Check jwr404's p = 19 section chain in one package: #4354, #4355, #4361, #4362, #4364 and all 28 two-power PRs hold; the sharp threshold is 2^-519 (no new κ)

research/gaussian-section-chain-check/README.md

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+# One check of jwr404's p = 19 section chain, with the sharp two-power threshold
+
+**Claim type: an independent certified finite check of one section. No new
+κ.** It covers 33 open PRs in one package, and adds the exact threshold that
+their two-power steps approach.
+
+## Object
+
+The section is the one from #4319 and my
+[#4336](https://github.com/CrocSwap/integer-mult-bounds/pull/4336): `p = 19`,
+`β₀ = −1/2`, bandwidth 2, `t_i = i`. The interiors `3..6` are eliminated, and
+the survivors `0, 1, 2, 7, 8, 9` are row-normalized to `H`.
+
+jwr404's chain then applies `HALO.md`'s polynomial inverse
+`P = p⁻²⁰ Σ_{j<J} Q^j H*`, with `Q = I − p⁻²⁰H*H` and `J = ⌈p⁴²⌉`, and
+bounds its error.
+
+| PR | Claim |
+| --- | --- |
+| #4354 | `281/100 < ‖H⁻¹‖₁ < 282/100` |
+| #4355 | `‖H⁻¹‖₂ < 319/100` |
+| #4361 | every absolute row and column sum of `H` is below `1226/625`, so `‖H‖₂ < 1226/625 < 2 < 19¹⁰` |
+| #4362 | every eigenvalue of `Q` lies in `(0, 1 − p⁻⁴⁰)` |
+| #4364 | `(1 − 19⁻⁴⁰)^(19⁴²) < e⁻³⁶¹`, so the error is below `(319/100)e⁻³⁶¹` |
+| #4366–#4369, #4374, #4379–#4400 (27 PRs) | `(319/100)e⁻³⁶¹ < 2^(−19C)` for `C = 1..27` |
+| #4401 | `(319/100)e⁻³⁶¹` is not below `2⁻⁵³²` |
+
+## Results
+
+Every claim holds, certified with exact rationals.
+
+| Quantity | Value |
+| --- | --- |
+| `‖H⁻¹‖₁` | `2.8138216884…`, inside `(2.81, 2.82)` |
+| `‖H⁻¹‖₂ ≤ √(‖·‖₁‖·‖_∞)` | `3.183009 < 3.19` |
+| Largest row or column sum of `H` | `1.9615957574 < 1226/625 = 1.9616` |
+| `‖H‖₂` | `≤ 1.961596` |
+| `p⁻²⁰(1226/625)² < 1` and `p⁻²⁰(100/319)² > p⁻⁴⁰` | exact, so the spectrum of `Q` lies in `(0, 1 − p⁻⁴⁰)` |
+| `19⁻⁴⁰ · 19⁴² = 361` | exact, so `(1 − x)^N < e^(−xN) = e⁻³⁶¹` |
+| The 27 two-power claims and #4401's negative | all hold, from `e³⁶¹` enclosed to 60 digits |
+
+**The sharp threshold.** `(319/100)e⁻³⁶¹ < 2^(−k)` holds exactly for
+`k ≤ 519`, and fails at `k = 520`. The chain's step of 19 stops at 513 and
+then fails at 532; the true boundary lies between them.
+
+**Two tight spots.**
+
+- The row and column sums of `H` clear `1226/625` by only `4.2·10⁻⁶`. The
+  largest sum is `1 + e^(−9π/722)`, from the forward step at vertex 0.
+- The round ceilings `2.82 · 3.61 = 10.1802` exceed `3.19² = 10.1761`.
+  #4355's 2-norm bound therefore needs the certified norms, not the rounded
+  ones, as its own README notes.
+
+## Method and independence
+
+Nothing is imported from jwr404's packages, and the method differs from
+theirs:
+
+- The entries come from `GAUGEFREE-INTERIORS.md`'s original form.
+- `H⁻¹` comes from the quotient identity: the survivor block of `A⁻¹` times
+  `diag(S)`.
+- `H` is the exact inverse of that, with one perturbation bound.
+
+The notes this reads are on main, pinned by SHA256 with phrase checks.
+
+## Not covered
+
+- Any other section, prime or bandwidth.
+- `HALO.md`'s all-size claims.
+- Whether `#4100`'s callers need more than the `2^(−C)` thresholds proved
+  here.
+
+## Reproduce
+
+```sh
+python3.12 -B research/gaussian-section-chain-check/verify.py
+```
+
+It prints `PASS_GAUSSIAN_SECTION_CHAIN_CHECK` in about 10 seconds, using only
+the standard library.
+
+## Open next questions
+
+None from this check.
+
+## Credit
+
+The chain, its section and its thresholds are jwr's (#4354, #4355,
+#4361–#4401). The kernel and lemmas are `#4100`'s `GAUSSIAN.md`, `HALO.md`,
+`INVERSE-CONDITIONING.md` and the vendored `GAUGEFREE-INTERIORS.md`.
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5)
+assistance. Apache-2.0.

research/gaussian-section-chain-check/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "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/GAUGEFREE-INTERIORS.md",
+      "usage": "input",
+      "git_blob": "20585e44fa4e49b13b5d8495bbf6ed254459a362",
+      "sha256": "d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"
+    }
+  ]
+}

research/gaussian-section-chain-check/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "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": "Independently certify every numerical claim of jwr404's p = 19 normalized-section chain (#4354, #4355, #4361, #4362, #4364, #4366-#4401) from exact section entries, and find the exact two-power threshold.",
+  "scope_change_reason": "The chain splits one section's conditioning and polynomial-inverse error into 33 open PRs with no outside check; one certified recomputation covers them together.",
+  "bottleneck_link": "HALO.md's polynomial inverse needs these conditioning and error bounds at #4100's precision caller.",
+  "budget_minutes": 30,
+  "stop_rule": "Stop once every claim is certified or fails.",
+  "outcome": "completed",
+  "evidence": "verify.py: ||H^-1||_1 = 2.8138 in (2.81, 2.82); ||H^-1||_2 <= 3.183 < 3.19; H row and column sums <= 1.9615957574 < 1226/625; Q spectrum in (0, 1-p^-40) and x N = 361 exact; e^361 enclosed to 60 digits proves all 27 claims (319/100) e^-361 < 2^-19C and #4401's negative; sharp threshold 519.",
+  "author": "Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance",
+  "ai_assistance": "Substantial Claude Code (Claude Opus 5.5) assistance in reading, verifier implementation and writing; no independent human or proof-kernel review.",
+  "open_questions": [],
+  "no_open_questions_reason": "Every chain claim holds and the sharp threshold is found."
+}

research/gaussian-section-chain-check/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/HALO.md",
+    "research/unconditional-batched-crt/INVERSE-CONDITIONING.md",
+    "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Pinned notes and phrases; alpha = 3; exact original-form entries; quotient-identity H^-1 and exact H with perturbation bounds; #4354 1-norm interval; #4355 2-norm bound; #4361 row and column sums; #4362 spectrum; #4364 exponent identity; 27 two-power claims and #4401's negative from a 60-digit e^361 enclosure; sharp threshold 519."
+    }
+  ]
+}