis=1)
+        margin = top[:, -1] - top[:, -2]
+        return np.array(self.vocab)[idx], margin
+
+    # ---- full model -----------------------------------------------------
+    def run(self, X, window=None, selective=True, blind_query=False, big=BIG):
+        window = self.window if window is None else window
+        E = self.embed(X)
+        Z = self.mamba(E, selective=selective)
+        read, A, lo = self.attention_last(Z, window, big=big,
+                                          blind_query=blind_query)
+        pred, margin = self.decode(read)
+        return pred, margin, A, lo, Z
+
+
+# --------------------------------------------------------------------------
+def all_sequences(V_ids, L):
+    """Every sequence in V^L as an (|V|^L, L) int array."""
+    n = len(V_ids) ** L
+    X = np.zeros((n, L), int)
+    ids = np.array(V_ids)
+    for pos in range(L):
+        rep = len(V_ids) ** (L - pos - 1)
+        X[:, pos] = np.tile(np.repeat(ids, rep), n // (rep * len(V_ids)))
+    return X
+
+
+MAX_ELEMS = 2.0e7        # cap on B * L * d floats held at once
+
+
+def evaluate(task, X, **kw):
+    """Chunked so that memory stays bounded at long L / large batches."""
+    tgt, nL, valid, jstar, _ = task.target(X)
+    X = X[valid]; tgt = tgt[valid]; nL = nL[valid]; jstar = jstar[valid]
+    n = len(X)
+    d = 5 * task.c + 1
+    step = max(1, int(MAX_ELEMS // (task.L * d)))
+    n_ok = n_att = n_ssm = 0
+    leak = 0.0
+    minmarg = np.inf
+    c = task.c
+    for lo_i in range(0, n, step):
+        hi = min(n, lo_i + step)
+        pred, margin, A, lo, Z = task.run(X[lo_i:hi], **kw)
+        n_ok += int(np.sum(pred == tgt[lo_i:hi]))
+        n_att += int(np.sum(lo + np.argmax(A, axis=1) == jstar[lo_i:hi]))
+        n_ssm += int(np.sum(np.all(
+            np.sign(Z[:, -1, 2 * c:3 * c]) == np.sign(task.code(nL[lo_i:hi])),
+            axis=1)))
+        leak = max(leak, float(np.max(1.0 - A.max(axis=1))))
+        minmarg = min(minmarg, float(np.min(margin)))
+    return {"n_inputs": int(n), "accuracy": n_ok / n,
+            "attention_argmax_correct": n_att / n,
+            "ssm_state_correct": n_ssm / n,
+            "max_softmax_leakage": leak,
+            "min_decode_margin": float(minmarg)}
+
+
+def main():
+    res = {"softmax_inverse_temperature_M": BIG}
+
+    # ---- correctness gate ------------------------------------------------
+    t = SelCopy([1, 2, 3], M=2, L=5)
+    X = all_sequences(t.vocab, 5)
+    tgt, nL, valid, jstar, running = t.target(X)
+    # re-derive F by a slow, independent reference
+    slow = []
+    nchk = min(5000, len(X))
+    for row in X[:nchk]:
+        n = 0
+        for v in row:
+            if t.is_num[v]:
+                n = int(v)
+        slow.append(int(row[len(row) - n]) if 0 < n <= len(row) else -1)
+    fast = [int(tgt[i]) if valid[i] else -1 for i in range(nchk)]
+    res["gate_reference_target_mismatches"] = int(sum(
+        1 for a, b in zip(slow, fast) if a != b))
+    cw = t.code(np.array(t.vocab))
+    res["gate_codes_injective"] = bool(len(np.unique(cw, axis=0)) == len(t.vocab))
+    res["gate_matmul_vs_einsum"] = float(np.max(np.abs(
+        cw @ cw.T - np.einsum("ij,kj->ik", cw, cw))))
+
+    # ---- exhaustive ------------------------------------------------------
+    exh = []
+    for offsets, M, L in [([1, 2, 3], 2, 5), ([1, 2, 3], 3, 6),
+                          ([1, 2, 3, 4], 3, 7)]:
+        task = SelCopy(offsets, M, L)
+        X = all_sequences(task.vocab, L)
+        r = evaluate(task, X)
+        r.update({"offsets": offsets, "M": M, "L": L, "V": task.V,
+                  "embed_dim": 5 * task.c + 1, "code_width_c": task.c,
+                  "log2_V": math.log2(task.V), "log2_L": math.log2(L),
+                  "window": task.window, "working_memory_ratio": task.window / L,
+                  "mamba_states": len(offsets) + 1,
+                  "exhaustive": True})
+        exh.append(r)
+    res["exhaustive"] = exh
+
+    # ---- random sweeps, including the paper's own scale -------------------
+    rnd = []
+    configs = [
+        ([1, 2, 3, 4, 5, 6], 4, 16, 200000),
+        ([5, 6, 7, 8, 9, 10], 26, 100, 200000),      # paper E.1 selective copy
+        ([5, 6, 7, 8, 9, 10], 194, 100, 100000),     # paper E.2, |V| = 200
+        ([5, 6, 7, 8, 9, 10], 994, 100, 100000),     # paper E.2, |V| = 1000
+        (list(range(1, 33)), 480, 1024, 30000),
+        (list(range(1, 65)), 960, 4096, 6000),
+    ]
+    for offsets, M, L, nsamp in configs:
+        task = SelCopy(offsets, M, L)
+        accs = []
+        row = None
+        for seed in range(3):
+            rng = np.random.default_rng(seed)
+            X = rng.choice(task.vocab, size=(nsamp // 3, L))
+            r = evaluate(task, X)
+            accs.append(r["accuracy"])
+            row = r
+        row["accuracy_mean_over_seeds"] = float(np.mean(accs))
+        row["accuracy_std_over_seeds"] = float(np.std(accs))
+        row["n_inputs_total"] = int(nsamp // 3 * 3)
+        row.update({"offsets_min_max": [min(offsets), max(offsets)],
+                    "n_offsets": len(offsets), "M": M, "L": L, "V": task.V,
+                    "embed_dim": 5 * task.c + 1, "code_width_c": task.c,
+                    "log2_V": math.log2(task.V), "log2_L": math.log2(L),
+                    "window": task.window,
+                    "working_memory_ratio": task.window / L,
+                    "mamba_states": len(offsets) + 1})
+        rnd.append(row)
+    res["random_sweeps"] = rnd
+    res["total_inputs_tested"] = int(sum(r["n_inputs"] for r in exh)
+                                     + sum(r["n_inputs_total"] for r in rnd))
+    res["min_accuracy_over_all_configs"] = min(
+        [r["accuracy"] for r in exh] + [r["accuracy_mean_over_seeds"] for r in rnd])
+
+    # ---- scaling of the embedding dimension ------------------------------
+    scal = []
+    for r in exh + rnd:
+        scal.append({"V": r["V"], "L": r["L"], "embed_dim": r["embed_dim"],
+                     "bound_max_log2": max(r["log2_V"], r["log2_L"]),
+                     "ratio": r["embed_dim"] / max(r["log2_V"], r["log2_L"])})
+    xs = np.array([math.log(s["bound_max_log2"]) for s in scal])
+    ys = np.array([math.log(s["embed_dim"]) for s in scal])
+    A = np.vstack([xs, np.ones_like(xs)]).T
+    coef, *_ = np.linalg.lstsq(A, ys, rcond=None)
+    pred = A @ coef
+    r2 = 1.0 - float(np.sum((ys - pred) ** 2)) / float(np.sum((ys - ys.mean()) ** 2))
+    res["embed_dim_scaling"] = {"rows": scal, "loglog_slope": float(coef[0]),
+                                "R2": r2, "max_ratio": max(s["ratio"] for s in scal)}
+
+    # ---- negative controls ------------------------------------------------
+    ctl = []
+    for offsets, M, L in [([5, 6, 7, 8, 9, 10], 26, 100),
+                          ([1, 2, 3, 4, 5, 6], 4, 16)]:
+        task = SelCopy(offsets, M, L)
+        rng = np.random.default_rng(0)
+        X = rng.choice(task.vocab, size=(60000, L))
+        base = evaluate(task, X)
+        short = evaluate(task, X, window=task.window - 1)
+        blind = evaluate(task, X, blind_query=True)
+        nonsel = evaluate(task, X, selective=False)
+        # predicted loss of the short window: instances whose offset is maximal
+        tgt, nL, valid, _, _ = task.target(X)
+        p_max = float(np.mean(nL[valid] == task.max_off))
+        ctl.append({
+            "offsets": [min(offsets), max(offsets)], "M": M, "L": L,
+            "full_construction_accuracy": base["accuracy"],
+            "control_window_minus_1_accuracy": short["accuracy"],
+            "control_window_minus_1_lower_bound": 1.0 - p_max,
+            "control_window_minus_1_excess_over_bound":
+                short["accuracy"] - (1.0 - p_max),
+            "frac_instances_needing_max_offset": p_max,
+            "control_no_ssm_query_accuracy": blind["accuracy"],
+            "control_nonselective_ssm_accuracy": nonsel["accuracy"],
+            "control_nonselective_ssm_state_correct": nonsel["ssm_state_correct"],
+            "chance_level": 1.0 / task.V,
+            "analytic_softmax_leakage_bound": float((task.window - 1)
+                                                    * math.exp(-2.0 * BIG)),
+        })
+    res["negative_controls"] = ctl
+
+    with open(os.path.join(OUT, "claim3.json"), "w") as f:
+        json.dump(res, f, indent=1, default=jsonable)
+
+    print("gate: reference mismatches      :", res["gate_reference_target_mismatches"])
+    print("gate: codes injective           :", res["gate_codes_injective"])
+    print("total inputs tested             :", res["total_inputs_tested"])
+    print("min accuracy over all configs   :", res["min_accuracy_over_all_configs"])
+    for r in exh:
+        print("  exhaustive V=%2d L=%2d n=%7d acc=%.6f att=%.6f leak=%.3e d=%d"
+              % (r["V"], r["L"], r["n_inputs"], r["accuracy"],
+                 r["attention_argmax_correct"], r["max_softmax_leakage"],
+                 r["embed_dim"]))
+    for r in rnd:
+        print("  random     V=%4d L=%5d n=%7d acc=%.6f d=%3d mem=%.4f"
+              % (r["V"], r["L"], r["n_inputs_total"], r["accuracy_mean_over_seeds"],
+                 r["embed_dim"], r["working_memory_ratio"]))
+    for c in ctl:
+        print("  control L=%d: full=%.4f  W-1=%.4f (bound %.4f)  noSSM=%.4f  nonsel=%.4f"
+              % (c["L"], c["full_construction_accuracy"],
+                 c["control_window_minus_1_accuracy"],
+                 c["control_window_minus_1_lower_bound"],
+                 c["control_no_ssm_query_accuracy"],
+                 c["control_nonselective_ssm_accuracy"]))
+
+
+if __name__ == "__main__":
+    main()

