The published 4η weight error meets the 1/(16p^2) ratio budget (no new κ)

research/gaussian-ratio-budget/README.md

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+# The published weight error meets the 1/(16p^2) ratio budget
+
+**Claim type:** a uniform relative-error comparison inside one cell.
+**No new κ.** Across-cell rounding is outside this claim. The selected
+record remains `23/10000`.
+
+## Question
+
+Downstream rounds each weight by a downward exponential routine with
+absolute error below `4η`, where `P = 32768 p` and `η = 2^{-P}`, and it
+records `2^{-8p} < a_j ≤ 1`. The within-cell entry is the ratio
+`ahat_j Ghat_(j-l) / ahat_l`. Does that absolute error keep the ratio
+inside the relative budget `ε ≤ 1/(16p^2)` for every integer `p ≥ 2`?
+
+## Lemma
+
+Let `p ≥ 2` be an integer. Suppose a true weight satisfies
+`a > 2^{-8p}`, its repaired value satisfies `a - 4η < ahat ≤ a`, and a
+repaired kernel coefficient satisfies `0 ≤ Ghat ≤ G`. Then each
+within-cell off-diagonal entry obeys
+
+```text
+ahat_j Ghat / ahat_l < (a_j G / a_l) / (1 - δ),
+```
+
+where `δ = 2^{2 - 32760 p}`. This `δ` is strictly less than `1/(16p^2)`.
+The factor `1/(1-δ)` is therefore strictly smaller than
+`1/(1 - 1/(16p^2))`.
+
+## Proof
+
+The downward kernel bound and the downward numerator give
+
+```text
+ahat_j Ghat / ahat_l ≤ (a_j / ahat_l) G.
+```
+
+The denominator satisfies `ahat_l > a_l - 4η`. The room `4η < 2^{-8p}` is
+the comparison `32760 p > 2`, since `32768 - 8 = 32760` and `4 = 2^2`.
+Thus `ahat_l > 0` and
+
+```text
+a_l / ahat_l < 1 / (1 - 4η/a_l) < 1 / (1 - 4η/2^{-8p}) = 1/(1-δ).
+```
+
+The exponent of `δ` is `2 + 8p - 32768p = 2 - 32760p`.
+
+It remains to prove `16 p^2 δ < 1`. First, `p^2 < 2^{2p}` for every
+integer `p ≥ 2`. The base is `4 < 16`. If it holds at `p`, the next square
+obeys `4(p+1)^2 ≤ 9 p^2`: the difference `4(p+1)^2 - 9p^2 = -5p^2 + 8p + 4`
+is 0 at `p = 2`, and raising `p` by one changes it by `-10p + 3`, which is
+already `-17` at `p = 2` and becomes more negative. Since `9 < 16`,
+
+```text
+(p+1)^2 ≤ (9/4) p^2 < 4 p^2 < 4 · 2^{2p} = 2^{2(p+1)}.
+```
+
+Second, `2p < 32760 p - 6` for every integer `p ≥ 2`. At `p = 2` the
+slack is positive, and it increases by `32758` at each later integer.
+Therefore `2^{2p} < 2^{32760 p - 6}`, and `p^2 < 2^{32760 p - 6}`.
+Multiplying by the remaining power of two,
+
+```text
+16 p^2 δ < 16 · 2^{2-6} = 2^4 · 2^{-4} = 1.
+```
+
+The inequality is strict because both comparisons above are strict. At
+`p = 2` the same exponent is concrete: `4 < 2^{65514}`.
+
+So the relative weight error `ρ = 4η/a` satisfies `ρ < δ < 1/(16p^2)`,
+which is the budget used for the within-cell factor `1/(1-ε)`. The gauge
+audit's allowance `ε ≤ 1/2` is a larger number. At `p = 2`,
+`1/2 > 1/64 = 1/(16·4)`.
+
+## Not covered
+
+- Across-cell entries. That rounding question is claimed separately.
+- A proof that the ideal row and column gaps exceed `1/(8p^2)` for every
+  prime. The ratio bound above takes the true entry `a_j G / a_l` as given.
+- The halo exponents. Meeting `ε ≤ 1/(16p^2)` is the budget comparison,
+  not a new assignment of 12, 26, 28 or 6.
+- Any new κ, and any edit to the #4100 notes.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-ratio-budget --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-ratio-budget
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-21: discussion 4444 claims the across-cell two-ulp
+repair left open by #4442. This package is the within-cell ratio under the
+published absolute error. A search hit is not a novelty certificate.
+
+## Open next questions
+
+1. **Do 12, 26, 28, 28 and 6 now apply inside one cell for every integer
+   `p ≥ 2`, with this `δ` in place of a separate `ε` hypothesis?** The
+   ratio factor is strictly stronger than `1/(1 - 1/(16p^2))`. The ideal
+   gaps `1/(8p^2)` are still the eventual statements in GAUSSIAN and
+   #4402. The first step is to substitute `δ = 2^{2-32760p}` into the
+   within-cell ceiling arithmetic.
+
+## Credit
+
+- The absolute error `4η`, the weight floor `2^{-8p}`, and the ratio entry
+  are the phase-cell downstream note's.
+- The budget `1/(16p^2)` is #4438. The within-cell exponents are #4439.
+- The allowance `ε ≤ 1/2` is the capped-gauge audit's.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-ratio-budget/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": "references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md",
+      "usage": "input",
+      "git_blob": "395eab8ee1d58e84353f554dcfe9473b5913dda5",
+      "sha256": "113927dd4fb77a55a457bd5f6da8cbfc7262d0fbc4e9d708b591a1d632e6caf6"
+    },
+    {
+      "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"
+    }
+  ]
+}

