Absorb the factor (9 p^2)^g by the same gate (no new κ)

research/gaussian-tensor-nine/README.md

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+# The factor (9 p^2)^g is absorbed by the same four-bit gate
+
+**Claim type:** an explicit absorption bound for one tensor error.
+**No new κ.** The per-axis ceiling `9 p^2` stays a hypothesis. The selected
+record remains `23/10000`.
+
+## Question
+
+#4493 shows that the within-cell formula is strictly below `9 p^2`, and asks
+whether the #4484 gate still absorbs
+
+```text
+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 to test
+whether three bits fail because `9 > 8`.
+
+## 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
+g (9 p^2)^g exp(K p - p^2) < 2^{-C p}.
+```
+
+The case `K = 0` is the monomial in the question. A global factor `exp(K p)`
+does not change the conclusion. Whenever `g log p = o(p)`, every fixed pair
+`K, C` satisfies the gate for all large `p`.
+
+Four bits are enough because `9 < 16`. Three bits are not a majorant of this
+shape: `9 > 8`, so
+
+```text
+(9 p^2)^g > 2^{3 g} p^{2 g}
+```
+
+for every integer `p ≥ 2` and `g ≥ 1`. A negative exponent written with `3`
+in place of `4` does not repair that comparison.
+
+The gate is not optional for every `g`. At `g = p^2` the same substituted
+error 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 values at 2, 3 and 4 are `2`, `6` and `24`,
+and the step uses `k + 1 ≥ 2`. The tail of `e` is then at most the geometric
+series of ratio `1/2`, so `e < 3 < 4` and `exp(K p) < 4^{K p} = 2^{2 K p}`
+whenever `K p > 0`. At `K = 0` the factor is 1.
+
+Since `9 < 16`, `(9 p^2)^g < 2^{4 g} p^{2 g}`. Also `p ≤ 2^L`, so
+`p^{2 g} ≤ 2^{2 g L}`. The hypothesis `16 g L ≤ p` forces `g < 2^L`, because
+`L ≥ 1` and therefore `g ≤ p/(16 L) ≤ p/16`. Hence
+
+```text
+g (9 p^2)^g exp(K p - p^2) < 2^{L + 4 g + 2 g L + 2 K p - p^2}.
+```
+
+It remains to show that the exponent is strictly less than `-C p`. For every
+integer `p ≥ 64`, `8 L ≤ p`. The base is `8 · 7 = 56 ≤ 64` at `p = 2^6`, and
+`8 ≤ 64`. If `8(m + 1) ≤ 2^m`, then `8(m + 2) ≤ 2^m + 8 ≤ 2^{m + 1}`. On each
+dyadic interval the largest ratio of `L` to `p` is at the left endpoint, so
+`8 L ≤ p` for every integer `p ≥ 64`.
+
+The gate `16 g L ≤ p` gives `2 g L ≤ p/8` and, with `L ≥ 1`, `4 g ≤ p/4`.
+The linear factor obeys `(2 K + C) p ≤ p^2 / 2` exactly when
+`p ≥ 4 K + 2 C`, because `2(2 K + C) = 4 K + 2 C`. Adding the four pieces
+produces at most
+
+```text
+p/8 + p/4 + p/8 + p^2/2 = p/2 + p^2/2.
+```
+
+The slack against `p^2` is `p(p - 1)/2`. At `p = 2` this is 1, and it
+increases by `p` at the next integer. The exponent is therefore at most
+`-C p - 1`.
+
+The three-bit comparison reverses. Since `9 > 8 = 2^3`,
+
+```text
+(9 p^2)^g > (8 p^2)^g = 2^{3 g} p^{2 g}.
+```
+
+At the legal point `p = 128`, `L = 7`, `g = 1`, both
+`L + 4 g + 2 g L - p^2` and `L + 3 g + 2 g L - p^2` are negative, while
+`9 p^2 > 8 p^2`. Negativity of the three-bit exponent is not an upper bound
+for the monomial.
+
+If `g log p = o(p)`, then `g log2 p ≤ p/32` for large `p`. For `p ≥ 3`,
+`L ≤ 2 log2 p`, so `p/(32 log2 p) ≤ p/(16 L)` and the gate `16 g L ≤ p`
+holds eventually. Every fixed `K` and `C` then satisfy `p ≥ 4 K + 2 C` as
+well.
+
+At `g = p^2`, `(9 p^2)^{p^2} > 4^{p^2} > exp(p^2)` because `9 p^2 ≥ 36 > 4`
+and `e < 4`. The substituted error is then greater than 1, which is not below
+`2^{-C p}`.
+
+## Not covered
+
+- A proof that each section inverse is at most `9 p^2`. This comparison
+  starts from that ceiling.
+- The telescoping monomial `g (9 p^2)^{g-1} p^6 exp(-p^2)`.
+- 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-tensor-nine --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-tensor-nine
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`. Upstream was fetched at 2026-10-22T01:47:06Z and was
+still that commit.
+
+Overlap snapshot: open PRs fetched 2026-10-22T01:48:03Z (348 records) and
+discussions fetched 2026-10-22T01:48:45Z (3371 records). A search for
+`(9 p^2)^g`, `three bits` and `L + 4 g` returned this draft and the #4493
+handoff, which leaves the absorption open, plus unrelated three-bit tape and
+encoding notes. A search miss is not a novelty certificate. #4484 is the
+four-bit gate. HALO is the written error.
+
+## Open next questions
+
+1. **Does the same gate absorb `g (9 p^2)^{g-1} p^6 exp(K p - p^2)`?** The
+   extra factor against this package is `p^4 / 9`, because
+   `g (9 p^2)^{g-1} p^6 = g (9 p^2)^g (p^4 / 9)`. The first step is to place
+   that factor in the exponent as `4 L` and clear it against the same eighths.
+
+## Credit
+
+- The four-bit gate and the exponent `L + 4 g + 2 g L + 2 K p - p^2` are #4484's.
+- The coefficient 9, and this question, are #4493'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-tensor-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-tensor-nine/TASK.json

