Attribute the three axis hypotheses on one core row (no new κ)

research/gaussian-axis-inheritance/README.md

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+# One core row inherits two of the three axis hypotheses
+
+**Claim type:** an attribution of three hypotheses to earlier packages.
+**No new κ.** The ideal-gap gauge stays a hypothesis. The selected record
+remains `23/10000`.
+
+## Question
+
+#4509 asks which of
+
+```text
+||B|| ≤ 9 p^2,
+||C|| ≤ 9 p^2 + ε,
+||B - C|| < 2 ε,
+```
+
+with `ε = p^6 exp(-p^2)`, a core row past `ceil(p^{48})` inherits from
+#4491 and from #4493.
+
+## Lemma
+
+Let `p ≥ 2` be an integer and let `A = ceil(p^{48})`. For an output kept at
+distance `A` from every face, the inheritance is:
+
+1. **Row identity, with no gauge.** #4491 proves `R < p^{26} < p^{48} = A`.
+   The section-polynomial row is the global polynomial row.
+
+2. **One section error, under the #4452 gauge.** #4491 then gives
+   `||P - H_sec^{-1}||_2 < ε`. That is the input `||B - P|| < ε`, with
+   `B = H_sec^{-1}`.
+
+3. **Inverse ceiling, under the same gauge.** #4493 proves `U < 9 p^2` as a
+   formula for every integer `p ≥ 2`. Under the gauge, the Neumann 1-norm
+   and ∞-norm are strictly below `U`, hence strictly below `9 p^2`. That
+   strict bound meets `||B|| ≤ 9 p^2`.
+
+4. **Approximant norm, under the same gauge.** #4509 turns the ceiling and
+   the one section error into `||P|| < 9 p^2 + ε`.
+
+5. **Two-inverse difference.** One core row supplies one section error. The
+   bound `||B - C|| < 2 ε` requires a second section inverse within `ε` of
+   the same polynomial.
+
+The ideal off-sums `1 - 1/(8 p^2)` remain a hypothesis. This package does
+not re-prove the radius, the coefficient 9, or the triangle split.
+
+## Proof
+
+The distance comparison is the exponent gap already recorded by #4491:
+`p^{48} = p^{26} p^{22}` and `p^{22} ≥ 2^{22} > 1`, so `p^{48} > p^{26}`.
+HALO's sentence that a row at distance `A` has its polynomial row exactly
+equal to the global row therefore applies. That sentence does not use the
+ideal gaps.
+
+The operator sentence in #4491 is conditional. It states
+`||P - H_sec^{-1}||_2 < p^6 exp(-p^2)` when the section meets the #4452
+gauge. That is one summand of the triangle
+
+```text
+B - C = (B - P) + (P - C).
+```
+
+The integer shadow `|(8 - 5)| = |(8 - 6)| + |(6 - 5)|` separates the pieces.
+The single piece `|(8 - 6)|` is strictly smaller than the whole difference,
+so one section error does not supply `||B - C|| < 2 ε`.
+
+#4493's formula bound is strict: `U < 9 p^2`. A number strictly below
+`9 p^2` satisfies the closed hypothesis `||B|| ≤ 9 p^2`. The same README
+states that this operator reading holds when the #4452 gauge holds, because
+the Neumann 1-norm and ∞-norm are already strictly below `U`. The formula
+comparison itself does not use the gauge.
+
+#4509 then adds that one error to the ceiling: `||P|| < 9 p^2 + ε`. The
+polynomial, not a second inverse, is the approximant in that statement.
+
+## Not covered
+
+- A proof of the ideal gaps.
+- A second section inverse within `ε` of the same polynomial.
+- 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-axis-inheritance --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-axis-inheritance
+```
+
+Base `f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc`, priority revision 7, selection
+refresh κ = `23/10000`. Upstream was fetched at 2026-10-22T02:57:30Z and was
+that commit.
+
+Overlap snapshot: open PRs fetched 2026-10-22T02:58:14Z (296 records) and
+discussions fetched 2026-10-22T02:58:59Z (3384 records). A search for
+`second section inverse`, `one core row inherits` and `H_sec` returned this
+draft, #4491, the ceiling-10 bridge, and the locality audit. That audit adds
+two `p^{10}` errors. It does not attribute the ceiling `9 p^2`. A search
+miss is not a novelty certificate. #4491 is the row match. #4493 is the
+coefficient 9. HALO is the row sentence.
+
+## Open next questions
+
+1. **Does a second core row supply the missing section error?** The
+   two-inverse difference needs `||C - P|| < ε` for another section inverse
+   against the same global polynomial. The first step is to apply the #4491
+   reading to that second row and add the two errors.
+
+## Credit
+
+- The row identity and the one section error are #4491's.
+- The formula `U < 9 p^2`, and its gauge-conditional operator reading, are
+  #4493's.
+- The triangle that turns one error and the ceiling into the approximant
+  norm is #4509's.
+- The three hypotheses are #4508'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-axis-inheritance/SOURCES.json