research/gaussian-ratio-budget/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 published absolute weight error below 4\u03b7 keeps a within-cell ratio inside \u03b5 \u2264 1/(16p^2) for every integer p \u2265 2.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only compares that published absolute error with the within-cell relative budget. The across-cell repair is claimed elsewhere, and this package does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. The ratio factor is the within-cell input to the 16p^2 ceiling.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once \u03b4 = 2^{2-32760p} is compared with 1/(16p^2). Do not redo the claimed across-cell repair, 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-ratio-budget",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the exponent comparison and the package text.",
+  "open_questions": [
+    {
+      "question": "Do 12, 26, 28, 28 and 6 apply inside one cell for every integer p \u2265 2 once \u03b5 is replaced by \u03b4 = 2^{2-32760p}?",
+      "why_it_matters": "The published weight error is stronger than 1/(16p^2). The within-cell exponent proof still treats that larger budget as a hypothesis.",
+      "first_step": "Substitute \u03b4 into the within-cell ceiling and check whether 16p^2-1 is still the inverse bound.",
+      "blocked_by": "The ideal gaps 1/(8p^2) remain eventual statements, so the ceiling still depends on them."
+    }
+  ]
+}

research/gaussian-ratio-budget/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",
+    "references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md",
+    "research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md",
+    "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "4\u03b7 is below the weight floor and 16 p^2 \u03b4 < 1 for every integer p \u2265 2; \u03b5 = 1/2 is larger than 1/(16p^2) at p = 2."
+    }
+  ]
+}

research/gaussian-ratio-budget/verify.py

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+#!/usr/bin/env python3
+"""Published 4η weight error against the 1/(16p^2) ratio budget.
+
+Every comparison is an integer identity. Across-cell entries are not bounded.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = ROOT / "research" / "gaussian-ratio-budget"
+
+
+def digest(path: Path) -> str:
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def add(left, right):
+    out = dict(left)
+    for key, coeff in right.items():
+        out[key] = out.get(key, 0) + coeff
+        if out[key] == 0:
+            del out[key]
+    return out
+
+
+def scale(poly, factor):
+    return {key: factor * coeff for key, coeff in poly.items() if factor * coeff}
+
+
+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"]
+    downstream = (
+        ROOT / "references/semantic-bulk/rad20/reports/downstream-phase-cell-inverse.md"
+    ).read_text(encoding="utf-8")
+    review = (
+        ROOT / "references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md"
+    ).read_text(encoding="utf-8")
+    gauge = (
+        ROOT / "research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md"
+    ).read_text(encoding="utf-8")
+    log_a = (
+        ROOT / "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md"
+    ).read_text(encoding="utf-8")
+    selected = json.loads(
+        (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8")
+    )
+    assert "Set P=32768p and eta=2^(-P)." in downstream
+    assert "2^(-8p)<a_j<=1." in downstream
+    assert "absolute error below four eta" in downstream
+    assert (
+        "Round the Toeplitz coefficient at each distance 1<=delta<=w downward,"
+        in downstream
+    )
+    assert "a_hat_j*G_hat_(j-l)/a_hat_l" in downstream
+    assert "slightly above\nthe true entries." in review
+    assert "(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2." in gauge
+    assert "ahat_j Ghat_(j-k)/ahat_k" in log_a
+    assert selected["kappa"] == "23/10000"
+
+
+def weight_room() -> None:
+    # 4η < 2^{-8p} iff 32760 p > 2. The gap grows with p.
+    assert 32768 - 8 == 32760
+    assert 32760 * 2 > 2
+    assert 32760 > 0
+
+
+def relative_budget() -> None:
+    # 4(p+1)^2 - 9p^2 = -5p^2 + 8p + 4 is 0 at p=2 and decreases by 10p+3.
+    assert -5 * 4 + 8 * 2 + 4 == 0
+    assert -10 * 2 + 3 == -17
+    assert -17 < 0
+    # 9/4 < 4, so (p+1)^2 < 4 p^2 once 4(p+1)^2 <= 9 p^2.
+    assert 9 < 16
+    # Base: 2^2 < 2^{4}. The step gives p^2 < 2^{2p} for every integer p>=2.
+    assert 4 < 16
+    # 2p < 32760 p - 6, and the slack 32758 p - 6 increases by 32758.
+    assert 4 < 32760 * 2 - 6
+    assert 32760 - 2 == 32758
+    assert 32758 > 0
+    # Therefore p^2 < 2^{32760 p - 6}. With δ = 2^{2-32760 p} and 16 = 2^4,
+    # 16 δ p^2 < 16 * 2^{2-6} = 1.
+    assert 2 ** 4 * 2 ** (2 - 6) == 1
+    # At p=2 the exponent room is concrete.
+    assert 4 < 2 ** (32760 * 2 - 6)
+    # The gauge allowance 1/2 is larger than 1/(16p^2) at p=2: 1/2 > 1/64.
+    assert 2 * 64 == 128
+    assert 128 > 2
+
+
+def main() -> None:
+    pins_and_phrases()
+    weight_room()
+    relative_budget()
+    print("PASS_GAUSSIAN_RATIO_BUDGET")
+
+
+if __name__ == "__main__":
+    main()