Keep the 10 p^2 row margin through a Schur complement (no new κ)

research/gaussian-schur-gap/README.md

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+# Schur complements keep a row margin of 1/(10 p^2)
+
+**Claim type:** one elimination lemma, plus the exponent arithmetic it feeds.
+**No new κ.** The ideal gaps stay hypotheses. The selected record remains
+`23/10000`.
+
+## Question
+
+#4462 transferred the exponents 11, 24, 26, 26 and 6 to the whole banded
+reference, with inverse bound `10 p^2` in its robust reading. Its open
+question asks whether those exponents survive the Schur complement. The
+first step named there is to restate the interior-elimination row gap with
+`10 p^2` in place of `16 p^2 - 1`.
+
+## Lemma
+
+Let `p ≥ 2` be an integer and let `δ = 1/(10 p^2)`. Suppose `H` has diagonal
+entries equal to 1, and every row and every column has absolute off-sum at
+most `1 - δ`. Let `S` be the Schur complement of any principal set whose
+pivots stay nonzero.
+
+Then every row and every column of `S` has margin at least `δ`, and absolute
+sum at most `2 - δ`. In particular
+
+```text
+||S^{-1}||_1 ≤ 10 p^2,    ||S^{-1}||_∞ ≤ 10 p^2,    ||S||_2 < 2,
+```
+
+and the same holds for `||S^{-1}||_2`. The polynomial used in #4452, with
+spectral endpoint 11, therefore satisfies
+`||P - S^{-1}||_2 < p^6 exp(-p^2)` on this complement. Its bandwidth is at
+most `(2 p^{24} - 1) W < p^{26}` for every integer `W ≤ p - 1`, and a row at
+distance `p^{26}` from a section edge agrees with the block row.
+
+So 11, 24, 26, 26 and 6 apply to every such Schur complement. Replacing
+`10 p^2` by `16 p^2 - 1` does not: at `p = 2` that ceiling squares to
+`3969 > 2048 = 2^{11}`.
+
+## Proof
+
+Write `δ` for the margin. The hypothesis gives, on the original matrix,
+margin `1 - off-sum ≥ δ` and absolute row sum `1 + off-sum ≤ 2 - δ`. The
+same holds for columns. Proceed by eliminating one index at a time. The
+active diagonal `d` and off-sum `o` of a remaining row satisfy `d - o ≥ δ`,
+and the active absolute row sum `d + o` is at most `2 - δ`. The pivot
+diagonal `a` satisfies `a ≥ δ`, so it is not zero. A coupling entry has
+absolute value `c`, and the pivot row has off-sum `o1 ≤ a - δ`.
+
+The new row is the old row minus `c/a` times the pivot row. The triangle
+inequality produces a lower bound `L` for the new margin and an upper bound
+`U` for the new absolute row sum:
+
+```text
+a L = a d - a (o - c) - c o1 = a (d - o) + c (a - o1),
+a U = a (d + o) + c (o1 - a).
+```
+
+The first right-hand side is at least `a δ`, so `L ≥ δ`. The second says
+`U = (d + o) - c (a - o1) / a ≤ d + o`. Thus the new margin is at least `δ`
+and the new absolute row sum is at most the old one, hence at most `2 - δ`.
+Columns are the rows of the transpose, so the same two identities apply to
+them. After the whole set has been eliminated, the active matrix is the
+block Schur complement `D - C A^{-1} B`.
+
+The bound `||S^{-1}||_∞ ≤ 1/δ` is the one-row estimate: if `S y = x` and
+`|y_k|` is maximal, the margin at `k` multiplies `|y_k|` by at most `||x||_∞`.
+The same estimate on `S^*` bounds the 1-norm. Therefore both inverse norms
+are at most `10 p^2`, and the 2-norm is at most `10 p^2`. Absolute row and
+column sums at most `2 - δ` give `||S||_2 < 2`.
+
+The eigenvalues of `S^* S` are then strictly larger than `p^{-11}` and
+strictly smaller than 4. Indeed `(10 p^2)^2 = 100 p^4`, and `100 < p^7` for
+every integer `p ≥ 2` because the minimum is `128` at `p = 2`. Also
+`4 < p^{11}`. The matrix `Q = I - p^{-11} S^* S` has spectrum in
+`[0, 1 - p^{-22})`. The length `p^{24}` contributes `p^2`. The length
+`p^{23}` contributes only `p`. The scalar step is strict because `1/2! < 1`.
+The normalizer 6 is legal because `10 p^2 < p^6`, while `10 p^2 > p^5` at
+`p = 2`, so 5 fails. The bandwidth identity is the same one used for the
+within-cell block:
+
+```text
+p^{26} - (2 p^{24} - 1)(p - 1) = p^{24} ((p - 1)^2 + 1) + p - 1 > 0.
+```
+
+At `p = 2` the loose radius equals `p^{25} + 1`. For every integer `p ≥ 3`
+the sharp radius at `W = p - 1` exceeds `p^{25}`.
+
+#4462's robust margin `(4p - 1)/(32 p^3)` is at least `7/(64 p^2)`, and
+`7/64 > 1/10`, so that margin is at least `δ`. Those matrices meet the
+hypothesis of this lemma. This package does not reprove the robust margin.
+
+| Role | With margin `1/(16 p^2 - 1)` | With margin `1/(10 p^2)` |
+| --- | --- | --- |
+| Inverse ceiling | `16 p^2 - 1` | `10 p^2` |
+| Spectral endpoint on the Schur complement | 12, because `63^2 > 2^{11}` | 11 |
+| Series, bandwidth, agreement, normalizer | 26, 28, 28, 6 | 24, 26, 26, 6 |
+
+## Not covered
+
+- A proof that the ideal gaps exceed `1/(8 p^2)` for every prime.
+- The core side `L = 2^{ceil(400 log2 p)}`. Agreement distance 26 does not
+  by itself replace that choice.
+- The broad matrix. HALO still applies `S` and does not invert it.
+- Any new κ. The #4100 notes keep their written gap, `p^{10}`, and `A = p^{48}`.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-schur-gap --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-schur-gap
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-22, from the snapshot fetched at 00:40 UTC, matched this
+draft, PR #4462, and the unrelated width-2 note
+`research/gaussian-origin-displayed`. #4462 asks for this elimination step
+and does not take it. No other open pull request assigns the margin
+`1/(10 p^2)` to a Schur complement. A search miss is not a novelty
+certificate. #4452 is the within-cell δ ceiling.
+
+## Open next questions
+
+1. **Can the core side exponent 400 fall?** Every Schur complement now has
+   the same inverse ceiling `10 p^2`, and the agreement distance on the
+   halo polynomial is `p^{26}`. HALO still chooses `L ≥ p^{400}` against
+   `A = p^{48}`. The first step is to substitute `A = p^{26}` into the two
+   sparse powers and name the smallest uniform integer that keeps both
+   negative whenever the group exponents are strictly less than 1.
+
+## Credit
+
+- The elimination sentence and the written core side are #4100's.
+- The inverse-ceiling `10 p^2`, and this question, are Ryanson Jonathan's #4462.
+- The exponent route is #4452. The across-cell contract it used is Mihir S's #4449.
+- The sharp bandwidth radius is Nick Petalas's halo audit, PR #4217.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-schur-gap/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/GAUSSIAN.md",
+      "usage": "input",
+      "git_blob": "adfb28c93e6b06178c901caaac39ede41a9e541b",
+      "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242"
+    },
+    {
+      "path": "research/unconditional-batched-crt/HALO.md",
+      "usage": "input",
+      "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b",
+      "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc"
+    },
+    {
+      "path": "research/unconditional-batched-crt/INVERSE-CONDITIONING.md",
+      "usage": "input",
+      "git_blob": "3853ffac91dfed5421b05533eb274a8646741071",
+      "sha256": "be3e4b729481b15a5ccc286b297908e4f9cef94c0c9144a28ee1cf58950d142a"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/FINE-PRIME.md",
+      "usage": "input",
+      "git_blob": "bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3",
+      "sha256": "47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"
+    }
+  ]
+}