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+{
+  "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-core-row/README.md",
+      "usage": "input",
+      "git_blob": "700035aa60ff1345dbb4dc7dd1c5a26b9eecc673",
+      "sha256": "e5f958c71589dfca1b287246f9af129fdf4da4f20c840e2ac20186ab95282b07"
+    },
+    {
+      "path": "research/gaussian-neumann-nine/README.md",
+      "usage": "input",
+      "git_blob": "54a8ce117a233e54e5057fd1edfa378e59527592",
+      "sha256": "189a0fb9ede1649ffea24ea22605a573366170e172ac563dd001c68d65b966ac"
+    },
+    {
+      "path": "research/unconditional-batched-crt/HALO.md",
+      "usage": "input",
+      "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b",
+      "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc"
+    }
+  ]
+}

research/gaussian-axis-inheritance/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": "Name which of the three axis hypotheses a core row past ceil(p^48) inherits from the polynomial-row match and from U < 9 p^2, keeping the ideal-gap gauge explicit.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only attributes those hypotheses. It does not prove the ideal gaps and does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the inheritance left open by the ceiling-9 polynomial split.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the three hypotheses are attributed. Do not prove a second section error, 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-axis-inheritance",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the attribution and the package text.",
+  "open_questions": [
+    {
+      "question": "Does a second core row supply the section error that the two-inverse difference still needs?",
+      "why_it_matters": "One row supplies ||B - P|| < epsilon under the gauge. ||B - C|| < 2 epsilon needs ||C - P|| < epsilon as well.",
+      "first_step": "Apply the #4491 reading to a second section inverse against the same global polynomial, then add the two errors.",
+      "blocked_by": "Both operator readings remain conditional on the ideal-gap gauge."
+    }
+  ]
+}

research/gaussian-axis-inheritance/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/gaussian-core-row/README.md",
+    "research/gaussian-neumann-nine/README.md",
+    "research/unconditional-batched-crt/HALO.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "A core row at distance p^48 clears p^26. Under the #4452 gauge, #4491 supplies one section error and #4493 supplies ||B|| < 9 p^2. One piece is strictly smaller than the two-piece difference, so ||B - C|| < 2 epsilon needs a second section error."
+    }
+  ]
+}

research/gaussian-axis-inheritance/verify.py

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+#!/usr/bin/env python3
+"""Name which axis hypotheses one core row inherits.
+
+Every comparison is an integer identity. The ideal-gap gauge 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-axis-inheritance"
+
+
+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")
+    core = (ROOT / "research/gaussian-core-row/README.md").read_text(encoding="utf-8")
+    nine = (ROOT / "research/gaussian-neumann-nine/README.md").read_text(encoding="utf-8")
+    selected = json.loads(
+        (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8")
+    )
+    assert "A=ceil(p^48)" in halo
+    assert "EXACTLY the corresponding row of the global P_i" in halo
+    assert "Keep outputs at distance A" in halo
+    assert "Its section-polynomial row is the global polynomial row." in core
+    assert "||P - H_sec^{-1}||_2 < p^6 exp(-p^2)," in core
+    assert "8 p^2 < U < 9 p^2." in nine
+    assert "an inverse bound only when the #4452 gauge holds" in nine
+    assert selected["kappa"] == "23/10000"
+
+
+def inheritance() -> None:
+    # p^48 = p^26 * p^22, and p^22 > 1, so the core distance clears the radius.
+    assert 48 - 26 == 22
+    assert 2**22 > 1
+    # A strict ceiling meets the closed hypothesis ||B|| <= 9 p^2.
+    assert 8 < 9
+    assert 8 <= 9
+    # One section error is one piece. The two-inverse difference is the sum of two pieces.
+    assert abs(8 - 5) == abs(8 - 6) + abs(6 - 5)
+    assert abs(8 - 5) > abs(8 - 6)
+    assert 1 + 1 == 2
+
+
+def main() -> None:
+    pins_and_phrases()
+    inheritance()
+    print("PASS_GAUSSIAN_AXIS_INHERITANCE")
+
+
+if __name__ == "__main__":
+    main()