Absorb the face factor 2 on the approximant monomial (no new κ)

research/gaussian-tensor-faces/README.md

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+# The face factor 2 spends one bit of slack
+
+**Claim type:** an explicit absorption bound for one constant factor.
+**No new κ.** The section ceiling `10 p^2` and the section error
+`p^6 exp(-p^2)` stay hypotheses. The selected record remains `23/10000`.
+
+## Question
+
+#4486 absorbs
+
+```text
+g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2)
+```
+
+under the gate `p ≥ max(64, 4K + 2C)` and `16 g ceil(log2 p) ≤ p`, with
+exponent at most `-C p - 4`. HALO partitions boundary outputs into `2g`
+rectangles, and a review note asks for a factor `2g` on the core error.
+Does that extra factor of 2 stay below `2^{-C p}` under the same 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 (10 p^2 + p^6 exp(-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 contributes
+one bit. The exponent is at most `-C p - 3`. Whenever `g log p = o(p)`,
+every fixed pair `K, C` satisfies the gate for all large `p`.
+
+This comparison does not identify HALO's `2g` rectangles with the tensor
+error. It charges the constant 2 against slack that is already present.
+At `g = p^2` the same 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) ≤ 2^{2 K p}`, with
+equality only at `K = 0`.
+
+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. From `8 L ≤ p`, `48 L = 6 · 8 L ≤ 6 p`.
+The quadratic `4 p^2 - 3 p - 4` equals 6 at `p = 2` and increases by `8 p + 1`
+at the next integer, so `6 p ≤ 8(p^2 - 1)`. Hence `6 L ≤ p^2 - 1`. Since
+`p ≤ 2^L`, `p^6 ≤ 2^{6 L} ≤ 2^{p^2 - 1}`, and
+
+```text
+p^6 exp(-p^2) < p^6 2^{-p^2} ≤ 1/2.
+```
+
+Thus `10 p^2 + p^6 exp(-p^2) < 10 p^2 + 1 < 16 p^2`, because `1 < 6 p^2`
+for every integer `p ≥ 2`. At `g = 1` the power `g - 1` is zero. For
+`g ≥ 2` the approximant base is strictly smaller than `16 p^2`, so
+
+```text
+2 g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2)
+  < 2 · g 2^{4g-4} p^{2g+4} 2^{2 K p - p^2}.
+```
+
+The gate forces `g ≤ p ≤ 2^L`, and `L ≥ 1`. Since `2 = 2^1`, the product is
+strictly less than
+
+```text
+2^{1 + 2 g L + 4 g + 5 L - 4 + 2 K p - p^2}.
+```
+
+Clear the eighths. The four pieces scale as `8 · 2 g L = 16 g L ≤ p`,
+`8 · 4 g = 2 · 16 g ≤ 2 p`, `8 · 5 L = 5 · 8 L ≤ 5 p`, and
+`8 (2 K + C) p ≤ 4 p^2` once `p ≥ 4 K + 2 C`, because
+`8 (2 K + C) = 4 (4 K + 2 C)`. The coefficients of `p` are `1 + 2 + 5 = 8`.
+Doubling the exponent without the new bit, plus `C p`, leaves at most
+`-(p^2 - 2 p + 8)`. That polynomial equals 8 at `p = 2` and increases by
+`2 p - 1`, so the doubled quantity is at most `-8`. The exponent without
+the face factor is at most `-C p - 4`. One bit lowers the remaining slack
+from 4 to 3, and the exponent above is at most `-C p - 3`.
+
+If `g log p = o(p)`, then `g log2 p ≤ p/32` for large `p`. For `p ≥ 3`,
+`L ≤ 2 log2 p`, so the gate holds eventually, and every fixed `K` and `C`
+satisfy `p ≥ 4 K + 2 C` as well.
+
+At `g = p^2` the product is twice a quantity larger than
+`g (10 p^2)^{g-1} p^6 exp(-p^2)`. The smaller monomial equals
+`g (10 p^2)^g exp(-p^2) · p^4 / 10`, and
+`(10 p^2)^{p^2} > 4^{p^2} > exp(p^2)` because `10 p^2 ≥ 40 > 4` and `e < 4`,
+while `p^6 / 10 > 1` because `p^6 ≥ 64 > 10`. That product is greater than 1,
+so the face-factor monomial is greater than 1 as well.
+
+## Not covered
+
+- A proof that the tensor difference is at most this monomial.
+- A proof that a section inverse is at most `10 p^2`, or that a section
+  error is less than `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-faces --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-tensor-faces
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-22 (open PRs fetched 2026-10-22T01:16:25Z, 340 records;
+discussions fetched 2026-10-22T01:17:10Z, 3365 records) names this draft, #4486
+(the approximant budget, which leaves the factor 2 open), #4484 and #4485,
+discussion comment 19078965, and the locality audit. That audit already says a
+factor 2 on `g p^{10 g} exp(-p^2)` is absorbed by `2^{-C p}`. It does not
+give this gate or the slack 3 for the approximant base. A broad search for
+`2g` also hits unrelated `D(2G)` claims. A search miss is not a novelty
+certificate. #4486 is the predecessor budget. HALO is the error sentence.
+
+## Open next questions
+
+1. **Is the tensor difference at most this monomial?** Another constant in
+   front of the same product is not the missing step. The first step is to
+   expand one axis of the tensor product and bound that slice by one section
+   error.
+
+## Credit
+
+- The tensor sentence, the `2g` rectangles, and the gate `g log p = o(p)` are #4100's.
+- The ceiling `10 p^2` is Ryanson Jonathan's #4462, as carried through #4481.
+- The section error `p^6 exp(-p^2)` is the within-cell bound of #4452.
+- The approximant budget is #4486. This package spends one bit of its slack.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-tensor-faces/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "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-faces/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 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2), times a global factor exp(K p), still falls below 2^{-C p} under the same gate as the approximant fold.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only charges the extra factor 2 against the existing exponent slack. It does not identify the tensor difference with this monomial, and it does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the face factor left open by the approximant absorption.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the factor 2 is compared with the exponent slack. Do not expand a tensor axis, 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-faces",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the absorption bound and the package text.",
+  "open_questions": [
+    {
+      "question": "Is the tensor difference at most 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2)?",
+      "why_it_matters": "The absorption bounds the monomial. It does not show that the core tensor error is that monomial.",
+      "first_step": "Expand one axis of the tensor product and bound that slice by one section error.",
+      "blocked_by": "A section inverse at most 10 p^2 and a section error below p^6 exp(-p^2) remain conditional on the ideal gaps."
+    }
+  ]
+}