exp4_ar_construction.py

@@ -0,0 +1,436 @@
+"""Claim 4 - Theorem 4.6: three-layer hybrid solves associative recall with
+decoding at 99% success, embedding dim O(max(log|V|,log L)), window ~O(|V|).
+
+Task (Definition 4.4).  V = M u {0,1}.  v(x) is the 0-1 subsequence of x and
+Phi(v) in M is the word token whose binary representation is v.  The answer is
+the token following the LAST occurrence of Phi(v(x)).
+
+Construction, transcribed from Appendix D.4:
+
+  Mamba layer.  W_A = I - S with S the shift, Delta(x) = 1{x in {0,1}}, so the
+  state is a shift register over the bit subsequence and
+      H_t = phi(n_t),  n_t = the token decoded from the bits seen so far.
+  Attention layer 1 (2 heads).  Head 1 has (B_i)_j = -inf * 1(j != i-1), i.e. it
+  copies the previous token into a second slot; head 2 is the identity.
+  Attention layer 2 (1 head), sliding window w.
+      q_i = M phi(n_i),  k_i = psi'(x_{i-1}),  v_i = psi'(x_i),
+  plus the paper's positional bias B so that the argmax lands on the LAST
+  matching position.  The readout is psi'(x_{i*+1}) = the answer.
+
+The 99% figure in Theorem 4.6 is a coverage statement: for uniform word tokens
+the last occurrence of the key lies inside a window of size ~O(|M|) with
+probability 1 - (1-1/|M|)^{n_w}.  We compute that probability in closed form,
+choose the window from it, and check the measured success against it.
+
+NEGATIVE CONTROLS.  (i) window = |M| must give ~1 - 1/e = 0.632.  (ii) removing
+the Mamba layer (no decoded key) must collapse to chance.  (iii) removing the
+positional bias must fail exactly on the instances where the key occurs more
+than once in the window.
+
+Run:  python3 exp4_ar_construction.py
+"""
+
+import json
+import math
+import os
+
+import numpy as np
+
+np.seterr(all="ignore")
+OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "outputs")
+os.makedirs(OUT, exist_ok=True)
+
+
+def jsonable(o):
+    if isinstance(o, np.integer):
+        return int(o)
+    if isinstance(o, np.floating):
+        return float(o)
+    if isinstance(o, np.ndarray):
+        return o.tolist()
+    raise TypeError(str(type(o)))
+
+
+class ARDecode:
+    """|M| word tokens with ids 0..|M|-1; bit tokens have ids |M| and |M|+1."""
+
+    def __init__(self, Mw, L):
+        assert Mw & (Mw - 1) == 0, "|M| must be a power of two"
+        self.Mw = Mw
+        self.L = L
+        self.ds = int(math.log2(Mw))            # bits per decoded word key
+        self.V = Mw + 2
+        # The proof's psi' encodes the *whole* vocabulary.  The Mamba register
+        # still consumes only ds bit tokens, but it is initialized at -1 and
+        # therefore yields the c-bit code of the decoded word (with leading
+        # zero bits) after those ds updates.  Using only ds bits for every
+        # token aliases the two bit-token ids to word ids and lets a trailing
+        # bit spuriously win the attention match.
+        self.c = int(math.ceil(math.log2(self.V)))
+        self.p = max(1, L.bit_length())
+        self.n_word = L - self.ds               # word positions 0..n_word-1
+        self.d = 4 * self.c + self.p + 2
+
+    # ---- data ------------------------------------------------------------
+    def sample(self, n, rng):
+        words = rng.integers(0, self.Mw, size=(n, self.n_word))
+        bits = rng.integers(0, 2, size=(n, self.ds))
+        X = np.concatenate([words, self.Mw + bits], axis=1)
+        return X, bits
+
+    def key_of(self, bits):
+        w = (1 << np.arange(self.ds - 1, -1, -1))
+        return (bits * w).sum(axis=1)
+
+    def target(self, X, bits):
+        """Token immediately after the last occurrence of the decoded key.
+
+        The task definition allows that successor to be either a word or a bit
+        token.  In particular, the last word may be the key and its successor
+        the first trailing bit.  Restricting successors to word tokens would
+        silently change Definition 4.4 and incorrectly count that valid case
+        as a failure.
+        """
+        n = len(X)
+        key = self.key_of(bits)
+        W = X[:, :self.n_word]
+        hit = (W == key[:, None])
+        any_hit = hit.any(axis=1)
+        istar = np.where(any_hit, self.n_word - 1 - hit[:, ::-1].argmax(axis=1), -1)
+        tgt = np.where(any_hit, X[np.arange(n), np.clip(istar + 1, 0, self.L - 1)], -1)
+        return key, istar, tgt, any_hit
+
+    # ---- codes -----------------------------------------------------------
+    def wcode(self, ids):
+        a = np.asarray(ids)
+        bits = ((a[..., None] >> np.arange(self.c - 1, -1, -1)) & 1)
+        return 2.0 * bits - 1.0
+
+    def pcode(self, ids):
+        a = np.asarray(ids)
+        bits = ((a[..., None] >> np.arange(self.p)) & 1)
+        return 2.0 * bits - 1.0
+
+    # ---- the three layers ------------------------------------------------
+    def embed(self, X):
+        n, L = X.shape
+        c, p, d = self.c, self.p, self.d
+        E = np.zeros((n, L, d))
+        is_word = (X < self.Mw)
+        E[:, :, 0:c] = self.wcode(X)
+        E[:, :, 3 * c] = np.where(~is_word, (X - self.Mw) * 2.0 - 1.0, 0.0)   # +-1 bit
+        E[:, :, 3 * c + 1] = (~is_word).astype(float)                         # gate g
+        pos = np.broadcast_to(np.arange(L), (n, L))
+        E[:, :, 3 * c + 2:3 * c + 2 + p] = self.pcode(pos)
+        return E
+
+    def mamba(self, E, active=True):
+        """Shift register over the bit subsequence -> phi(n_t) in block b3."""
+        n, L, d = E.shape
+        c = self.c
+        out = E.copy()
+        H = -np.ones((n, c))                     # empty register
+        for t in range(L):
+            g = E[:, t, 3 * c + 1:3 * c + 2]
+            b = E[:, t, 3 * c:3 * c + 1]
+            if active:
+                Hs = np.concatenate([H[:, 1:], b], axis=1)   # S H + bit
+                H = (1.0 - g) * H + g * Hs
+            out[:, t, 2 * c:3 * c] = H
+        return out
+
+    def attn1(self, Z, exact_shift=True):
+        """Previous-token head: (B_i)_j = -inf 1(j != i-1).  Writes b1."""
+        n, L, d = Z.shape
+        c = self.c
+        out = Z.copy()
+        prev = np.zeros((n, L, c))
+        prev[:, 1:, :] = Z[:, :-1, 0:c]
+        out[:, :, c:2 * c] = prev
+        if not exact_shift:
+            return out
+        return out
+
+    def attn1_via_softmax(self, Z):
+        """Same head written as a real masked softmax attention, for the gate."""
+        n, L, d = Z.shape
+        c = self.c
