Apply the smaller halo exponents inside one cell, with normalizer 6 (no new κ)

research/gaussian-cell-exponents/README.md

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+# The smaller halo exponents hold inside one cell
+
+**Claim type:** conditional exponent arithmetic for one within-cell block.
+**No new κ.** The global rounded matrix, whose across-cell entries are not
+this gauge ratio, is outside the claim.
+
+## Question
+
+#4406 showed that HALO's exponents can fall to 12, 26, 28, 28 and 7 when both
+2-norms are at most `16 p^2`. #4438 gave a within-cell column budget
+`ε ≤ 1/(16p^2)`. Do those five exponents apply to the rounded block
+
+```text
+A = diag(ahat) Ghat diag(ahat)^{-1}
+```
+
+once that budget holds?
+
+## Lemma
+
+Let `p ≥ 2` be an integer and let `W` be an integer at most `p-1`. Assume the
+ideal row and column off-sums are at most `1 - 1/(8p^2)`, the conclusions
+GAUSSIAN and #4402 state eventually, and assume the within-cell rounding of
+#4438: `Ghat_0 = 1`, `0 ≤ Ghat ≤ exp(-a ρ h^2)`, and
+`(1-ε) a ≤ ahat ≤ a` with `ε ≤ 1/(16p^2)`.
+
+Then both rounded off-sums are at most `(16p^2-2)/(16p^2-1)`, so both gaps
+are at least `1/(16p^2-1)`. The Neumann series gives
+
+```text
+||A^{-1}||_∞ ≤ 16 p^2 - 1,    ||A^{-1}||_1 ≤ 16 p^2 - 1,
+```
+
+and therefore `||A^{-1}||_2 ≤ 16 p^2 - 1`. Each absolute row and column sum is
+strictly less than 2, so `||A||_2 < 2`.
+
+For this block the polynomial
+
+```text
+P = p^{-12} sum_{j < p^{26}} Q^j A^*,    Q = I - p^{-12} A^* A,
+```
+
+satisfies `||P - A^{-1}||_2 < p^6 exp(-p^2)`. Its bandwidth is at most
+`(2 p^{26} - 1) W < p^{28}`, and the looser count `(2 p^{26} + 1) W` has the
+same ceiling. A row at distance `p^{28}` from a section edge of this block
+agrees with the block row. Dividing by `p^6` puts the inverse ceiling strictly
+below one.
+
+So 12, 26, 28, 28 and 7 all apply. The first four remain the smallest uniform
+integers with that property. The normalizer 7 is not smallest: the ceiling
+`16 p^2 - 1` is already strictly below `p^6`, while it is above `p^5` at
+`p = 2`. The smallest normalizer is 6.
+
+## Proof
+
+The gauge ratio multiplies every off-diagonal entry by at most `1/(1-ε)`, on
+a row and on a column. With `ε = 1/(16p^2)` and an ideal off-sum at most
+`1 - 1/(8p^2)`,
+
+```text
+off ≤ (1 - 1/(8p^2)) / (1 - 1/(16p^2)) = (16p^2 - 2)/(16p^2 - 1).
+```
+
+The complementary gap is `1/(16p^2-1)`. Adding the diagonal produces the row
+sum `(32p^2-3)/(16p^2-1)`, and `2(16p^2-1) - (32p^2-3) = 1`, so the sum is
+strictly below 2. The same bound is the column sum. The 2-norm comparison
+`||·||_2 ≤ sqrt(||·||_1 ||·||_∞)` is the one already used for these matrices.
+It gives `||A||_2 < 2` and `||A^{-1}||_2 ≤ 16 p^2 - 1`.
+
+The inverse ceiling is 63 at `p = 2`. Then `63^2 = 3969 ≤ 4096 = 2^{12}` and
+`3969 > 2048 = 2^{11}`. The forward 2-norm is below 2, so the upper end
+`p^{12}` is loose, but `p^{-12}` is still a legal lower bound on the
+eigenvalues of `A^* A` and `p^{-11}` is not. Thus `Q` has spectrum in
+`[0, 1-p^{-24}]`. The scalar step in the exponential series is strict because
+`1/2! < 1`, and `p^{26} p^{-24} = p^2`. The shorter length `p^{25}` leaves a
+factor `p^{11}` against `(16p^2-1)^2`, and `2048 < 3969` at `p = 2`.
+
+The error is therefore strictly less than `(16p^2-1) exp(-p^2)`. Since
+`p^4 - 16 = (p-2)(p+2)(p^2+4)`, one has `16p^2 - 1 < p^6`, with room 1 at
+`p = 2`. The multiplier `p^5` fails because `63 > 32`.
+
+The bandwidth identity is unchanged:
+
+```text
+p^{28} - (2 p^{26} - 1)(p-1) = p^{26} ((p-1)^2 + 1) + p - 1 > 0.
+```
+
+At `p = 2` the loose radius `(2J+1)W` equals `p^{27}+1`. For every integer
+`p ≥ 3` the sharp radius at `W = p-1` exceeds `p^{27}`. Exponent 27 is not a
