Absorb the restricted monomial by the four-bit gate (no new κ)

research/gaussian-restricted-gate/README.md

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+# The restricted monomial keeps the four-bit gate
+
+**Claim type:** an explicit absorption bound for one scalar monomial.
+**No new κ.** The ceiling `9 p^2` and the section error `p^6 exp(-p^2)`
+stay hypotheses. The selected record remains `23/10000`.
+
+## Question
+
+#4517 bounds the kept core outputs by
+
+```text
+2 g M^{g-1} ε,
+```
+
+and asks whether that scalar, at `M = 9 p^2` and `ε = p^6 exp(-p^2)`,
+still lies below `2^{-C p}`. The named step is to replace the base
+`9 p^2 + ε` in #4506 by `9 p^2` and recheck the four-bit gate.
+
+## Lemma
+
+Let `p ≥ 2`, `g ≥ 1`, `K ≥ 0` and `C ≥ 0` be integers, and let
+`L = ceil(log2 p)`. If
+
+```text
+p ≥ max(64, 4K + 2C)    and    16 g L ≤ p,
+```
+
+then
+
+```text
+2 g (9 p^2)^{g-1} p^6 exp(K p - p^2) < 2^{-C p}.
+```
+
+The case `K = 0` is the monomial in the question. The factor 2 is one bit.
+The exponent is at most `-C p - 2`. Whenever `g log p = o(p)`, every fixed
+pair `K, C` satisfies the gate for all large `p`.
+
+Four bits still majorize the coefficient, because `9 < 16`. Three bits do
+not: `9 > 8`. Removing `ε` from the base does not change that count. It
+removes the separate room `p^6 exp(-p^2) < 1/2` used by #4506.
+
+At `g = p^2` the monomial is greater than 1 for every integer `p ≥ 2`.
+
+## Proof
+
+The series `1 + 1 + 1/2 = 5/2 > 2` gives `e > 2`, so `exp(-p^2) < 2^{-p^2}`.
+For `k ≥ 2`, `k! ≥ 2^{k-1}`. The tail of `e` is then at most a geometric
+series of ratio `1/2`, so `e < 4` and `exp(K p) < 2^{2 K p}` whenever
+`K p > 0`. At `K = 0` the factor is 1.
+
+#4496 places the telescoping monomial strictly below `2^{E_0}`, where
+
+```text
+E_0 = 2 g L + 4 g + 5 L - 3 + 2 K p - p^2,
+```
+
+and proves `E_0 ≤ -C p - 3` under this gate. The identity behind that
+exponent is `g (9 p^2)^{g-1} p^6 = g (9 p^2)^g (p^4 / 9)`, with
+`p^4 / 9 < 2^{4 L - 3}` because `9 > 8` and `p ≤ 2^L`. The factor 2 in
+the present monomial multiplies that bound by `2^1`, so the new exponent is
+
+```text
+E = E_0 + 1 = 2 g L + 4 g + 5 L - 2 + 2 K p - p^2.
+```
+
+Clear eight times `E + C p`. The pieces scale as `8 · 2 g L = 16 g L ≤ p`,
+`8 · 4 g = 32 g ≤ 2 p` because `L ≥ 1` turns the gate into `16 g ≤ p`,
+`8 · 5 L = 40 L ≤ 5 p` because `8 L ≤ p` for every integer `p ≥ 64`, and
+`8 (2 K + C) p ≤ 4 p^2` once `p ≥ 4 K + 2 C`. The constant `-2` contributes
+`-16`. The sum is at most
+
+```text
+-4 (p^2 - 2 p + 4).
+```
+
+At `p = 2` the polynomial equals 4, and the step to the next integer is
+`2 p - 1 ≥ 1`. Eight times `E + C p` is therefore at most `-16`, and
+`E ≤ -C p - 2`. The monomial is strictly less than `2^E`, hence strictly
+less than `2^{-C p}`.
+
+The same induction `8(m + 1) ≤ 2^m` for `m ≥ 6` gives `8 L ≤ p` on
+`p ≥ 64`. Its base is `8 · 7 = 56 ≤ 64`.
+
+The three-bit comparison still reverses: `(9 p^2)^g > 2^{3 g} p^{2 g}`.
+At the legal point `p = 128`, `L = 7`, `g = 1`, both the three-bit and
+four-bit exponents are negative, while `9 p^2 > 8 p^2`.
+
+At `g = p^2` the monomial is twice the #4496 monomial. That smaller
+monomial is already greater than 1, because `(9 p^2)^{p^2} > 4^{p^2}`
+and `4 > e`, while `p^4 / 9 > 1`. The extra factor 2 keeps it above 1.
+
+## Not covered
+
+- A proof of the restricted tensor bound in #4517. This package absorbs
+  the scalar only after `M = 9 p^2` and `ε = p^6 exp(-p^2)` are granted.
+- A proof that `||C_i|| ≤ 9 p^2`, or that `||P_glob - C_i||_2 < ε`.
+- Either ideal-gap hypothesis.
+- Editing the #4100 notes. They still say `g p^{10 g} exp(-p^2)`.
+- Any new κ.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-restricted-gate --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-restricted-gate
+```
+
+Base `f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc`, priority revision 7, selection
+refresh κ = `23/10000`. Upstream was fetched at 2026-10-22T03:18:46Z and was
+that commit.
+
+Overlap snapshot: open PRs fetched 2026-10-22T03:21:17Z (305 records) and
+discussions fetched 2026-10-22T03:21:59Z (3397 records). A search for
+`restricted bound`, `(9 p^2)^{g-1}` and `Cp - 2` returned #4496, which
+absorbs the monomial without the factor 2, and #4506, which absorbs the
+factor 2 only after the base has grown by `ε`. The other hits are the
+predecessor gate and unrelated restricted bounds. No result charges the
+factor 2 against the fold slack of 3. A search miss is not a novelty
+certificate. #4496 is the fold. #4506 is the larger base. HALO is the
+written error.
+
+## Open next questions
+
+1. **Is `||P_glob - C_i||_2` below `p^6 exp(-p^2)`, or only the written
+   `p^{10} exp(-p^2)`?** The restricted monomial uses the section error.
+   The global tail in HALO is `p^{10}`. Replacing the section error by
+   that tail would add `4 L` to the exponent. The first step is to compare
+   `P - H^{-1} = -Q^J H^{-1}` with the within-cell `p^6` bound, and not
+   to put `p^{10}` into this gate yet.
+
+## Credit
+
+- The fold exponent `-C p - 3`, and the factor `p^4 / 9`, are #4496's.
+- The face-factor inequality with base `9 p^2 + ε` is #4506's.
+- The restricted scalar `2 g M^{g-1} ε`, and this question, are #4517's.
+- The four-bit gate is #4495's.
+- The written error `g p^{10 g} exp(-p^2)` is #4100's.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-restricted-gate/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "research/gaussian-tensor-nine-faces/README.md",
+      "usage": "input",
+      "git_blob": "03ac0ffa1ed344a62902cd9b2320e26d60cca003",
+      "sha256": "bb733a49416eddd574d64a9e9092ef1ba8dc0fb22f2eea15f7f3ce0d280d7aee"
+    },
+    {
+      "path": "research/gaussian-tensor-nine-fold/README.md",
+      "usage": "input",
+      "git_blob": "af78553b992a9f1801423b3ba42d984cdbe12c4d",
+      "sha256": "8e003f1ec809b5c7b8f6552592ccd32d1d806ba55b58bae60fff220b959b2d05"
+    },
+    {
+      "path": "research/unconditional-batched-crt/HALO.md",
+      "usage": "input",
+      "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b",
+      "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc"
+    }
+  ]
+}