research/gaussian-schur-gap/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": "Decide whether a unit-diagonal matrix with row and column margins at least 1/(10 p^2) passes that margin, and the exponents 11, 24, 26, 26 and 6, to every principal Schur complement.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only restates the interior-elimination gap at the 10 p^2 ceiling. It does not lower the core side 400 and does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. The Schur margin is the elimination input to the halo polynomial, still under the eventual ideal-gap hypothesis.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the Schur margin is compared with 10 p^2 and with 16 p^2-1. Do not retune the core side 400, 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-schur-gap",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the elimination identities and the package text.",
+  "open_questions": [
+    {
+      "question": "Can HALO's core side exponent 400 fall once every Schur complement has inverse bound 10 p^2 and the agreement distance is p^{26}?",
+      "why_it_matters": "The written core side is still p^{400} against agreement distance p^{48}. The elimination gap no longer forces the larger inverse ceiling.",
+      "first_step": "Substitute A = p^{26} into HALO's two sparse powers and name the smallest uniform integer that keeps both negative for group exponents strictly less than 1.",
+      "blocked_by": "The ideal gaps 1/(8 p^2) remain eventual statements, so the margin hypothesis is still conditional."
+    }
+  ]
+}

research/gaussian-schur-gap/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/HALO.md",
+    "research/unconditional-batched-crt/INVERSE-CONDITIONING.md",
+    "research/unconditional-batched-crt/vendor/FINE-PRIME.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "One Schur step preserves a margin of 1/(10 p^2), so 11, 24, 26, 26 and 6 apply to the complement; 10, 23, 25 and 5 fail, and 16 p^2-1 does not give endpoint 11 at p=2."
+    }
+  ]
+}