Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ)

research/gaussian-delta-exponents/README.md

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+# The published weight error lowers four halo exponents inside one cell
+
+**Claim type:** conditional exponent arithmetic for one within-cell block.
+**No new κ.** The ideal gaps stay hypotheses. The selected record remains
+`23/10000`.
+
+## Question
+
+#4451 showed that the published absolute weight error is a relative error
+below `δ = 2^{2-32760p}`, which is stronger than `1/(16p^2)`. Do the
+exponents 12, 26, 28, 28 and 6 still apply inside one cell, and is
+`16p^2-1` still the inverse ceiling?
+
+## Lemma
+
+Let `p ≥ 2` be an integer and let `W` be an integer at most `p-1`. Assume
+each ideal row off-sum and each ideal column off-sum is at most
+`1 - 1/(8p^2)`. Inside the cell, assume `0 ≤ Ghat ≤ G` and
+`a - 4η < ahat ≤ a` whenever `a > 2^{-8p}`, with `P = 32768p` and
+`η = 2^{-P}`.
+
+Let `D = 2^{32760 p}` and
+
+```text
+U = 8 p^2 (D - 4) / (D - 32 p^2).
+```
+
+Then both rounded off-sums are strictly below `(1 - 1/(8p^2)) / (1 - δ)`,
+every absolute row sum and column sum is strictly below 2, and the Neumann
+bounds on `||A^{-1}||_1` and `||A^{-1}||_∞` are strictly less than `U`. In
+particular `8 p^2 < U < p^6` and `U < 16 p^2 - 1`. At `p = 2`, `U > 32 = p^5`.
+
+For this block the polynomial
+
+```text
+P = p^{-11} sum_{j < p^{24}} Q^j A^*,    Q = I - p^{-11} A^* A,
+```
+
+satisfies `||P - A^{-1}||_2 < p^6 exp(-p^2)`. Its bandwidth is at most
+`(2 p^{24} - 1) W < p^{26}`, and the looser count `(2 p^{24} + 1) W` has
+the same ceiling. A row at distance `p^{26}` from a section edge agrees
+with the block row.
+
+So 12, 26, 28, 28 and 6 still apply. The normalizer 6 remains the smallest
+uniform integer with that role. The other four are no longer smallest:
+11, 24, 26 and 26 replace them. The value `16p^2-1` still upper-bounds the
+inverse, and it is strictly larger than `U`.
+
+## Proof
+
+The denominator repair gives `a / ahat < 1/(1-δ)` with
+`δ = 4/D = 2^{2-32760p}`, because `4η < 2^{-8p}` is the exponent identity
+`32768 - 8 = 32760`. A downward kernel does not increase the entry. An
+ideal off-sum at most `1 - 1/(8p^2)` therefore rounds to an off-sum
+strictly below
+
+```text
+(1 - 1/(8p^2)) / (1 - δ) = (D / (D - 4)) (1 - 1/(8p^2)).
+```
+
+The complementary gap is strictly larger than `(1/(8p^2) - δ) / (1 - δ)`,
+so the Neumann bound is strictly less than `U`. Since `D > 32 p^2`, the
+rounded off-sum is also strictly below 1, and both matrix norms that enter
+the 2-norm comparison are strictly below 2.
+
+For every integer `p ≥ 2`, `2^{2p} ≥ p^2 + 1`. The base is `16 ≥ 5`. The
+step is `4(p^2 + 1) - (p + 1)^2 = 3p^2 - 2p + 3`, and after `p = 2 + t`
+every coefficient of that quadratic is positive. Also `2p + 6 ≤ 32760 p`:
+at `p = 2` the numbers are `10 ≤ 65520`, and the slack increases by
+`32758`. Therefore `D ≥ 2^{2p+6} = 64 · 2^{2p} ≥ 64(p^2 + 1)`.
+
+Clearing the positive denominator shows `U > 8 p^2` exactly when
+`32 p^2 > 4`. At `p = 2` that lower bound is `32 = p^5`, so `U > p^5` and
+`U^2 > (8p^2)^2 = p^{10}`. The same clearing gives `U < p^6` once
+`32(p^6 - 1) < D(p^4 - 8)`. At `p = 2` this is `2016 < 8 D` with
+`D ≥ 2^{16}`. For `p ≥ 3`, the comparison
+`64(p^2 + 1)(p^4 - 8) - 32(p^6 - 1)` equals `32(p^6 + 2p^4 - 16p^2 - 15)`.
+After `p = 3 + t` every coefficient is positive, and `p^4 ≥ 81 > 16`.
+
+The looser multiple `V = 8 p^2 D / (D - 32 p^2)` satisfies `U < V`. Then
+`V^2 < p^{11}` once `64 D^2 < p^7 (D - 32 p^2)^2`. At `p = 2` this is
+`D^2 < 2(D - 128)^2`, and `D^2 - 512 D + 32768` is positive for
+`D ≥ 2^{16}` because `2^{32} > 2^{25}`. For `p ≥ 3`, `32 p^2/D < 1/2`, so
+the squared factor exceeds `1/4` and `p^7/4 ≥ 2187/4 > 64`. Thus
+`U^2 < p^{11}`. The eigenvalues of `A^* A` are strictly larger than
+`p^{-11}` and strictly smaller than `4`. Since `4 < p^{11}`, they are
+strictly smaller than `p^{11}`. The matrix `Q = I - p^{-11} A^* A` 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 bandwidth identity is
+
+```text
+p^{26} - (2 p^{24} - 1)(p - 1) = p^{24} ((p - 1)^2 + 1) + p - 1 > 0.
+```
+
+At `p = 2` the loose radius `(2 p^{24} + 1) W` equals `p^{25} + 1`. For
+every integer `p ≥ 3`,
+`(2 p^{24} - 1)(p - 1) - p^{25} = p^{24}(p - 2) - (p - 1)`, and
+`p^2 - (p - 1) = p^2 - p + 1 > 0`, so the sharp radius exceeds `p^{25}`.
+Exponent 25 is not a uniform ceiling. Exponent 26 is, for the bandwidth
+and for the agreement distance.
+
+Finally `V < 16 p^2 - 1`. The cleared difference
+`2(8p^2 - 1)(p^2 + 1) - p^2(16p^2 - 1)` equals `15p^2 - 2`, which is
+positive for every integer `p ≥ 2`. Combined with `D ≥ 64(p^2 + 1)`, this
+is `(8p^2 - 1) D > 32 p^2 (16p^2 - 1)`. At `p = 2` the same comparison is
+`31 D > 8064`. The older ceiling remains legal and is strictly coarser
+than `U`.
+
+| Role | Under `1/(16p^2)` | Under `δ` | Smallest |
+| --- | ---: | ---: | --- |
+| Spectral endpoint | 12 | 11 | 10 fails at `p=2` |
+| Series length | 26 | 24 | 23 fails at `p=2` |
+| Bandwidth | 28 | 26 | 25 fails |
+| Agreement distance | 28 | 26 | 25 fails |
+| Normalizer | 6 | 6 | 5 fails at `p=2` |
+
+## Not covered
+
+- Across-cell entries. #4449 gives `0 ≤ H ≤ n` for those entries. This
+  package does not move the global reference.
+- A proof that the ideal gaps exceed `1/(8p^2)` for every prime.
+- The core exponent 400. It still dominates an agreement distance of 26.
+- Any new κ. The #4100 notes keep 20, 42, 45, 48 and 100.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-delta-exponents --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-delta-exponents
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-21, using the snapshot fetched at 21:26 UTC and a
+later look at the newest handoff comments and open pull requests, matched
+this draft, PR #4451, and an unrelated digit string in
+`research/capped-fusion-t74-k554`. It did not find another assignment of
+11, 24, 26, 26 and 6. #4451 is the relative-error comparison. #4439 is the
+coarser budget `1/(16p^2)`. A search miss is not a novelty certificate.
+
+## Open next questions
+
+1. **Do 11, 24, 26, 26 and 6 survive across a cell?** #4449 puts each
+   across-cell entry in `[0, n]`, with absolute error below `4η`. The
+   within-cell factor is now `1/(1-δ)`. The first step is the same column
+   split as #4442, with `δ` in place of `1/(16p^2)`.
+
+## Credit
+
+- HALO's locality witness and its written exponents are #4100's.
+- The sharp radius `(2J-1)W` is Nick Petalas's halo audit, PR #4217.
+- The in-cell ideal column gap is Ryanson Jonathan's #4402.
+- The relative budget `1/(16p^2)` is #4438. The coarser exponents are #4439.
+- The published error `δ = 2^{2-32760p}` is #4451.
+- The across-cell repair `0 ≤ H ≤ n` is Mihir S's #4449.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-delta-exponents/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": "references/semantic-bulk/rad20/reports/downstream-phase-cell-inverse.md",
+      "usage": "input",
+      "git_blob": "d72345a0aa748443c4efcddf87aa10d524664841",
+      "sha256": "3cdf492bcb719db4295658604dc202d6a2c5a008b87de43ec54f0107c64db730"
+    },
+    {
+      "path": "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md",
+      "usage": "input",
+      "git_blob": "3bcf490a8c164d44aff5b7e2ec2e0f4bf07d7a73",
+      "sha256": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272"
+    },
+    {
+      "path": "research/gaussian-column-gap/README.md",
+      "usage": "input",
+      "git_blob": "7f8b5d9cbad96a1cd46432a6cf47422e2ced7fb4",
+      "sha256": "7e589bec5c03d788ac366c85080af9fe26dad254d289bec93553398380b46dab"
+    },
+    {
+      "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"
+    }
+  ]
+}