research/gaussian-restricted-gate/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Absorb 2 g (9 p^2)^{g-1} p^6 exp(K p - p^2) by the same four-bit gate, with the approximant increment removed from the base.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only rechecks that scalar gate. It does not prove the restricted tensor bound, the ideal gaps, or a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the 2^{-C p} comparison left open by the core-row operator lift.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the restricted monomial is below 2^{-C p}. Do not prove ||C_i|| <= 9 p^2, do not replace p^6 by the global tail, do not edit the #4100 notes, and do not claim a new kappa.",
+  "outcome": "completed",
+  "evidence": "python3.12 -B scripts/research_package.py --package research/gaussian-restricted-gate",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the one-bit slack check and the package text.",
+  "open_questions": [
+    {
+      "question": "Is ||P_glob - C_i||_2 below p^6 exp(-p^2), or only the written p^{10} exp(-p^2)?",
+      "why_it_matters": "The restricted monomial uses the section error p^6. The global tail in HALO is p^{10}, and replacing it would add 4 L to the exponent.",
+      "first_step": "Compare the global identity P - H^{-1} = -Q^J H^{-1} with the within-cell p^6 bound, and do not put p^{10} into this gate yet.",
+      "blocked_by": "The section ceiling 9 p^2 and both ideal-gap gauges remain hypotheses, and the global inverse norm is not the within-cell Neumann ceiling."
+    }
+  ]
+}

research/gaussian-restricted-gate/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/gaussian-tensor-nine-faces/README.md",
+    "research/gaussian-tensor-nine-fold/README.md",
+    "research/unconditional-batched-crt/HALO.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Under p >= max(64, 4K+2C) and 16 g L <= p, the factor 2 adds one bit to the fold exponent. Eight times the new exponent is at most -16, so the exponent is at most -C p - 2. Three bits still fail because 9 > 8. At g = p^2 the monomial exceeds 1."
+    }
+  ]
+}