+uniform ceiling. Exponent 28 is.
+
+Dividing by `p^6` is strict, as above. Dividing by `p^7` remains strict and
+is the value #4406 recorded for the closed ceiling `16 p^2`.
+
+## Not covered
+
+- Across-cell entries. They are the downward original entries, not
+  `ahat_j Ghat_(j-k)/ahat_k`, so this does not move HALO's exponents for the
+  global reference.
+- A proof that the ideal gaps exceed `1/(8p^2)` for every prime. That remains
+  the eventual statement in GAUSSIAN and #4402.
+- The core exponent 400. It still dominates a halo of 28, and it is not retuned.
+- Any new κ. The selected record remains `23/10000`. The #4100 notes keep
+  20, 42, 45, 48 and 100.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-cell-exponents --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-cell-exponents
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-21 did not find a within-cell assignment of these
+exponents. #4406 is the conditional arithmetic for a closed ceiling `16 p^2`.
+#4438 is the relative-error budget. A search miss is not a novelty certificate.
+
+## Open next questions
+
+1. **Do 12, 26, 28, 28 and 6 survive across a cell?**
+   The within-cell block now meets the ceiling those exponents use, and the
+   normalizer may be 6 rather than 7. The global reference is still blocked:
+   its across-cell entries are the downward approximation of the original
+   entry, and no column bound for that rule is written. The first step is a
+   column off-sum for those entries under the same `ε`.
+
+## 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-error budget is #4438.
+- The closed-ceiling arithmetic is #4406.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-cell-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": "research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md",
+      "usage": "input",
+      "git_blob": "a2d1c5ae923db7649627dbf419434e36267e57c6",
+      "sha256": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f"
+    },
+    {
+      "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-cell-exponents/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 the exponents 12, 26, 28, 28 and 7 apply to the rounded within-cell block once the relative weight error is at most 1/(16p^2).",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only treats the within-cell gauge block. It does not move the exponents of the global rounded reference.",
+  "bottleneck_link": "The selected frontier is unchanged. These exponents matter for HALO only where both 2-norms of the matrix being inverted are at most 16 p^2.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once each exponent has a yes or no for this block, including whether 7 is smallest. Do not bound across-cell entries, 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-cell-exponents",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the ceiling arithmetic, the minimality checks and the package text.",
+  "open_questions": [
+    {
+      "question": "Do the exponents 12, 26, 28, 28 and 6 survive across a cell?",
+      "why_it_matters": "The within-cell block meets the ceiling, with normalizer 6. Callers invert the global reference, which also has across-cell entries.",
+      "first_step": "Bound the column off-sum of the downward original entries used across a cell, under the same relative-error budget.",
+      "blocked_by": "Those entries are not the ratio ahat_j Ghat_(j-k)/ahat_k, so the within-cell transfer does not apply."
+    }
+  ]
+}

research/gaussian-cell-exponents/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.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": "Both gaps give the ceiling 16p^2-1; 12, 26 and 28 stay smallest; the normalizer falls to 6."
+    }
+  ]
+}