+        idx = np.arange(L)
+        mask = (idx[:, None] - 1) == idx[None, :]
+        S = np.where(mask, 0.0, -np.inf)
+        S[0, :] = np.where(idx == 0, 0.0, -np.inf)          # position 0 -> itself
+        A = np.exp(S - S.max(axis=1, keepdims=True))
+        A /= A.sum(axis=1, keepdims=True)
+        prev = np.einsum("ij,njc->nic", A, Z[:, :, 0:c])
+        prev[:, 0, :] = 0.0
+        out = Z.copy()
+        out[:, :, c:2 * c] = prev
+        return out
+
+    def attn2(self, Z, window, use_bias=True, blind_query=False):
+        """Windowed recall head evaluated at the last position."""
+        n, L, d = Z.shape
+        c, p = self.c, self.p
+        big = 100.0 * L
+        lo = max(0, L - window)
+        q = big * Z[:, -1, 2 * c:3 * c]
+        if blind_query:
+            q = np.zeros_like(q)
+        K = Z[:, lo:L, c:2 * c]
+        V = Z[:, lo:L, 0:c]
+        S = np.einsum("nc,ntc->nt", q, K)
+        if use_bias:
+            # Appendix D.4 permits an arbitrary positional bias B whose only
+            # job is to select the *last* tied key.  Use the literal position
+            # order as B, scaled so adjacent tied locations differ by big/(4L)
+            # (making finite softmax choose the last one), while its complete
+            # range is at most big/4.  A wrong binary key loses at least
+            # 2*big, so B cannot turn a non-match into a match.  The earlier
+            # signed-bit surrogate was not monotone in position and therefore
+            # was not a faithful realization of the stated B.
+            posval = np.arange(lo, L, dtype=float)[None, :] / float(L + 1)
+            S = S + (big / 4.0) * posval
+        S = S - S.max(axis=1, keepdims=True)
+        A = np.exp(S)
+        A /= A.sum(axis=1, keepdims=True)
+        read = np.einsum("nt,ntc->nc", A, V)
+        return read, A, lo
+
+    def decode(self, read):
+        # psi' in Appendix D.4 is a code for the full vocabulary, not only
+        # word tokens.  This is required when the answer is a trailing bit.
+        cw = self.wcode(np.arange(self.V))
+        sims = read @ cw.T
+        idx = np.argmax(sims, axis=1)
+        top = np.sort(sims, axis=1)
+        return idx, top[:, -1] - top[:, -2]
+
+    def run(self, X, window, mamba_on=True, use_bias=True, blind_query=False,
+            softmax_shift=False):
+        E = self.embed(X)
+        Z = self.mamba(E, active=mamba_on)
+        Z = self.attn1_via_softmax(Z) if softmax_shift else self.attn1(Z)
+        read, A, lo = self.attn2(Z, window, use_bias=use_bias,
+                                 blind_query=blind_query)
+        pred, margin = self.decode(read)
+        return pred, margin, A, lo, Z
+
+
+# --------------------------------------------------------------------------
+def required_window(Mw, ds, delta=0.01):
+    """Smallest window w such that P(key's last occurrence is in-window) >= 1-delta."""
+    p = 1.0 / Mw
+    n_w = math.ceil(math.log(delta) / math.log(1.0 - p))
+    return int(n_w + ds + 1), int(n_w)
+
+
+def analytic_success(Mw, n_eligible, n_in_window):
+    p = 1.0 / Mw
+    p_exists = 1.0 - (1.0 - p) ** n_eligible
+    p_in = 1.0 - (1.0 - p) ** min(n_eligible, n_in_window)
+    # p_in is the unconditional coverage probability and hence a conservative
+    # certificate even if one treats inputs with no key occurrence as failures.
+    return p_in, p_exists
+
+
+def eligible_keys_in_window(task, window):
+    """How many word-key positions have their successor in the last window."""
+    lo = max(0, task.L - window)
+    return max(0, task.n_word - max(0, lo - 1))
+
+
+def evaluate(task, X, bits, window, **kw):
+    key, istar, tgt, ok = task.target(X, bits)
+    Xs, ts = X[ok], tgt[ok]
+    # Stream all large/exhaustive grids.  This changes no inputs or arithmetic
+    # and prevents a full 8^5 x bit-grid from allocating several simultaneous
+    # dense attention tensors.
+    n = len(Xs)
+    chunk = 10000
+    n_acc = n_ssm = n_att = 0
+    leak = 0.0
+    min_margin = np.inf
+    c = task.c
+    keyc = task.wcode(key[ok])
+    istars = istar[ok]
+    for start in range(0, n, chunk):
+        stop = min(n, start + chunk)
+        pred, margin, A, lo, Z = task.run(Xs[start:stop], window, **kw)
+        n_acc += int(np.sum(pred == ts[start:stop]))
+        n_ssm += int(np.sum(np.all(
+            np.sign(Z[:, -1, 2 * c:3 * c]) == keyc[start:stop], axis=1)))
+        att_pos = lo + np.argmax(A, axis=1)
+        n_att += int(np.sum(att_pos == (istars[start:stop] + 1)))
+        leak = max(leak, float(np.max(1.0 - A.max(axis=1))) )
+        min_margin = min(min_margin, float(np.min(margin)))
+    return {"n_instances": int(n),
+            "frac_target_defined": float(np.mean(ok)),
+            "success": n_acc / n,
+            "ssm_key_decoded_correctly": n_ssm / n,
+            "attention_argmax_correct": n_att / n,
+            "max_softmax_leakage": leak,
+            "min_decode_margin": min_margin}
+
+
+def all_sequences_small(task):
+    """Exhaustive enumeration for tiny |M|, L (word part x bit part)."""
+    Mw, nw, ds = task.Mw, task.n_word, task.ds
+    nwords = Mw ** nw
+    W = np.zeros((nwords, nw), int)
+    for pos in range(nw):
+        rep = Mw ** (nw - pos - 1)
+        W[:, pos] = np.tile(np.repeat(np.arange(Mw), rep), nwords // (rep * Mw))
+    B = np.zeros((2 ** ds, ds), int)
+    for pos in range(ds):
+        rep = 2 ** (ds - pos - 1)
+        B[:, pos] = np.tile(np.repeat(np.arange(2), rep), (2 ** ds) // (rep * 2))
+    Wr = np.repeat(W, len(B), axis=0)
+    Br = np.tile(B, (len(W), 1))
+    X = np.concatenate([Wr, task.Mw + Br], axis=1)
+    return X, Br
+
+
+def main():
+    res = {}
+
+    # ---- correctness gates ------------------------------------------------
+    t = ARDecode(4, 8)
+    X, bits = t.sample(2000, np.random.default_rng(0))
+    Za = t.attn1(t.mamba(t.embed(X)))
+    Zb = t.attn1_via_softmax(t.mamba(t.embed(X)))
+    res["gate_shift_head_vs_softmax_attention"] = float(np.max(np.abs(Za - Zb)))
+    key, istar, tgt, ok = t.target(X, bits)
+    slow = []
+    for r in range(200):
+        k = int(key[r]); w = X[r, :t.n_word]
+        pos = [i for i in range(t.n_word) if w[i] == k]
+        slow.append(int(X[r, pos[-1] + 1]) if pos else -1)
+    fast = [int(tgt[r]) if ok[r] else -1 for r in range(200)]
+    res["gate_reference_target_mismatches"] = int(
+        sum(1 for a, b in zip(slow, fast) if a != b))
+
+    # ---- exhaustive, full window -------------------------------------------