research/gaussian-restricted-gate/verify.py

@@ -0,0 +1,119 @@
+#!/usr/bin/env python3
+"""Absorb the restricted monomial by the same four-bit gate.
+
+Every comparison is an integer identity. The ceiling and the section error
+stay hypotheses.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-restricted-gate"
+
+
+def digest(path: Path) -> str:
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def ceil_log2(value: int) -> int:
+    level = 0
+    power = 1
+    while power < value:
+        power *= 2
+        level += 1
+    return level
+
+
+def pins_and_phrases() -> None:
+    sources = json.loads((PACKAGE / "SOURCES.json").read_text(encoding="utf-8"))
+    for row in sources["files"]:
+        if row["usage"] == "input":
+            assert digest(ROOT / row["path"]) == row["sha256"], row["path"]
+    halo = (ROOT / "research/unconditional-batched-crt/HALO.md").read_text(encoding="utf-8")
+    fold = (ROOT / "research/gaussian-tensor-nine-fold/README.md").read_text(encoding="utf-8")
+    faces = (ROOT / "research/gaussian-tensor-nine-faces/README.md").read_text(encoding="utf-8")
+    selected = json.loads(
+        (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8")
+    )
+    assert "g p^(10g) exp(-p^2)," in halo
+    assert "Keep outputs at distance A" in halo
+    assert "The resulting exponent is at most" in fold
+    assert "`-C p - 3`" in fold
+    assert (
+        "2 g (9 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2) < 2^{-C p}."
+        in faces
+    )
+    assert selected["kappa"] == "23/10000"
+
+
+def series_and_bits() -> None:
+    # e > 5/2 > 2 and e < 4. Four bits clear 9; three bits do not.
+    assert 2 + 2 + 1 > 4
+    assert 3 < 4
+    assert 2 >= 2 and 6 >= 4 and 24 >= 8
+    assert 8 < 9 < 16
+    assert 2**3 == 8
+    assert 2**4 == 16
+    # The factor 2 is one bit, taken from the fold slack of 3.
+    assert 2**1 == 2
+    assert 8 * (-3) == -24
+    assert 8 * (-2) == -16
+    assert -24 + 8 == -16
+
+
+def log_and_budget() -> None:
+    # 8 L <= p for every integer p >= 64, by the same induction as #4495.
+    assert 8 * 7 <= 64
+    assert 8 <= 64
+    assert 2**6 + 8 <= 2**7
+    assert ceil_log2(64) == 6
+    assert ceil_log2(128) == 7
+    # Eight times the new pieces: 2 g L, 4 g, 5 L, and the constant -2.
+    assert 8 * 2 == 16
+    assert 8 * 4 == 32
+    assert 32 == 2 * 16
+    assert 8 * 5 == 40
+    assert 40 == 5 * 8
+    assert 1 + 2 + 5 == 8
+    # (2K + C) p <= p^2 / 2 becomes 8(2K + C) p <= 4 p^2 when p >= 4K + 2C.
+    assert 8 * (2 * 3 + 5) == 4 * (4 * 3 + 2 * 5)
+    # p^2 - 2 p + 4 is 4 at p = 2 and then increases by 2 p - 1.
+    assert 2 * 2 - 2 * 2 + 4 == 4
+    assert -4 * 4 == -16
+    assert (3 * 3 - 2 * 3 + 4) - (2 * 2 - 2 * 2 + 4) == 2 * 2 - 1
+
+
+def witness_and_failures() -> None:
+    prime = 128
+    level = ceil_log2(prime)
+    groups = 1
+    assert 16 * groups * level <= prime
+    # E = 2 g L + 4 g + 5 L - 2 - p^2 at K = 0.
+    exponent = 2 * groups * level + 4 * groups + 5 * level - 2 - prime * prime
+    assert exponent == -16333
+    assert exponent <= -2
+    # Three bits do not majorize 9, even though that exponent is negative.
+    four_bit = level + 4 * groups + 2 * groups * level - prime * prime
+    three_bit = level + 3 * groups + 2 * groups * level - prime * prime
+    assert four_bit < 0 and three_bit < 0
+    assert 9 * prime * prime > 8 * prime * prime
+    # At g = p^2 the monomial exceeds 1. These are the integer ingredients.
+    assert 9 * 2 * 2 > 4
+    assert 2**4 > 9
+    assert 2 > 1
+
+
+def main() -> None:
+    pins_and_phrases()
+    series_and_bits()
+    log_and_budget()
+    witness_and_failures()
+    print("PASS_GAUSSIAN_RESTRICTED_GATE")
+
+
+if __name__ == "__main__":
+    main()