Independently check #4470's arithmetic (κ = 0.00302015): invoice, local change, root bracket, tails (no new κ)

research/pr4470-arithmetic-check/README.md

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+# Independent arithmetic check of #4470 (conditional κ = 0.00302015)
+
+**Claim type:** an independent arithmetic check with certified interval logarithms. **No new κ.** It
+covers 1337isnot1337's [#4470](https://github.com/CrocSwap/integer-mult-bounds/pull/4470) (head
+`b9a13c16`), which proposes conditional κ = 0.00302015 through joint helper boundaries. It follows my #4455,
+which checks the p17 chain #4440–#4453, and reuses that package's interval helpers, copied here so this
+package is self-contained.
+
+## What is checked
+
+1. **The p16 invoice.** 64,620 paid jobs and 10,928 productive fields. Rank masses 239,886 (target) and
+   239,134 (paid), surplus 752.
+2. **The local change against #4447.** The B/480 paid profile shared by #4447 and #4453 (per #4453's
+   table) changes from {1:2254, 2:514, 3:444, 4:103, 5:40, 8:8} to {1:2237, 2:388, 3:494, 4:147, 5:36, 8:8}.
+   That is **53 fewer paid jobs** per local call. The helper profile changes from
+   {4:26, 5:2, 7:29, 8:42, 9:309} to {4:28, 7:7, 8:47, 9:326}. Both rank masses rise by **37**.
+3. **The surplus-gain polynomial.** It is `17 + 126·2^t − 50·3^t − 42·4^t + 2·5^t − 22·7^t + 5·8^t + 17·9^t`,
+   derived independently from the two profiles. It vanishes at `t = 1`, and **its derivative is certified negative on
+   `[0, 1]`** by interval bounds on 256 subintervals. So the gain is positive on `(0, 1)`, matching #4470's
+   own derivative-cover proof by independent code.
+4. **The raw root.** With certified intervals, `T_t > P_t` at 0.0030201659770 and `T_t < P_t` at
+   0.0030201659771, which reproduces #4470's enclosure.
+5. **The tails** at the tested saving `604031/200000000`, `u = 1/5`, cutoff 20, floor 21:
+   - `α < 1 − 10⁻⁹`;
+   - `γ = 0.956176646498 < 0.97`;
+   - `δ = 0.958284435309 < 0.97`.
+
+   All three match #4470's printed values. Since `0.97^{1024} < 2^{−32}`, the normalized paid bound
+   `1/(1 + 1024·10⁻⁹) + 4·2^{−32} = 2097152001953127/2097154147483648 < 1` also reproduces exactly.
+
+6. **The logarithmic compiler**, at ambient `145^{1024}` (the largest target rank), `θ̄ = 63/64` and
+   `p = 1000`:
+   - `θ = log 20/log 21 < 63/64`;
+   - `2^{b−a}r^b ≤ m^a`;
+   - `θ̄^p < η`;
+   - `m^{1−t} < 2^p`;
+   - `z − t = 1/1400000000` exactly;
+   - the stated `H = 64,215,200,000,000` satisfies `(1+H)^p < 2^{(z−t)H}`.
+
+7. **The outer ledger.** All nine outer rows reproduce **exactly** from #4470's printed parameters
+   with the FINE-PRIME row formulas:
+
+   | Row | Formula |
+   | --- | --- |
+   | main transform | `εq` |
+   | global router | `a` |
+   | fused Gaussian | `1−ε+qβ` |
+   | group bit router | `1−ε+aβ` |
+   | group scans | `1−ε+β` |
+   | polynomial products | `ε` |
+   | sparse union movement | `ξ+1/2−ε` |
+   | micro-box faces | `79−ε−β` |
+   | boundary algebraic matrix | `3/32+ξ−ε` |
+
+   The binding row is the main transform, at `κ + 4195771787/1960000000000000000`.
+
+## Not covered
+- The native word and the controller replay.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/pr4470-arithmetic-check --check-only
+python3.12 -B scripts/research_package.py --package research/pr4470-arithmetic-check
+```
+
+The verifier prints `PASS_PR4470_ARITHMETIC_CHECK` in about 2 seconds, using only the standard library.
+
+## Open next questions
+
+1. **None arithmetic.** What remains for #4470 is the native word and the controller replay.
+
+## Credit
+
+- The construction and the claim are 1337isnot1337's (Cinna Davis, #4470).
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.