research/gaussian-delta-exponents/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 the within-cell halo exponents fall from 12, 26, 28, 28 and 6 to 11, 24, 26, 26 and 6 once the relative budget is the published \u03b4 = 2^{2-32760p}.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only substitutes the published weight error into the within-cell ceiling. Across-cell transfer stays open, and this package does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. A smaller within-cell inverse ceiling is the input to the halo polynomial, still under the eventual ideal-gap hypothesis.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once 11, 24, 26, 26 and 6 are compared with the \u03b4 ceiling. Do not transfer them across a cell, 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-delta-exponents",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the exponent comparison and the package text.",
+  "open_questions": [
+    {
+      "question": "Do 11, 24, 26, 26 and 6 survive across a cell for every integer p \u2265 2, using the published absolute error below 4\u03b7 together with 0 \u2264 H \u2264 n?",
+      "why_it_matters": "The within-cell ceiling is now U, which is stricter than 16p^2-1. The across-cell block still uses the coarser budget.",
+      "first_step": "Repeat the column split from #4442 with \u03b4 in place of 1/(16p^2).",
+      "blocked_by": "The ideal gaps 1/(8p^2) remain eventual statements, so both ceilings still depend on them."
+    }
+  ]
+}

research/gaussian-delta-exponents/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "references/semantic-bulk/rad20/reports/downstream-phase-cell-inverse.md",
+    "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md",
+    "research/gaussian-column-gap/README.md",
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/HALO.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Under the ideal-gap hypothesis the Neumann ceiling is strictly below p^6 and its square is strictly below p^11, so 11, 24, 26, 26 and 6 apply; 10, 23, 25 and 5 fail."
+    }
+  ]
+}