Bound one tensor axis by two section errors (no new κ)

research/gaussian-tensor-axis/README.md

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+# One tensor axis is at most two section errors
+
+**Claim type:** a telescoping identity with an explicit one-axis norm bound.
+**No new κ.** The per-axis norm bounds stay hypotheses. The selected record
+remains `23/10000`.
+
+## Question
+
+#4487 bounds the scalar monomial
+
+```text
+2 g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2)
+```
+
+and asks whether the tensor difference is at most that monomial. The first
+step is one axis of the product.
+
+## Lemma
+
+Let `g ≥ 1`, and let `B_i` and `C_i` be bounded operators on Hilbert spaces.
+Write `ε = p^6 exp(-p^2)` for an integer `p ≥ 2`. Suppose that for every axis
+
+```text
+||B_i|| ≤ 10 p^2,
+||C_i|| ≤ 10 p^2 + ε,
+||B_i - C_i|| < 2 ε,
+```
+
+and that `||X ⊗ Y|| = ||X|| ||Y||`. Then
+
+```text
+⊗_i B_i - ⊗_i C_i = sum_{k=1}^g T_k,
+```
+
+where
+
+```text
+T_k = (⊗_{i<k} B_i) ⊗ (B_k - C_k) ⊗ (⊗_{i>k} C_i),
+```
+
+an empty product being the identity. Every slice satisfies
+
+```text
+||T_k|| < 2 (10 p^2 + ε)^{g-1} ε,
+```
+
+and the triangle inequality gives
+
+```text
+||⊗_i B_i - ⊗_i C_i|| < 2 g (10 p^2 + ε)^{g-1} ε.
+```
+
+The factor 2 is the sum of two section errors on the active axis. HALO's
+`2g` rectangles are a volume partition, not this factor. The scalar model
+checked below is the one-dimensional case of the same product. The gate that
+puts the right-hand side below `2^{-C p}` is #4487; it is not re-proved here.
+
+## Proof
+
+For `g = 1` the identity is `B_1 - C_1 = T_1`. If it holds for `g - 1` factors,
+then
+
+```text
+(⊗_{i<g} B_i) ⊗ B_g - (⊗_{i<g} C_i) ⊗ C_g
+  = (⊗_{i<g} B_i) ⊗ (B_g - C_g)
+    + ((⊗_{i<g} B_i) - (⊗_{i<g} C_i)) ⊗ C_g.
+```
+
+The second sum expands by the inductive hypothesis, and its slices are the
+slices `T_k` for `k < g`. The first sum is `T_g`.
+
+The norm hypothesis multiplies across a pure tensor, so
+
+```text
+||T_k|| = (∏_{i<k} ||B_i||) · ||B_k - C_k|| · (∏_{i>k} ||C_i||)
+  < (10 p^2)^{k-1} (10 p^2 + ε)^{g-k} · 2 ε.
+```
+
+For integers `u ≥ v ≥ 0` and `1 ≤ k ≤ g`, the exponents satisfy
+`(k - 1) + (g - k) = g - 1`, and `v ≤ u` gives
+`v^{k-1} u^{g-k} ≤ u^{g-1}`. Apply this with `v = 10 p^2` and
+`u = 10 p^2 + ε`. Each slice is strictly smaller than `2 u^{g-1} ε`. Adding
+the `g` slices preserves the strict inequality.
+
+The cap `||B_i - C_i|| < 2 ε` is the sum of two strict section errors. At
+`g = 1` it is necessary: a difference of size `3` already exceeds the cap `2`
+in the scalar model. The executable check uses the integer caps `10`, `11`
+and `2` in place of `10 p^2`, `10 p^2 + ε` and `2 ε`.
+
+## Not covered
+
+- A proof that any axis inverse, approximant, or section error meets the
+  three norm hypotheses.
+- The comparison of the monomial with `2^{-C p}`.
+- 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-axis --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-tensor-axis
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-22 (open PRs fetched 2026-10-22T01:21:17Z, 341 records;
+discussions fetched 2026-10-22T01:22:02Z, 3366 records) names this draft, #4487
+(the scalar monomial, which leaves the identification open), and the locality
+audit. That audit already gives the telescoping shape with a factor 2 in
+front of `p^{10}`. It does not use the ceiling `10 p^2` or the section error
+`p^6 exp(-p^2)`. A search for "one axis" also hits unrelated BIT and
+expansion notes. A search miss is not a novelty certificate. #4487 is the
+scalar monomial. HALO is the tensor sentence.
+
+## Open next questions
+
+1. **Does one core axis meet the three norm hypotheses?** The telescope uses
+   `||B_i|| ≤ 10 p^2`, `||C_i|| ≤ 10 p^2 + ε`, and `||B_i - C_i|| < 2 ε`.
+   The first step is to split `B_i - C_i` through the section polynomial and
+   apply one section error on each side.
+
+## Credit
+
+- The tensor sentence is #4100's.
+- The shape of the telescoping sum, with a factor 2 in front of `p^{10}`, is
+  the locality-witness audit.
+- 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 scalar monomial is #4487.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-tensor-axis/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-axis/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": "Expand one axis of the tensor product and bound that slice, then the sum of the slices, by 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2).",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only records the telescoping identity under three axis norm hypotheses. It does not prove those hypotheses and does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the tensor identification left open by the face-factor absorption.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once one slice and the sum of the slices are bounded by the monomial. Do not prove the axis norms, do not re-prove the 2^{-C p} gate, 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-axis",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the telescoping identity and the package text.",
+  "open_questions": [
+    {
+      "question": "Does one core axis satisfy ||B_i|| <= 10 p^2, ||C_i|| <= 10 p^2 + p^6 exp(-p^2), and ||B_i - C_i|| < 2 p^6 exp(-p^2)?",
+      "why_it_matters": "The telescope bounds the tensor difference only after those three inequalities.",
+      "first_step": "Split B_i - C_i through the section polynomial and apply one section error on each side.",
+      "blocked_by": "The section inverse bound and the section error remain conditional on the ideal gaps."
+    }
+  ]
+}

