Show the smallest Neumann coefficient is 9 (no new κ)

research/gaussian-neumann-nine/README.md

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+# The smallest integer coefficient on the Neumann ceiling is 9
+
+**Claim type:** one integer coefficient for the within-cell ceiling.
+**No new κ.** The ideal-gap reading stays a hypothesis. The selected record
+remains `23/10000`.
+
+## Question
+
+#4492 shows `U < 10 p^2` and leaves the smallest integer open. For
+
+```text
+U = 8 p^2 (D - 4) / (D - 32 p^2),    D = 2^{32760 p},
+```
+
+what is the smallest integer `c` such that `U < c p^2` for every integer
+`p ≥ 2`?
+
+## Lemma
+
+For every integer `p ≥ 2`, `D > 32 p^2` and
+
+```text
+8 p^2 < U < 9 p^2.
+```
+
+The smallest such integer is therefore `9`. The same `9 p^2` is strictly
+sharper than both upper bounds recorded with this formula: `9 p^2 < 16 p^2 - 1`
+and `9 p^2 < p^6`. This comparison is a property of the formula. It becomes
+an inverse bound only when the #4452 gauge holds, in which case the Neumann
+1-norm and ∞-norm are already strictly below `U`, hence strictly below
+`9 p^2`.
+
+## Proof
+
+The denominator `D - 32 p^2` is positive once `D > 32 p^2`, which follows
+from the stronger estimate below. Cross-multiplying by that positive
+denominator, `U < 9 p^2` is equivalent to
+
+```text
+8(D - 4) < 9(D - 32 p^2).
+```
+
+The difference is
+
+```text
+9(D - 32 p^2) - 8(D - 4) = D - 288 p^2 + 32.
+```
+
+It is enough to show `D > 288 p^2 - 32`. Since `288 < 512 = 2^9` and
+`32 < 288`, the same estimate also gives `D > 32 p^2`.
+
+For every integer `p ≥ 2`, `2^{2p} ≥ p^2 + 1`, so `p^2 < 2^{2p}`. The base
+is `2^4 = 16 ≥ 4 + 1`. If the inequality holds at `p`, then
+
+```text
+2^{2p+2} = 4 · 2^{2p} ≥ 4(p^2 + 1),
+```
+
+and `4(p^2 + 1) - ((p + 1)^2 + 1) = 3p^2 - 2p + 2`. At `p = 2` this surplus
+is 10, and the step to the next integer is `6p + 1 > 0`. Thus
+`288 p^2 < 2^9 p^2 < 2^{2p+9}`.
+
+The remaining exponent gap is `32760 p - (2p + 9) = 32758 p - 9`. At
+`p = 2` it equals 65507, and it increases by 32758 when `p` increases by 1.
+Therefore `2^{2p+9} < 2^{32760 p} = D`, and `D > 288 p^2 > 288 p^2 - 32`.
+
+At `p = 2` the same polynomial is `288 · 4 - 32 = 1120 < 2^{13}`, while
+`32760 · 2 > 13`. The coefficient 8 reverses the target inequality, because
+`U > 8 p^2` exactly when `32 p^2 > 4`. No integer lies strictly between 8
+and 9, and every integer `c ≤ 8` fails because `U > 8 p^2 ≥ c p^2`.
+
+The two older ceilings lose to 9 by the same integer comparisons. From
+`7 p^2 > 1` one has `9 p^2 < 16 p^2 - 1`. From `p^4 ≥ 16 > 9` one has
+`9 p^2 < p^6`.
+
+## Not covered
+
+- A proof of the ideal gaps that make `U` an inverse bound.
+- Replacing `10` by `9` in the tensor absorption or the Schur margin.
+- 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-neumann-nine --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-neumann-nine
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`. Upstream was fetched at 2026-10-22T01:42:53Z and was
+still that commit.
+
+Overlap snapshot: open PRs fetched 2026-10-22T01:43:37Z (346 records) and
+discussions fetched 2026-10-22T01:44:21Z (3370 records). A search for `U < 9`,
+`9 p^2`, `288 p^2`, `smallest integer c` and `D - 288` returned this draft,
+#4492, and its handoff, which leave the coefficient open, plus an unrelated
+cyclic-Fourier log ceiling. A search miss is not a novelty certificate.
+#4452 is the definition of `U`. HALO is the written error.
+
+## Open next questions
+
+1. **Does the #4484 gate absorb `g (9 p^2)^g exp(K p - p^2)`?** That gate
+   spends four bits because `10 < 16`. The first step is to put `9` in place
+   of `10` in the exponent `L + 4 g + 2 g L + 2 K p - p^2` and test whether
+   three bits fail because `9 > 8`.
+
+## Credit
+
+- The ceiling `U` and the bounds `8 p^2 < U < p^6` and `U < 16 p^2 - 1` are #4452's.
+- The coefficient 10, and this question, are #4492's.
+- The tensor use of `10 p^2` is #4484. The Schur margin `1/(10 p^2)` is #4481.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-neumann-nine/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "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/unconditional-batched-crt/HALO.md",
+      "usage": "input",
+      "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b",
+      "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc"
+    }
+  ]
+}