+    exh = []
+    for Mw, L in [(4, 8), (4, 9), (8, 8)]:
+        task = ARDecode(Mw, L)
+        X, bits = all_sequences_small(task)
+        r = evaluate(task, X, bits, window=L)
+        r.update({"Mw": Mw, "L": L, "V": task.V, "ds": task.ds,
+                  "embed_dim": task.d, "window": L, "exhaustive": True,
+                  "n_enumerated": int(len(X))})
+        exh.append(r)
+    res["exhaustive_full_window"] = exh
+
+    # ---- the theorem's window, at scale -------------------------------------
+    rows = []
+    for Mw, L in [(8, 100), (16, 200), (32, 400), (64, 800), (128, 1600)]:
+        task = ARDecode(Mw, L)
+        w_req, n_w = required_window(Mw, task.ds, delta=0.01)
+        n_in_window = eligible_keys_in_window(task, w_req)
+        pred_succ, p_exists = analytic_success(Mw, task.n_word, n_in_window)
+        accs = []
+        row = None
+        for seed in range(5):
+            rng = np.random.default_rng(1000 + seed)
+            X, bits = task.sample(20000, rng)
+            row = evaluate(task, X, bits, window=w_req)
+            accs.append(row["success"])
+        row["success_mean"] = float(np.mean(accs))
+        row["success_std"] = float(np.std(accs))
+        row["n_instances_total"] = int(sum(1 for _ in range(5)) * row["n_instances"])
+        row.update({"Mw": Mw, "L": L, "V": task.V, "ds": task.ds,
+                    "embed_dim": task.d,
+                    "bound_max_log2": max(math.log2(task.V), math.log2(L)),
+                    "window_required": w_req,
+                    "window_over_Mw": w_req / Mw,
+                    "window_over_L": w_req / L,
+                    "analytic_success": pred_succ,
+                    "abs_gap_vs_analytic": abs(float(np.mean(accs)) - pred_succ),
+                    "empirical_mean_meets_99pct": bool(np.mean(accs) >= 0.99),
+                    "analytic_certificate_meets_99pct": bool(pred_succ >= 0.99),
+                    "seeds": 5})
+        rows.append(row)
+    res["theorem_window"] = rows
+    res["min_success_at_theorem_window"] = min(r["success_mean"] for r in rows)
+    res["max_gap_vs_analytic"] = max(r["abs_gap_vs_analytic"] for r in rows)
+    res["all_meet_99pct"] = all(r["analytic_certificate_meets_99pct"] for r in rows)
+
+    # ---- window sweep + negative controls -----------------------------------
+    ctl = []
+    for Mw, L in [(16, 200), (32, 400)]:
+        task = ARDecode(Mw, L)
+        rng = np.random.default_rng(7)
+        X, bits = task.sample(40000, rng)
+        w_req, _ = required_window(Mw, task.ds, delta=0.01)
+        full = evaluate(task, X, bits, window=L)
+        thm = evaluate(task, X, bits, window=w_req)
+        tiny = evaluate(task, X, bits, window=Mw)
+        pred_tiny, _ = analytic_success(Mw, task.n_word,
+                                        eligible_keys_in_window(task, Mw))
+        blind = evaluate(task, X, bits, window=w_req, blind_query=True)
+        nomamba = evaluate(task, X, bits, window=w_req, mamba_on=False)
+        nobias = evaluate(task, X, bits, window=w_req, use_bias=False)
+        # instances where the key occurs more than once inside the window
+        key, istar, tgt, ok = task.target(X, bits)
+        Wr = X[ok][:, :task.n_word]
+        lo = max(0, L - w_req)
+        inw = Wr[:, max(0, lo - 1):task.n_word]
+        nrep = (inw == key[ok][:, None]).sum(axis=1)
+        p_multi = float(np.mean(nrep > 1))
+        ctl.append({
+            "Mw": Mw, "L": L, "window_required": w_req,
+            "full_window_success": full["success"],
+            "theorem_window_success": thm["success"],
+            "control_window_eq_Mw_success": tiny["success"],
+            "control_window_eq_Mw_analytic": pred_tiny,
+            "control_window_eq_Mw_gap": abs(tiny["success"] - pred_tiny),
+            "one_minus_1_over_e": 1.0 - 1.0 / math.e,
+            "control_blind_query_success": blind["success"],
+            "control_no_mamba_success": nomamba["success"],
+            "control_no_mamba_key_decoded": nomamba["ssm_key_decoded_correctly"],
+            "control_no_positional_bias_success": nobias["success"],
+            "frac_key_repeated_in_window": p_multi,
+            "control_no_bias_predicted_ceiling": 1.0 - p_multi,
+            "chance_level": 1.0 / Mw,
+        })
+    res["negative_controls"] = ctl
+
+    # ---- window sweep --------------------------------------------------------
+    task = ARDecode(32, 400)
+    rng = np.random.default_rng(3)
+    X, bits = task.sample(30000, rng)
+    sweep = []
+    for w in [8, 16, 32, 64, 96, 128, 154, 200, 300, 400]:
+        r = evaluate(task, X, bits, window=w)
+        pa, _ = analytic_success(32, task.n_word,
+                                 eligible_keys_in_window(task, w))
+        sweep.append({"window": w, "success": r["success"], "analytic": pa,
+                      "abs_gap": abs(r["success"] - pa)})
+    res["window_sweep_Mw32_L400"] = sweep
+    res["window_sweep_max_gap"] = max(s["abs_gap"] for s in sweep)
+
+    with open(os.path.join(OUT, "claim4.json"), "w") as f:
+        json.dump(res, f, indent=1, default=jsonable)
+
+    print("gate shift-head vs softmax attn :", res["gate_shift_head_vs_softmax_attention"])
+    print("gate reference mismatches       :", res["gate_reference_target_mismatches"])
+    for r in exh:
+        print("  exhaustive |M|=%d L=%d n=%6d success=%.6f ssm=%.4f"
+              % (r["Mw"], r["L"], r["n_enumerated"], r["success"],
+                 r["ssm_key_decoded_correctly"]))
+    for r in rows:
+        print("  |M|=%3d L=%4d w=%4d (%.2f|M|, %.3fL) success=%.5f+-%.5f  analytic=%.5f  d=%d"
+              % (r["Mw"], r["L"], r["window_required"], r["window_over_Mw"],
+                 r["window_over_L"], r["success_mean"], r["success_std"],
+                 r["analytic_success"], r["embed_dim"]))
+    for c in ctl:
+        print("  ctl |M|=%d: full=%.4f thm=%.4f w=|M|:%.4f (analytic %.4f, 1-1/e=%.4f) blind=%.4f nomamba=%.4f nobias=%.4f"
+              % (c["Mw"], c["full_window_success"], c["theorem_window_success"],
+                 c["control_window_eq_Mw_success"], c["control_window_eq_Mw_analytic"],
+                 c["one_minus_1_over_e"], c["control_blind_query_success"],
+                 c["control_no_mamba_success"], c["control_no_positional_bias_success"]))
+    print("window sweep max |measured-analytic| :", res["window_sweep_max_gap"])
+
+
+if __name__ == "__main__":
+    main()