@@ -0,0 +1,74 @@
+{
+  "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": "Decide whether the #4484 gate absorbs g (9 p^2)^g exp(K p - p^2), and whether three bits fail because 9 > 8.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only substitutes the coefficient 9 into that gate. It does not prove a 9 p^2 operator bound and does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the absorption left open by the coefficient 9.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the factor (9 p^2)^g is compared with the four-bit gate. Do not absorb the telescoping p^6 factor, 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-tensor-nine",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the four-bit comparison and the package text.",
+  "open_questions": [
+    {
+      "question": "Does the same gate absorb g (9 p^2)^{g-1} p^6 exp(K p - p^2)?",
+      "why_it_matters": "The tensor telescope still carries one section error p^6 exp(-p^2) against g-1 factors of the coefficient 9.",
+      "first_step": "Place the extra factor p^4 / 9 in the exponent as 4 L and clear it against the same eighths.",
+      "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-tensor-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": "Under p >= max(64, 4K+2C) and 16 g L <= p, g (9 p^2)^g exp(Kp - p^2) < 2^{-C p} because 9 < 16 leaves the exponent L + 4 g + 2 g L + 2 K p - p^2 at most -C p - 1; three bits fail because 9 > 8."
+    }
+  ]
+}

research/gaussian-tensor-nine/verify.py

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+#!/usr/bin/env python3
+"""The factor (9 p^2)^g is absorbed by the same four-bit gate.
+
+Every comparison is an integer identity. The per-axis ceiling is a hypothesis.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-tensor-nine"
+
+
+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")
+    selected = json.loads(
+        (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8")
+    )
+    assert "g p^(10g) exp(-p^2)," in halo
+    assert "exp(O(p)) leaves this below 2^-Cp for every fixed C eventually, provided" in halo
+    assert "g log p=o(p)." in halo
+    assert selected["kappa"] == "23/10000"
+
+
+def series_bounds() -> None:
+    # e > 1 + 1 + 1/2 = 5/2 > 2, and e < 3 < 4.
+    assert 5 > 4
+    assert 3 < 4
+    assert 2 >= 2
+    assert 6 >= 4
+    assert 24 >= 8
+    assert 3 >= 2
+    # Four bits clear 9, and three bits do not: 8 < 9 < 16.
+    assert 8 < 9
+    assert 9 < 16
+    assert 2**3 == 8
+    assert 2**4 == 16
+
+
+def log_induction() -> None:
+    # For m >= 6, 8(m+1) <= 2^m, so 8 ceil(log2 p) <= p for every integer p >= 64.
+    assert 8 * 7 <= 64
+    assert 8 <= 64
+    assert 2**6 + 8 <= 2**7
+    assert ceil_log2(64) == 6
+    assert ceil_log2(65) == 7
+    assert ceil_log2(128) == 7
+    assert 8 * ceil_log2(128) <= 128
+
+
+def budget() -> None:
+    # The eighths 1+2+1 make a half, and p^2 - p is at least 2 for every integer p >= 2.
+    assert 1 + 2 + 1 == 4
+    assert 4 - 2 == 2
+    assert (3**2 - 3) - (2**2 - 2) == 2 * 2
+    # p >= 4K + 2C is the cleared form of (2K+C) p <= p^2 / 2.
+    assert 2 * (2 * 3 + 5) == 4 * 3 + 2 * 5
+    # Witness: p=128, L=7, g=1, K=0, C=0 meets the gate and the four-bit exponent is negative.
+    prime = 128
+    level = ceil_log2(prime)
+    groups = 1
+    assert 16 * groups * level <= prime
+    four = level + 4 * groups + 2 * groups * level - prime * prime
+    three = level + 3 * groups + 2 * groups * level - prime * prime
+    assert four < 0
+    assert three < 0
+    # The negative three-bit exponent is not an upper bound: 9 p^2 > 8 p^2.
+    assert 9 * prime * prime > 8 * prime * prime
+    assert 9 * 2**2 > 2**3 * 2**2
+    # g = p^2 leaves the substituted error above 1, since 9 p^2 > 4 > e.
+    # The value 9 p^2 is smallest at p = 2 and increases by 9(2p+1).
+    assert 9 * 4 > 4
+    assert 9 * (3**2) - 9 * (2**2) == 9 * (2 * 2 + 1)
+
+
+def main() -> None:
+    pins_and_phrases()
+    series_bounds()
+    log_induction()
+    budget()
+    print("PASS_GAUSSIAN_TENSOR_NINE")
+
+
+if __name__ == "__main__":
+    main()