research/gaussian-tensor-faces/VERIFY.json

@@ -0,0 +1,21 @@
+{
+  "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 ceil(log2 p) <= p, 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(Kp - p^2) is below 2^{-C p} with exponent slack 3; g = p^2 leaves that error above 1."
+    }
+  ]
+}

research/gaussian-tensor-faces/verify.py

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+#!/usr/bin/env python3
+"""The face factor 2 spends one bit of the telescoping slack.
+
+Every comparison is an integer identity. The section ceiling and the section
+error are hypotheses.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-tensor-faces"
+
+
+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 "These 2g sets are disjoint rectangles." 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
+    # 10 p^2 + 1 < 16 p^2. The face factor is the next power of two.
+    assert 6 * 4 > 1
+    assert 10 < 16
+    assert 2 ** 4 == 16
+    assert 2 ** 1 == 2
+    assert 4 - 1 == 3
+    assert 4 * 1 - 4 == 0
+
+
+def section_error_room() -> None:
+    # 8 L <= p implies 48 L <= 6 p, and 6 p <= 8(p^2 - 1) for every integer p >= 2.
+    assert 6 * 8 == 48
+    assert 8 * (2 ** 2 - 1) - 6 * 2 == 2 * (4 * 2 ** 2 - 3 * 2 - 4)
+    assert 4 * 2 ** 2 - 3 * 2 - 4 == 6
+    assert (4 * 3 ** 2 - 3 * 3 - 4) - (4 * 2 ** 2 - 3 * 2 - 4) == 8 * 2 + 1
+    assert ceil_log2(128) == 7
+    assert 6 * 7 <= 128 ** 2 - 1
+    assert 2 * (2 ** 3) == 2 ** 4
+
+
+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(2) == 1
+    assert ceil_log2(64) == 6
+    assert ceil_log2(65) == 7
+    assert ceil_log2(128) == 7
+    assert 8 * ceil_log2(128) <= 128
+
+
+def budget() -> None:
+    assert 8 * 2 == 16
+    assert 8 * 4 == 2 * 16
+    assert 8 * 5 == 5 * 8
+    assert 1 + 2 + 5 == 8
+    assert 4 * (4 * 3 + 2 * 5) == 8 * (2 * 3 + 5)
+    assert 2 ** 2 - 2 * 2 + 8 == 8
+    assert (3 ** 2 - 2 * 3 + 8) - (2 ** 2 - 2 * 2 + 8) == 2 * 2 - 1
+    assert 8 >= 2
+    # The doubled slack 8 is four bits; one bit remains three.
+    assert 8 // 2 - 1 == 3
+    prime = 128
+    level = ceil_log2(prime)
+    groups = 1
+    assert 16 * groups * level <= prime
+    slack = (
+        1
+        + 2 * groups * level
+        + 4 * groups
+        + 5 * level
+        - 4
+        - prime * prime
+    )
+    assert slack <= -3
+    # g = p^2 still leaves the smaller inverse-ceiling monomial above 1.
+    assert 10 * 4 > 4
+    assert 2 ** 6 > 10
+
+
+def main() -> None:
+    pins_and_phrases()
+    series_bounds()
+    section_error_room()
+    log_induction()
+    budget()
+    print("PASS_GAUSSIAN_TENSOR_FACES")
+
+
+if __name__ == "__main__":
+    main()