index.html

@@ -1,19 +0,0 @@
-<!doctype html>
-<html>
-	<head>
-		<meta charset="utf-8" />
-		<meta name="viewport" content="width=device-width" />
-		<title>My static Space</title>
-		<link rel="stylesheet" href="style.css" />
-	</head>
-	<body>
-		<div class="card">
-			<h1>Welcome to your static Space!</h1>
-			<p>You can modify this app directly by editing <i>index.html</i> in the Files and versions tab.</p>
-			<p>
-				Also don't forget to check the
-				<a href="https://huggingface.co/docs/hub/spaces" target="_blank">Spaces documentation</a>.
-			</p>
-		</div>
-	</body>
-</html>

logbook.json

@@ -0,0 +1,71 @@
+{
+  "schema_version": 1,
+  "title": "Reproduction audit: Hybrid Sequence Models",
+  "emoji": "\ud83d\udd2c",
+  "proposed_target_slug": "repro-hybrid-seq-82ejxjzg6r",
+  "publication_status": "local-only; no Space created or modified",
+  "paper": {
+    "arxiv_id": "2603.08859v1",
+    "openreview_id": "82EJxJzG6r"
+  },
+  "updated_at": "2026-07-28T00:00:00+00:00",
+  "root": {
+    "slug": "index",
+    "title": "Reproduction audit",
+    "file": "pages/index.md",
+    "children": [
+      {
+        "slug": "executive-summary",
+        "title": "Executive summary",
+        "file": "pages/executive-summary/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-1-theorem-3-3-literal-bound",
+        "title": "Claim 1: Theorem 3.3 proves that any k-layer state-space model solving the function-composition tasks under injectivity conditions must have total log state-space size scaling as \u03a9(m\u00b7log|V| \u2212 q\u00b7log|Y|), linear in the hidden dimension m (Theorem 3.3).",
+        "file": "pages/claim-1-theorem-3-3-literal-bound/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-2-theorem-3-7-window-bound",
+        "title": "Claim 2: Theorem 3.7 proves that sliding-window Transformers solving the same tasks under a local-sensitivity condition require total window size scaling with the context-dependency range R (Theorem 3.7).",
+        "file": "pages/claim-2-theorem-3-7-window-bound/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-3-theorem-4-3-selective-copy",
+        "title": "Claim 3: Theorem 4.3 constructs a two-layer hybrid (Mamba + attention) model that solves the selective copying task using embedding dimension O(max(log|V|, log L)) and working memory \u00d5(N), versus \u03a9(L) required by pure Transformers (Theorem 4.3).",
+        "file": "pages/claim-3-theorem-4-3-selective-copy/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-4-theorem-4-6-associative-recall",
+        "title": "Claim 4: Theorem 4.6 constructs a three-layer hybrid model that achieves 99% accuracy on the associative recall task using embedding dimension O(max(log|V|, log L)) and window size \u00d5(|V|) (Theorem 4.6).",
+        "file": "pages/claim-4-theorem-4-6-associative-recall/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-5-figure-4-selective-copy-learning",
+        "title": "Claim 5: On the selective copying task, the learned hybrid model reaches perfect accuracy with roughly 2,000 parameters while pure Transformer/SSM models need roughly 12,000 parameters to match it, a 6x parameter gap (Figure 4).",
+        "file": "pages/claim-5-figure-4-selective-copy-learning/page.md",
+        "children": []
+      },
+      {
+        "slug": "claim-6-figures-5-6-associative-recall-learning",
+        "title": "Claim 6: On multi-key associative recall, the hybrid model reaches 60% accuracy using 6x fewer parameters than pure Transformers, which plateau near 40% accuracy on single-key associative recall (Figures 5-6).",
+        "file": "pages/claim-6-figures-5-6-associative-recall-learning/page.md",
+        "children": []
+      }
+    ]
+  },
+  "routes_built": {
+    "claim_pages": [
+      "pages/claim-1-theorem-3-3-literal-bound/page.md",
+      "pages/claim-2-theorem-3-7-window-bound/page.md",
+      "pages/claim-3-theorem-4-3-selective-copy/page.md",
+      "pages/claim-4-theorem-4-6-associative-recall/page.md",
+      "pages/claim-5-figure-4-selective-copy-learning/page.md",
+      "pages/claim-6-figures-5-6-associative-recall-learning/page.md"
+    ]
+  }
+}

make_manifest.py

@@ -0,0 +1,29 @@
+"""Write a sorted, recursive SHA-256 manifest for the local package."""
+
+from __future__ import annotations
+
+import hashlib
+from pathlib import Path
+
+
+ROOT = Path(__file__).resolve().parent
+EXCLUDED_PARTS = {".git", "__pycache__"}
+EXCLUDED_NAMES = {"MANIFEST.sha256"}
+
+
+def main() -> None:
+    entries = []
+    for path in sorted(ROOT.rglob("*")):
+        if not path.is_file() or path.name in EXCLUDED_NAMES:
+            continue
+        relative = path.relative_to(ROOT)
+        if any(part in EXCLUDED_PARTS for part in relative.parts):
+            continue
+        entries.append(f"{hashlib.sha256(path.read_bytes()).hexdigest()}  {relative.as_posix()}")
+    text = "\n".join(entries) + "\n"
+    (ROOT / "MANIFEST.sha256").write_text(text)
+    print(f"manifest entries: {len(entries)}")
+
+
+if __name__ == "__main__":
+    main()

official_claims.json

@@ -0,0 +1,6 @@
+{
+  "source": "/Users/sshpro/icml-pending/claims_anchored.json",
+  "source_key": "82EJxJzG6r",
+  "note": "Literal registered claim text copied without edits before evidence was attached.",
+  "claims_file": "CLAIMS.json"
+}

official_validator.py

@@ -0,0 +1,35 @@
+"""Authoritative read-only validator for this *local* evidence package.
+
+"Official" here means the package's release gate.  It is explicitly not
+represented as a validator maintained by ICML, OpenReview, the paper authors,
+or a remote reproduction campaign.
+"""
+
+from __future__ import annotations
+
+import json
+import subprocess
+import sys
+from pathlib import Path
+
+
+ROOT = Path(__file__).resolve().parent
+
+
+def main() -> None:
+    subprocess.run([sys.executable, "validate_logbook.py", "--require-replay"], cwd=ROOT, check=True)
+    report = {
+        "validator": "official_validator.py",
+        "authority": "authoritative release gate for this local package only",
+        "not_claimed": ["ICML validator", "OpenReview validator", "paper-author validator", "remote campaign validator"],
+        "semantic_v4": "SEMANTIC_V4.json",
+        "validation": "VALIDATION.json",
+        "passed": True,
+    }
+    (ROOT / "OFFICIAL_VALIDATOR_RUN.json").write_text(json.dumps(report, indent=2) + "\n")
+    subprocess.run([sys.executable, "make_manifest.py"], cwd=ROOT, check=True)
+    print("official local package validator: PASS (semantic-v4 12/12)")
+
+
+if __name__ == "__main__":
+    main()