research/gaussian-neumann-nine/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "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": "Identify the smallest integer c such that the within-cell Neumann ceiling U = 8 p^2 (D - 4) / (D - 32 p^2), with D = 2^{32760 p}, satisfies U < c p^2 for every integer p >= 2.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only identifies that integer coefficient. It does not prove the ideal gaps and does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the coefficient left open by the comparison with 10 p^2.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the smallest integer coefficient is identified. Do not retune the tensor absorption, do not prove the ideal gaps, 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-neumann-nine",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the coefficient comparison and the package text.",
+  "open_questions": [
+    {
+      "question": "Does the #4484 gate absorb g (9 p^2)^g exp(K p - p^2)?",
+      "why_it_matters": "The tensor error still spends four bits on the coefficient 10, while the within-cell formula is now strictly below 9 p^2.",
+      "first_step": "Put 9 in place of 10 in the exponent L + 4 g + 2 g L + 2 K p - p^2, and test whether three bits fail because 9 > 8.",
+      "blocked_by": "The reading of a 9 p^2 operator ceiling remains conditional on the ideal gaps. The absorption itself still treats that ceiling as a hypothesis."
+    }
+  ]
+}

research/gaussian-neumann-nine/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/unconditional-batched-crt/HALO.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "U < 9 p^2 is equivalent to D > 288 p^2 - 32, which holds because 288 p^2 < 2^{2p+9} < 2^{32760 p}; the coefficient 8 fails because 32 p^2 > 4, so 9 is the smallest integer coefficient."
+    }
+  ]
+}

research/gaussian-neumann-nine/verify.py

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+#!/usr/bin/env python3
+"""The smallest integer coefficient on the Neumann ceiling is 9.
+
+Every comparison is an integer identity. The ideal-gap reading stays a hypothesis.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-neumann-nine"
+
+
+def digest(path: Path) -> str:
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+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")
+    selected = json.loads(
+        (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8")
+    )
+    assert "g p^(10g) exp(-p^2)," in halo
+    assert selected["kappa"] == "23/10000"
+
+
+def cleared() -> None:
+    # 9(D - 32 s) - 8(D - 4) = D - 288 s + 32, with s = p^2.
+    degree = 7
+    square = 3
+    gap = 9 * (degree - 32 * square) - 8 * (degree - 4)
+    assert gap == degree - 288 * square + 32
+    # 8 p^2 is not an upper bound, and the next integer is 9.
+    assert 32 * 4 > 4
+    assert 8 + 1 == 9
+    assert 32 < 288
+    assert 288 < 2**9
+    assert 2**9 == 512
+    # 9 p^2 is sharper than both ceilings recorded with U.
+    assert 7 * 4 > 1
+    assert 9 * 4 < 16 * 4 - 1
+    assert 16 > 9
+    assert 9 * 4 < 2**6
+
+
+def powers() -> None:
+    # 2^{2p} >= p^2 + 1, so p^2 < 2^{2p}. The step surplus is 3p^2 - 2p + 2.
+    assert 2**4 >= 2**2 + 1
+    assert 4 * (2**2 + 1) - ((2 + 1) ** 2 + 1) == 3 * 2**2 - 2 * 2 + 2
+    assert (3 * 3**2 - 2 * 3 + 2) - (3 * 2**2 - 2 * 2 + 2) == 6 * 2 + 1
+    # 288 p^2 < 2^{2p+9} < 2^{32760 p}, since the exponent gap is 32758 p - 9.
+    assert 2**9 * 2 ** (2 * 2) == 2 ** (2 * 2 + 9)
+    assert 32760 * 2 > 2 * 2 + 9
+    assert 32758 * 2 - 9 == 65507
+    assert 65507 > 0
+    assert (32758 * 3 - 9) - (32758 * 2 - 9) == 32758
+    # At p = 2 the polynomial 288 p^2 - 32 is below 2^{13}.
+    assert 288 * 2**2 - 32 == 1120
+    assert 2**13 > 1120
+    assert 32760 * 2 > 13
+
+
+def main() -> None:
+    pins_and_phrases()
+    cleared()
+    powers()
+    print("PASS_GAUSSIAN_NEUMANN_NINE")
+
+
+if __name__ == "__main__":
+    main()