Independently check #4412's and #4414's strictification inequalities: all hold; each tested saving sits within 1e-8 of its invoice's limit (no new κ)

research/pr4412-pr4414-moment-check/README.md

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+# Independent check of the strictification inequalities in #4412 and #4414
+
+**Claim type: an independent certified finite check. No new κ.** This
+accepts neither PR. It checks none of their native audits, outer ledgers or
+inherited premises.
+
+## Object
+
+1337isnot1337's two predecessors of #4416 (whose check is
+[#4420](https://github.com/CrocSwap/integer-mult-bounds/pull/4420)) each
+tensor a native mixed unit, then split paid jobs at a cutoff `J`. With
+`α = P_τ/T_τ`, `γ = P_1/(b^C T_τ)` and `X_τ = Σ n_r r^τ`:
+
+| PR | conditional κ | saving `C` | cutoff | needs |
+| --- | --- | --- | --- | --- |
+| [#4412](https://github.com/CrocSwap/integer-mult-bounds/pull/4412) | 0.00247755 | `0.00247756` | `J = 4^n` | `α < 1 − 10⁻⁹`, `γ < 999/1000`; with `n = 15000`, `1/(1+n/10⁹) + 2(999/1000)^n < 1` |
+| [#4414](https://github.com/CrocSwap/integer-mult-bounds/pull/4414) | 0.00267987 | `0.00267988` | `J = ⌊(19/4)^v⌋` | `α < 1 − 10⁻⁸`, `γ < 9999/10000`; with `v = 150000`, `1/(1+v/10⁸) + 4/2¹⁵ < 1` |
+
+No outside check existed for either.
+
+## Results
+
+Every link holds, recomputed with exact rationals and enclosure widths below
+`10⁻³⁰`. The enclosures overlap each PR's recorded intervals.
+
+| | #4412 | #4414 |
+| --- | --- | --- |
+| Invoice: targets / paid calls, rank-one moments | 3,528 / 25,774; 47,075 vs 46,851 | 6,648 / 48,442; 108,769 vs 108,257 |
+| `α` | `0.999999996840`; `1 − α = 3.16·10⁻⁹` | `0.999999982435`; `1 − α = 1.76·10⁻⁸` |
+| `γ` | `0.9989653420` (base 4) | `0.9998629803` (base 19/4) |
+| Closing bound | surplus `1.4392544772·10⁻⁵`, inside the recorded interval | exactly `16386003/16408576`, gap `22573/16408576` |
+| **Limit saving `C*` (α = 1)** | `0.002477561641`, **`1.6·10⁻⁹` above the tested saving** | `0.002679889969`, **`1.0·10⁻⁸` above** |
+
+`γ` stays below 1 at each `C*`. So, as at #4416, each tested saving sits at
+its own invoice's ceiling. Larger powers cannot raise either κ; only a
+different invoice can.
+
+Other checks:
+
+- #4414's `(9999/10000)^10000 < 1/2` is exact.
+- #4414's control holds: the integer cutoff base 4 fails the tail, since
+  `γ > 1` there.
+- #4412's cutoff base 4 lies below its minimum target rank 5.
+
+## Not covered
+
+- The native audits behind `T` and `P`. **Tightest margin:** #4412's `α`
+  clears `1 − 10⁻⁹` by only `2.2·10⁻⁹`.
+- The tensor and splitting proofs, the compiler ranks, the outer rows and
+  every inherited premise.
+
+## Reproduce
+
+```sh
+python3.12 -B research/pr4412-pr4414-moment-check/verify.py
+```
+
+It prints `PASS_PR4412_PR4414_MOMENT_CHECK` in about 2 minutes, using only
+the standard library.
+
+## Open next questions
+
+1. **Rebuild the three p6/p7 invoices from their native words** (#4412,
+   #4414 and #4416).
+   - *Why it matters:* every `α` margin is below `10⁻⁸`, while one paid call
+     moves `α` by more than `10⁻⁶`.
+   - *First step:* census the native words with separate code.
+   - *Blocked by:* nothing.
+
+## Credit
+
+The constructions and invoices are 1337isnot1337's (#4412, #4414, with
+OpenAI Codex assistance), building on zxv's #4359, #4405 and Jacob Alan
+Sussman's Theorem 1G. Prepared by Ryanson Jonathan with substantial Claude
+Code (Claude Opus 5.5) assistance. Apache-2.0.

research/pr4412-pr4414-moment-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"
+    }
+  ]
+}

research/pr4412-pr4414-moment-check/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "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": "Independently certify the strictification inequalities of #4412 (saving 0.00247756) and #4414 (saving 0.00267988) from their recorded native invoices, and compute each invoice's limit saving.",
+  "scope_change_reason": "The card asks for independent coverage of new moment claims; #4412 and #4414 are open conditional-kappa predecessors of #4416 with no outside check.",
+  "bottleneck_link": "Each raises the conditional kappa; their moment inequalities must hold before their native and outer premises are worth review.",
+  "budget_minutes": 30,
+  "stop_rule": "Stop once every stated inequality of both PRs is certified or fails.",
+  "outcome": "completed",
+  "evidence": "verify.py: invoices bound to each native inventory; #4412 alpha = 0.99999999684 < 1-1e-9, gamma = 0.99897 < 999/1000, closing surplus 1.4392544772e-5 inside its interval; #4414 alpha = 0.99999998243 < 1-1e-8, gamma = 0.99986 < 9999/10000, bound 16386003/16408576 and gap exact, base-4 control fails; limit savings 0.002477561641 and 0.002679889969, 1.6e-9 and 1.0e-8 above the tested savings, with gamma < 1 there.",
+  "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": [
+    {
+      "question": "Rebuild the p6/p7 native invoices of #4412, #4414 and #4416 with separate code.",
+      "why_it_matters": "Every alpha margin is below 1e-8 while one paid call moves alpha by more than 1e-6.",
+      "first_step": "Census the native words independently.",
+      "blocked_by": "None."
+    }
+  ]
+}

research/pr4412-pr4414-moment-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": 600,
+      "checks": "Vendored TENSOR.md and RESULT.json hashes and phrases for both PRs; invoice binding and aggregates; certified alpha and gamma at each tested saving overlapping the recorded intervals; closing bounds and gaps; #4414 decay comparison and base-4 control; limit savings with gamma < 1 there."
+    }
+  ]
+}