outputs/claim1.json

@@ -0,0 +1,747 @@
+{
+ "task": "F(u,v) = u_v with Q = [m]; G(u)=u is injective (Asm 3.2)",
+ "grid": [
+  {
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+   "V": 2,
+   "n_prefixes": 4,
+   "log2_states_required": 2.0,
+   "theory_m_log2_V": 2.0,
+   "abs_residual": 0.0,
+   "pairs_checked": 6,
+   "pairs_separated": 6,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 3,
+   "control_acc_closed_form": 0.875,
+   "control_acc_bruteforce": 0.875,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0
+    ],
+    [
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 3,
+   "V": 2,
+   "n_prefixes": 8,
+   "log2_states_required": 3.0,
+   "theory_m_log2_V": 3.0,
+   "abs_residual": 0.0,
+   "pairs_checked": 28,
+   "pairs_separated": 28,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 7,
+   "control_acc_closed_form": 0.9583333333333334,
+   "control_acc_bruteforce": 0.9583333333333334,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 4,
+   "V": 2,
+   "n_prefixes": 16,
+   "log2_states_required": 4.0,
+   "theory_m_log2_V": 4.0,
+   "abs_residual": 0.0,
+   "pairs_checked": 120,
+   "pairs_separated": 120,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 15,
+   "control_acc_closed_form": 0.984375,
+   "control_acc_bruteforce": 0.984375,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "theory_m_log2_V": 5.0,
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+   "pairs_separated": 496,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 31,
+   "control_acc_closed_form": 0.99375,
+   "control_acc_bruteforce": 0.99375,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "pairs_checked": 2016,
+   "pairs_separated": 2016,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 63,
+   "control_acc_closed_form": 0.9973958333333334,
+   "control_acc_bruteforce": 0.9973958333333334,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 7,
+   "V": 2,
+   "n_prefixes": 128,
+   "log2_states_required": 7.0,
+   "theory_m_log2_V": 7.0,
+   "abs_residual": 0.0,
+   "pairs_checked": 8128,
+   "pairs_separated": 8128,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 127,
+   "control_acc_closed_form": 0.9988839285714286,
+   "control_acc_bruteforce": 0.9988839285714286,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "log2_states_required": 8.0,
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+   "pairs_separated": 32640,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 255,
+   "control_acc_closed_form": 0.99951171875,
+   "control_acc_bruteforce": 0.99951171875,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 9,
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+   "n_prefixes": 512,
+   "log2_states_required": 9.0,
+   "theory_m_log2_V": 9.0,
+   "abs_residual": 0.0,
+   "pairs_checked": 130816,
+   "pairs_separated": 130816,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 511,
+   "control_acc_closed_form": 0.9997829861111112
+  },
+  {
+   "m": 10,
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+   "n_prefixes": 1024,
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+   "abs_residual": 0.0,
+   "control_states": 1023,
+   "control_acc_closed_form": 0.99990234375
+  },
+  {
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+   "theory_m_log2_V": 3.169925001442312,
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+   "pairs_separated": 36,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
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+   "control_acc_closed_form": 0.9444444444444444,
+   "control_acc_bruteforce": 0.9444444444444444,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0
+    ],
+    [
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "log2_states_required": 4.754887502163468,
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+   "abs_residual": 0.0,
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+   "pairs_separated": 351,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 26,
+   "control_acc_closed_form": 0.9876543209876543,
+   "control_acc_bruteforce": 0.9876543209876543,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 4,
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+   "n_prefixes": 81,
+   "log2_states_required": 6.339850002884624,
+   "theory_m_log2_V": 6.339850002884624,
+   "abs_residual": 0.0,
+   "pairs_checked": 3240,
+   "pairs_separated": 3240,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 80,
+   "control_acc_closed_form": 0.9969135802469136,
+   "control_acc_bruteforce": 0.9969135802469136,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
+   "m": 5,
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+   "log2_states_required": 7.924812503605781,
+   "theory_m_log2_V": 7.92481250360578,
+   "abs_residual": 8.881784197001252e-16,
+   "pairs_checked": 29403,
+   "pairs_separated": 29403,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 242,
+   "control_acc_closed_form": 0.9991769547325103,
+   "control_acc_bruteforce": 0.9991769547325103,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "log2_states_required": 9.509775004326936,
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+   "abs_residual": 0.0,
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+   "control_acc_closed_form": 0.9997713763145862
+  },
+  {
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+   "log2_states_required": 11.094737505048093,
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+   "abs_residual": 0.0,
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+   "control_acc_closed_form": 0.9999346789470246
+  },
+  {
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+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 15,
+   "control_acc_closed_form": 0.96875,
+   "control_acc_bruteforce": 0.96875,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0
+    ],
+    [
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "log2_states_required": 6.0,
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+   "pairs_checked": 2016,
+   "pairs_separated": 2016,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
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+   "control_acc_closed_form": 0.9947916666666666,
+   "control_acc_bruteforce": 0.9947916666666666,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "pairs_checked": 32640,
+   "pairs_separated": 32640,
+   "all_pairs_separated": true,
+   "min_separating_queries": 1,
+   "control_states": 255,
+   "control_acc_closed_form": 0.9990234375,
+   "control_acc_bruteforce": 0.9990234375,
+   "control_acc_gap": 0.0,
+   "control_witness_pair": [
+    [
+     0,
+     0,
+     0,
+     0
+    ],
+    [
+     0,
+     0,
+     0,
+     1
+    ]
+   ]
+  },
+  {
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+   "n_prefixes": 1024,
+   "log2_states_required": 10.0,
+   "theory_m_log2_V": 10.0,
+   "abs_residual": 0.0,
+   "control_states": 1023,
+   "control_acc_closed_form": 0.9998046875
+  },
+  {
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+   "n_prefixes": 4096,
+   "log2_states_required": 12.0,
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+   "abs_residual": 0.0,
+   "control_states": 4095,
+   "control_acc_closed_form": 0.9999593098958334
+  }
+ ],
+ "max_residual_log2_states": 8.881784197001252e-16,
+ "max_control_gap": 0.0,
+ "all_pairs_separated_everywhere": true,
+ "linear_in_m_fit": {
+  "2": {
+   "fitted_slope": 1.0,
+   "theory_slope_log2_V": 1.0,
+   "slope_abs_err": 0.0,
+   "intercept": 1.6770126117963062e-15,
+   "R2": 1.0,
+   "max_abs_residual": 1.7763568394002505e-15
+  },
+  "3": {
+   "fitted_slope": 1.5849625007211565,
+   "theory_slope_log2_V": 1.584962500721156,
+   "slope_abs_err": 4.440892098500626e-16,
+   "intercept": -4.520058619021783e-15,
+   "R2": 1.0,
+   "max_abs_residual": 3.552713678800501e-15
+  },
+  "4": {
+   "fitted_slope": 2.0,
+   "theory_slope_log2_V": 2.0,
+   "slope_abs_err": 0.0,
+   "intercept": -1.3107111616019684e-15,
+   "R2": 1.0,
+   "max_abs_residual": 1.7763568394002505e-15
+  }
+ },
+ "exhaustive_accuracy_curves": [
+  {
+   "m": 2,
+   "V": 2,
+   "n_prefixes": 4,
+   "partitions_enumerated": "Bell(4)",
+   "A_star": {
+    "1": 0.5,
+    "2": 0.75,
+    "3": 0.875,
+    "4": 1.0
+   },
+   "min_states_acc_1.0": 4,
+   "min_states_acc_0.875": 3,
+   "min_states_acc_0.9": 4,
+   "min_states_acc_0.5": 1,
+   "A_star_at_n_minus_1": 0.875,
+   "closed_form_at_n_minus_1": 0.875
+  },
+  {
+   "m": 3,
+   "V": 2,
+   "n_prefixes": 8,
+   "partitions_enumerated": "Bell(8)",
+   "A_star": {
+    "1": 0.5,
+    "2": 0.75,
+    "3": 0.7916666666666666,
+    "4": 0.8333333333333334,
+    "5": 0.875,
+    "6": 0.9166666666666666,
+    "7": 0.9583333333333334,
+    "8": 1.0
+   },
+   "min_states_acc_1.0": 8,
+   "min_states_acc_0.875": 5,
+   "min_states_acc_0.9": 6,
+   "min_states_acc_0.5": 1,
+   "A_star_at_n_minus_1": 0.9583333333333334,
+   "closed_form_at_n_minus_1": 0.9583333333333334
+  },
+  {
+   "m": 2,
+   "V": 3,
+   "n_prefixes": 9,
+   "partitions_enumerated": "Bell(9)",
+   "A_star": {
+    "1": 0.3333333333333333,
+    "2": 0.5555555555555556,
+    "3": 0.6666666666666666,
+    "4": 0.7222222222222222,
+    "5": 0.7777777777777778,
+    "6": 0.8333333333333334,
+    "7": 0.8888888888888888,
+    "8": 0.9444444444444444,
+    "9": 1.0
+   },
+   "min_states_acc_1.0": 9,
+   "min_states_acc_0.875": 7,
+   "min_states_acc_0.9": 8,
+   "min_states_acc_0.5": 2,
+   "A_star_at_n_minus_1": 0.9444444444444444,
+   "closed_form_at_n_minus_1": 0.9444444444444444
+  }
+ ],
+ "exhaustive_min_states_equals_Vm": true,
+ "max_curve_control_gap": 0.0,
+ "lemma_3_5": [
+  {
+   "k": 2,
+   "sizes": [
+    2,
+    3
+   ],
+   "product_bound": 6,
+   "reachable_product_states": 2,
+   "sequences_tested": 510,
+   "behaviour_mismatches": 0,
+   "bound_respected": true
+  },
+  {
+   "k": 3,
+   "sizes": [
+    2,
+    2,
+    2
+   ],
+   "product_bound": 8,
+   "reachable_product_states": 2,
+   "sequences_tested": 254,
+   "behaviour_mismatches": 0,
+   "bound_respected": true
+  },
+  {
+   "k": 2,
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+   "behaviour_mismatches": 0,
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+ ],
+ "lemma_3_5_tight_witnesses": 3,
+ "printed_bound_audit": {
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+  "max_literal_bound_over_admissible_grid": 0.0,
+  "literal_bound_ever_positive": false,
+  "min_appendix_bound_on_instantiation": 0.6628711136008072,
+  "appendix_slope_in_m": {
+   "2": {
+    "fitted_slope": 0.3314355568004036,
+    "closed_form_slope": 0.3314355568004036,
+    "abs_err": 0.0
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+    "closed_form_slope": 1.2064355568004035,
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+    "closed_form_slope": 2.0814355568004035,
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+    "closed_form_slope": 2.9564355568004035,
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+    "closed_form_slope": 3.8314355568004035,
+    "abs_err": 4.440892098500626e-16
+   }
+  },
+  "note": "literal = m*log2|V| - q*log2|Y| (Thm 3.3 as printed); appendix = m*log2|V| - q*(H2(1/8)+log2|Y|/8) (proof, err<1/8)"
+ },
+ "one_state_guessing_accuracy": {
+  "2": 0.5,
+  "3": 0.3333333333333333,
+  "4": 0.25,
+  "8": 0.125,
+  "26": 0.038461538461538464,
+  "32": 0.03125
+ }
+}

outputs/claim2.json

@@ -0,0 +1,1307 @@
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+  "note": "Accelerate RuntimeWarnings are spurious; verified here"
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+}

outputs/claim3.json

@@ -0,0 +1,349 @@
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outputs/claim3_native.json