research/gaussian-tensor-axis/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": "The scalar telescope equals the sum of its axis slices; each slice is at most u^{g-1} times the cap 2 when the factors obey v <= u; a g = 1 difference of 3 exceeds that cap."
+    }
+  ]
+}

research/gaussian-tensor-axis/verify.py

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+#!/usr/bin/env python3
+"""One tensor axis is at most two section errors times the other factors.
+
+Every comparison is an integer identity. The axis norm bounds are hypotheses.
+The scalar model is the one-dimensional case of the same product.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-tensor-axis"
+
+
+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 "tensor product of the section inverses" in halo
+    assert "g p^(10g) exp(-p^2)," in halo
+    assert "These 2g sets are disjoint rectangles." in halo
+    assert selected["kappa"] == "23/10000"
+
+
+def telescope() -> None:
+    # g = 3 scalars. Caps: |b| <= 10, |c| <= 11, |b - c| <= 2.
+    left = 8 * 9 * 6 - 7 * 11 * 5
+    axis_1 = (8 - 7) * 11 * 5
+    axis_2 = 8 * (9 - 11) * 5
+    axis_3 = 8 * 9 * (6 - 5)
+    assert left == axis_1 + axis_2 + axis_3
+    # g = 1 is the single axis.
+    assert 5 - 2 == 5 - 2
+    # The factor 2 is the sum of two section errors.
+    assert 1 + 1 == 2
+
+
+def one_axis_cap() -> None:
+    # v <= u gives v^{k-1} u^{g-k} <= u^{g-1}, because the exponents add to g-1.
+    assert 10 <= 11
+    assert (2 - 1) + (3 - 2) == 3 - 1
+    assert 10 ** 1 * 11 ** 1 <= 11 ** 2
+    assert 11 ** 2 <= 11 ** 2
+    assert 10 ** 2 <= 11 ** 2
+    # Each displayed slice sits under u^{g-1} times the axis cap 2.
+    assert abs((8 - 7) * 11 * 5) <= 11 ** 2 * 2
+    assert abs(8 * (9 - 11) * 5) <= 10 ** 1 * 11 ** 1 * 2
+    assert abs(8 * 9 * (6 - 5)) <= 10 ** 2 * 2
+    assert abs(8 * (9 - 11) * 5) <= 11 ** 2 * 2
+    # Without the cap |b - c| <= 2, the g = 1 slice can exceed it.
+    assert abs(5 - 2) > 2
+
+
+def main() -> None:
+    pins_and_phrases()
+    telescope()
+    one_axis_cap()
+    print("PASS_GAUSSIAN_TENSOR_AXIS")
+
+
+if __name__ == "__main__":
+    main()