research/pr4470-arithmetic-check/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"
+    }
+  ]
+}

research/pr4470-arithmetic-check/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 7,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does a new paid-profile or mixture-ceiling claim have independent source-bound coverage of its actual changed ledger?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-mixed-fourth-entry/README.md",
+      "research/staged-fourth-mixture-ceiling/README.md",
+      "research/r437-two-window-carrier/README.md"
+    ],
+    "methods": [
+      "Independently reconstruct one changed paid ledger or pooled dual with exact source and model binding.",
+      "Check a genuinely uncovered moment implication with a different enclosure method and semantic controls."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Choose one uncovered implication in #4088 literal mixed fill, #4091 pooled dual/model binding or #4067 changed frame/host bill. #4091 pins an earlier #4088 head: validate word, capacity and stock equivalence before applying that ceiling to the later fill.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Independently reconstruct one changed ledger or pooled certificate with a positive case and a semantic failure control. Compare exact sources and name what existing checks miss. Keep this separate from supplier-boundary-audit (ordinary/outer dispatch) and uniform-transfer-audit (all-size proof). Stop after the chosen implication; more digits of an unchanged root are not additional mechanism coverage.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Calling an aggregate histogram or source archive an independently accepted physical supplier."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering",
+      "4088",
+      "4091",
+      "4067",
+      "staged544-mixed-fourth-entry",
+      "staged-fourth-mixture-ceiling",
+      "r437-two-window-carrier"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: mixed local/owner/phase ledger and moment reported.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Independent binding of changed construction to paid histogram, distinct from full admission.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.",
+        "evidence": "research/r437-two-window-carrier/README.md",
+        "remaining": "Independent full cost delta; remaining caller/precision/stock obligations are separate.",
+        "status": "author-checked",
+        "source_commit": "a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Independent arithmetic check of #4470 (conditional kappa 0.00302015).",
+  "scope_change_reason": "New paid-profile claim needing independent coverage.",
+  "bottleneck_link": "Independent coverage of the newest conditional kappa claim.",
+  "budget_minutes": 20,
+  "stop_rule": "Stop at invoice, root and tails; compiler and outer rows left.",
+  "outcome": "completed",
+  "evidence": "Invoice totals, local change (53 fewer jobs, +37 masses), gain polynomial, certified root bracket and alpha/gamma/delta all reproduce #4470's printed values. Compiler conditions and stated H reproduce. Outer ledger: all nine rows reproduce; binding main_transform = kappa + 2.14e-9.",
+  "author": "Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance",
+  "ai_assistance": "Substantial Claude Code (Claude Opus 5.5) assistance; no independent human review.",
+  "open_questions": [
+    {
+      "question": "Native word and controller replay of #4470.",
+      "why_it_matters": "Only non-arithmetic parts remain.",
+      "first_step": "Replay the 16-pair controller as in #4457.",
+      "blocked_by": "None."
+    }
+  ]
+}

research/pr4470-arithmetic-check/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "#4470: p16 invoice totals and surplus 752, local change vs #4447 (53 fewer jobs, masses +37), surplus-gain polynomial (derivative certified negative on [0,1]), certified raw root bracket (0.0030201659770, 0.0030201659771), tails alpha/gamma/delta within limits and matching printed values. Logarithmic compiler conditions at ambient 145^1024 and sufficiency of H = 64,215,200,000,000. All nine outer rows from the printed parameters (FINE-PRIME formulas), binding row main_transform above kappa."
+    }
+  ]
+}