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outputs/claim4.json

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outputs/claim4_native.json

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outputs/claim5.json

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+    {
+      "parameters_approx": 6000.0,
+      "pure_tf": 0.727,
+      "pure_ssm": 0.485,
+      "tf_to_ssm": 0.822,
+      "ssm_to_tf": 1.0
+    },
+    {
+      "parameters_approx": 12000.0,
+      "pure_tf": 0.923,
+      "pure_ssm": 0.931,
+      "tf_to_ssm": 0.908,
+      "ssm_to_tf": 1.0
+    }
+  ],
+  "literal_table_comparison": {
+    "hybrid_ssm_to_tf_at_approximately_2000": 0.999,
+    "pure_tf_at_approximately_12000": 0.923,
+    "pure_ssm_at_approximately_12000": 0.931,
+    "parameter_ratio_12000_over_2000": 6.0,
+    "strict_table_value_is_exactly_one": false
+  },
+  "source_caption_says_perfect": true,
+  "source_results_says_roughly_six_times": true,
+  "assessment": "The authored table supports the approximate 6x/near-perfect comparison (.999 versus .923/.931), but its printed .999 is not literally 1.000 while the caption calls it perfect. This is source evidence, not an independent learned-model reproduction.",
+  "training_route": {
+    "official_train_file": "source/official-code/micro_hf/train_utils.py",
+    "official_train_file_sha256": "ec4ccc0fa03f636f141904484f54763576e741c6f3586b1405343bb844c7c040",
+    "hard_coded_cuda_calls": 1,
+    "torch_cuda_available_on_this_host": false,
+    "result_artifacts_present": false,
+    "reason_no_local_training_rerun": "The official micro_hf entry point hard-codes CUDA tensors; this host has no CUDA device. The checkout contains figures and lrs.json metadata but no per-run evaluation JSON/CSV/checkpoint results."
+  }
+}

outputs/claim6.json

@@ -0,0 +1,58 @@
+{
+  "kind": "exact_authored_source_audit",
+  "source": "source/authored/sections/experiments.tex",
+  "source_sha256": "fcd88332cbe90aff62337c3add333f71fc542736b3e459015a28f0b08ae3c55b",
+  "mkar_table_heading_line": 63,
+  "mkar_caption_line": 80,
+  "single_key_figure_result_line": 60,
+  "mkar_table_rows": [
+    {
+      "parameters_approx": 1000.0,
+      "pure_tf": 0.124,
+      "pure_ssm": 0.158,
+      "tf_to_ssm": 0.131,
+      "ssm_to_tf": 0.144
+    },
+    {
+      "parameters_approx": 2000.0,
+      "pure_tf": 0.159,
+      "pure_ssm": 0.173,
+      "tf_to_ssm": 0.183,
+      "ssm_to_tf": 0.512
+    },
+    {
+      "parameters_approx": 6000.0,
+      "pure_tf": 0.23,
+      "pure_ssm": 0.356,
+      "tf_to_ssm": 0.286,
+      "ssm_to_tf": 0.99
+    },
+    {
+      "parameters_approx": 12000.0,
+      "pure_tf": 0.668,
+      "pure_ssm": 0.517,
+      "tf_to_ssm": 0.524,
+      "ssm_to_tf": 0.989
+    }
+  ],
+  "literal_table_checks": {
+    "hybrid_ssm_to_tf_at_approximately_2000": 0.512,
+    "hybrid_ssm_to_tf_at_approximately_6000": 0.99,
+    "pure_tf_at_approximately_12000": 0.668,
+    "sixfold_parameter_ratio_from_2000_to_12000": 6.0,
+    "hybrid_reaches_0_60_at_approximately_2000": false,
+    "hybrid_reaches_0_60_at_approximately_6000": true,
+    "ratio_for_6000_vs_12000": 2.0
+  },
+  "source_caption_claims_60pct_and_six_times": true,
+  "single_key_statement_is_a_different_task": true,
+  "assessment": "The authored MKAR table is internally insufficient for the literal 60%-at-6x conjunction: at the 6x row it reports .512 (<.60), while at .990 the nearest shown pure-TF row is only 2x larger. The caption asserts the headline, but raw points needed to locate an unshown 60% crossing were not released. The <=40% statement is explicitly about associative recall with decoding (Figure 5), not MKAR (Figure 6).",
+  "training_route": {
+    "official_train_file": "source/official-code/micro_hf/train_utils.py",
+    "official_train_file_sha256": "ec4ccc0fa03f636f141904484f54763576e741c6f3586b1405343bb844c7c040",
+    "hard_coded_cuda_calls": 1,
+    "torch_cuda_available_on_this_host": false,
+    "result_artifacts_present": false,
+    "reason_no_local_training_rerun": "The official micro_hf entry point hard-codes CUDA tensors; this host has no CUDA device. The checkout contains figures and lrs.json metadata but no per-run evaluation JSON/CSV/checkpoint results."
+  }
+}

pages/claim-1-theorem-3-3-literal-bound/page.md

@@ -0,0 +1,16 @@
+# Theorem 3.3 proves that any k-layer state-space model solving the function-composition tasks under injectivity conditions must have total log state-space size scaling as Ω(m·log|V| − q·log|Y|), linear in the hidden dimension m (Theorem 3.3).
+
+**Assessment: falsified as a literal positive linear lower-bound claim.**
+
+Assumption 3.2 makes G: V^m → Y^q injective, so cardinality gives m·log₂|V| ≤ q·log₂|Y|. The printed right-hand side m·log|V| − q·log|Y| is therefore never positive under its own assumption. The appendix instead derives a different Fano bound under error < 1/8; that is not the printed probability-1/2 theorem.
+
+## Evidence
+
+- Exact authored statement: source/authored/sections/func_comp_and_construct.tex:26–28.
+- 1,296 admissible cardinality configurations: maximum printed RHS = 0.0.
+- Exhaustive binary selective-copy partitions: one state attains exactly 1/2 success for m=2 and m=3, matching the theorem's printed threshold.
+- The appendix source records its stronger, different prerequisite as err < 1/8 (source/authored/appendix/missing_proof_lb.tex).
+
+## Scope and limitations
+
+This is a literal-statement audit, not a claim that no corrected lower bound can be proved. The exact selective-copy enumeration is finite; the cardinality implication itself is general.

pages/claim-2-theorem-3-7-window-bound/page.md

@@ -0,0 +1,16 @@
+# Theorem 3.7 proves that sliding-window Transformers solving the same tasks under a local-sensitivity condition require total window size scaling with the context-dependency range R (Theorem 3.7).
+
+**Assessment: supported by explicit paper-task witnesses and exact receptive-field checks.**
+
+For the selective-copy witnesses, changing only a token outside the causal receptive field leaves the terminal logits bit-identical. On the two-point witness distribution this forces accuracy 1/2, below 2/3. A full-window control separates the same pair.
+
+## Evidence
+
+- 960 real float64 causal-attention stacks; outside-RF violations = 0.
+- Maximum terminal-logit change outside RF = 0.0 for the hard witnesses.
+- Full-window control separates 100% of tested witness configurations.
+- The source proof explicitly identifies the terminal dependency as the last sum_i W_i tokens (appendix/missing_proof_lb.tex).
+
+## Scope and limitations
+
+Finite numerical checks cannot prove the universal theorem. They directly exercise its stated quantities (window, R, indistinguishability and success threshold) on the paper's selective-copy task rather than a proxy.

pages/claim-3-theorem-4-3-selective-copy/page.md

@@ -0,0 +1,16 @@
+# Theorem 4.3 constructs a two-layer hybrid (Mamba + attention) model that solves the selective copying task using embedding dimension O(max(log|V|, log L)) and working memory Õ(N), versus Ω(L) required by pure Transformers (Theorem 4.3).
+
+**Assessment: supported for the literal construction; official notebook does not establish its universal scope.**
+
+A direct transcription of Appendix D.2/D.5 achieves exact final-position selective copying across exhaustive and large deterministic sweeps, with a window-minus-one and no-SSM destructive controls. The separately executed author notebook is included unchanged as provenance and scores only .825 on the notebook generator's 1,024 final-position examples, so it is not used as evidence of the theorem's 'every input' quantifier.
+
+## Evidence
+
+- Independent construction: 1,506,370 valid inputs; minimum accuracy = 1.000000.
+- Includes L=100, |V|=32 and longer L=1,024/4,096 sweeps; these are labelled construction checks, not learned-model runs.
+- Native notebook baseline/control: 0.825195 / 0.054688 on 1024 eligible examples.
+- The destructive construction controls remove one attention position, zero the query, or disable selective state update.
+
+## Scope and limitations
+
+The independent checker is a faithful finite implementation of the written construction, not a mechanized proof. The native notebook's aggregate failure is retained rather than overwritten or relabelled as a success.

pages/claim-4-theorem-4-6-associative-recall/page.md

@@ -0,0 +1,16 @@
+# Theorem 4.6 constructs a three-layer hybrid model that achieves 99% accuracy on the associative recall task using embedding dimension O(max(log|V|, log L)) and window size Õ(|V|) (Theorem 4.6).
+
+**Assessment: supported by a full-vocabulary construction and exact coverage certificate; native notebook is only a limited check.**
+
+The Appendix D.4 construction was implemented with the stated full-vocabulary binary code, last-match positional bias, and a finite softmax separation. Its exact iid coverage formula is at least .99 for each measured scale; the 5×20,000-instance point estimates track that certificate (one mean, .989820, is below .99 and is not relabelled as an empirical pass). The implementation also exhausts three small full-window domains.
+
+## Evidence
+
+- All analytic coverage certificates meet 99%: True.
+- Minimum five-seed empirical mean = 0.989820; maximum gap from exact coverage = 0.000619.
+- Exhaustive full-window domains (|M|, L)=(4,8),(4,9),(8,8) all score 1.0.
+- Native authored notebook baseline/control: 0.927521 / 0.026261; it has no windowed 99%-coverage test.
+
+## Scope and limitations
+
+The native notebook is a direct one-construction execution, not the paper's probabilistic window experiment. The 99% support comes from the written construction plus an exact iid coverage calculation and finite tests; it is not a claimed learned-model training rerun.

pages/claim-5-figure-4-selective-copy-learning/page.md

@@ -0,0 +1,15 @@
+# On the selective copying task, the learned hybrid model reaches perfect accuracy with roughly 2,000 parameters while pure Transformer/SSM models need roughly 12,000 parameters to match it, a 6x parameter gap (Figure 4).
+
+**Assessment: authored-source supported, but not independently reproduced.**
+
+The exact authored Figure 4 table gives SSM→TF .999 at approximately 2,000 parameters and pure TF/SSM .923/.931 at approximately 12,000 parameters, a sixfold nominal parameter ratio. The caption calls .999 'perfect', so the strict word perfect and printed value are internally inconsistent at the shown precision.
+
+## Evidence
+
+- Pinned source table: hybrid@2k=0.999; pure TF/SSM@12k=0.923/0.931.
+- The TeX table, caption, and source-file SHA are in outputs/claim5.json.
+- The official micro_hf training path contains hard-coded CUDA transfers and this machine has no CUDA device; no local run is presented as a reproduction.
+
+## Scope and limitations
+
+The repository did not contain per-run scalar results, checkpoints, or an author-pinned code commit in the arXiv archive. This page reports the paper's own exact table, not an independent empirical confirmation.

pages/claim-6-figures-5-6-associative-recall-learning/page.md

@@ -0,0 +1,15 @@
+# On multi-key associative recall, the hybrid model reaches 60% accuracy using 6x fewer parameters than pure Transformers, which plateau near 40% accuracy on single-key associative recall (Figures 5-6).
+
+**Assessment: not independently established; the authored MKAR table conflicts with the literal numeric conjunction.**
+
+The Figure 6 table's sixfold row is approximately 2,000 versus 12,000 parameters, where SSM→TF is .512—not 60%. At the first shown hybrid result above 60% (.990 at approximately 6,000), the nearest shown pure-TF row is approximately 12,000, only 2× larger. The caption asserts 60%-at-6×, but the raw points needed to locate a different crossing were not released.
+
+## Evidence
+
+- Pinned MKAR values: hybrid@2k=0.512, hybrid@6k=0.990, pure TF@12k=0.668.
+- The <=40% sentence is from Figure 5's associative recall with decoding, a different task from Figure 6's MKAR; it is not treated as an MKAR plateau measurement.
+- The exact TeX/table SHA and CUDA-only non-rerun route are recorded in outputs/claim6.json.
+
+## Scope and limitations
+
+This is an authored-source/data audit. It does not infer a curve between unreleased points or substitute the Figure 5 task for MKAR. A GPU rerun would require compatible CUDA hardware and a declared protocol; neither is claimed here.

pages/executive-summary/page.md

@@ -0,0 +1,14 @@
+# Executive summary
+
+This is a six-claim audit of arXiv:2603.08859v1 / OpenReview `82EJxJzG6r`. Claim text is frozen in `CLAIMS.json`; evidence does not rewrite scope.
+
+| claim | subject | assessment | headline |
+| --- | --- | --- | --- |
+| 1 | Theorem 3.3 literal lower bound | falsified | printed RHS non-positive under injectivity |
+| 2 | Theorem 3.7 window lower bound | supported | paper-task witnesses and real attention stacks |
+| 3 | Theorem 4.3 selective copy | supported | independent construction; native notebook limitation retained |
+| 4 | Theorem 4.6 associative recall | supported | full-vocabulary construction and exact coverage |
+| 5 | Figure 4 learned selective copy | source-supported | reported table only; .999/perfect rounding conflict |
+| 6 | Figures 5–6 learned recall | not established | MKAR table conflicts with 60%-at-6x conjunction |
+
+The package distinguishes (1) native author-notebook execution, (2) an independent implementation of the written constructions, and (3) exact authored-table evidence. GPU-only learned-model training was not rerun on this non-CUDA host.

pages/index.md

@@ -0,0 +1,5 @@
+# Reproduction audit: Expressivity-Efficiency Tradeoffs for Hybrid Sequence Models
+
+Paper: arXiv:2603.08859v1 · OpenReview `82EJxJzG6r`
+
+Six registered claims are preserved verbatim in `CLAIMS.json`. Run `python3 run_all.py` for paired deterministic replays, page generation, validation, and a recursive manifest.

replay_a/claim1.json

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+    "closed_form_slope": 3.8314355568004035,
+    "abs_err": 4.440892098500626e-16
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+  },
+  "note": "literal = m*log2|V| - q*log2|Y| (Thm 3.3 as printed); appendix = m*log2|V| - q*(H2(1/8)+log2|Y|/8) (proof, err<1/8)"
+ },
+ "one_state_guessing_accuracy": {
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+  "4": 0.25,
+  "8": 0.125,
+  "26": 0.038461538461538464,
+  "32": 0.03125
+ }
+}

replay_a/claim2.json

@@ -0,0 +1,1307 @@
+{
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+  "note": "Accelerate RuntimeWarnings are spurious; verified here"
+ },
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replay_a/claim3.json

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replay_a/claim3_native.json

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replay_a/claim4.json

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replay_a/claim4_native.json

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replay_a/claim5.json

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replay_a/claim6.json

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+    "abs_err": 4.440892098500626e-16
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+  "note": "literal = m*log2|V| - q*log2|Y| (Thm 3.3 as printed); appendix = m*log2|V| - q*(H2(1/8)+log2|Y|/8) (proof, err<1/8)"
+ },
+ "one_state_guessing_accuracy": {
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+  "4": 0.25,
+  "8": 0.125,
+  "26": 0.038461538461538464,
+  "32": 0.03125
+ }
+}

replay_b/claim2.json

@@ -0,0 +1,1307 @@
+{
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+  "note": "Accelerate RuntimeWarnings are spurious; verified here"
+ },
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replay_b/claim3.json

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replay_b/claim3_native.json

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replay_b/claim4.json

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