Prove the exact 406-merge threshold for the frozen joint invoice

research/joint-cross-merge-threshold/README.md

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+# Exactly 406 cross-side merges for the frozen joint invoice
+
+This contribution is produced by Nick Petalas. It is a scoped mathematical
+proof with a finite exact accounting certificate, under Apache-2.0.
+
+For the complete frozen role ledger of Mark / Matarener's
+[PR #5298](https://github.com/CrocSwap/integer-mult-bounds/pull/5298), the minimum
+number of disjoint cross-side paid-job merges that passes the signed moment at
+`C = 7/2000` is **406**. Each merge replaces one complex rank `a` and one BIT
+rank `b` by the paid rank `a+b`. Every productive target and every unmerged paid
+job stays in the bill. All roles are optimistically eligible. No job can be
+used twice, and merged jobs cannot merge again.
+
+The constructive witness is 222 pairs of ranks `(72,63)`, two pairs `(72,56)`,
+and 182 pairs `(64,56)`. Write `M = sum targets^(1-C) - sum paid^(1-C)`.
+The exact verifier encloses the optimal margins in these rational intervals:
+
+```text
+405 merges: -0.197285824316 <= M <= -0.197285824315
+406 merges:  0.088428560458 <= M <=  0.088428560459
+```
+
+The general pairing theorem below makes 405 an exclusion for every allowed
+matching, rather than a failed heuristic search. The application and exact threshold are the new result.
+The predecessor's ledger and mathematical conclusions are cited premises and
+are not re-audited here. This changes the `joint-query-controller` card to a
+precise accounting question at the predecessor's explicit whole-job boundary.
+
+## Optimal pairing theorem
+
+Let `0 < alpha < 1` and let both sides be finite lists of positive ranks.
+For exactly `k` disjoint cross-side merges, the smallest paid alpha-moment
+comes from selecting the largest `k` ranks on each side and pairing them in
+that same descending order. This holds for arbitrary positive real ranks.
+
+To prove it, define the saving from one merge by
+
+```text
+Delta(a,b) = a^alpha + b^alpha - (a+b)^alpha.
+```
+
+It is positive by strict concavity. Its derivative in `a` is
+`alpha [a^(alpha-1) - (a+b)^(alpha-1)] > 0`, and the derivative in `b` is
+positive too. Thus replacing a selected rank by a larger unselected rank
+cannot reduce the saving. Repeating the exchange selects the largest `k`
+ranks on both sides, including ties in any order.
+
+The mixed derivative is
+`Delta_ab = alpha(1-alpha)(a+b)^(alpha-2) > 0`.
+For `a >= a'` and `b >= b'`, integrate this derivative over the rectangle to
+obtain
+
+```text
+Delta(a,b) + Delta(a',b') >= Delta(a,b') + Delta(a',b).
+```
+
+Exchanging inverted pairs therefore increases or preserves the total saving.
+A finite sequence of exchanges produces the descending pairing. Since the
+unmerged original bill is fixed, maximizing this saving minimizes the complete
+paid moment. Further disjoint merges strictly improve the margin, so failure
+at the optimal 405-pair matching excludes every matching with at most 405
+pairs. The explicit 406-pair witness proves sharpness.
+
+This is a specialization of the classical supermodular rearrangement argument.
+The contribution claims the exact source-bound threshold and its application,
+not discovery of rearrangement or of concave merge gains.
+
+## Exact arithmetic and evidence
+
+The stdlib-only verifier reads the full predecessor `RESULT.json` as a frozen
+input, with SHA256
+`e712e5373ecbfea0c1b7c115b52eea4fb533c6c7c2ebae3aa98a2dd1f058721c`.
+It computes the new complete moments from that ledger, without replaying the
+predecessor's roots, six models, native words or other conclusions.
+
+The exponent is the exact rational `alpha=1993/2000`. Powers use rational
+outward bounds, without floating point, Decimal, logarithm library calls or a
+root grid. Write `r=2^s u` with `1<=u<2`. For `z=(u-1)/(u+1)`, sum the first
+64 terms of `log u = 2 sum z^(2j+1)/(2j+1)` on a `2^-120` grid, rounding each
+term outward. The remaining positive tail is bounded by
+`2 z^129/[129(1-z^2)]`. Apply the same bound to `log 2`.
+
+For `x=C log r`, the verifier sums `exp(x)` through degree 24 and bounds its
+tail by the degree-25 term times `26/25`, valid for `0<=x<=1`. Monotonicity and
+reciprocal interval arithmetic then bound `r exp(-C log r)`.
+All later additions, subtractions and comparisons use exact rational numbers.
+
+An independent certificate rounds every needed power interval outward to a
+`2^-36` grid and checks the defining integer-power inequalities for all 145
+ranks from 1 through 145. If its endpoints are `l/2^36` and `u/2^36`, it checks
+`l^2000 <= r^1993 * 2^72000 <= u^2000`. The coarser, integer-certified bounds
+separately prove both moment signs. Thus the sharp 406 threshold has a finite
+certificate that does not require trusting the logarithm/exponential enclosure
+implementation. The general matching theorem remains a written proof.
+
+Finite corroboration enumerates all 1,200 choices of two three-job multisets
+over ranks 1 through 4 and a merge count 1 through 3. All 13,200 labeled partial
+matchings agree with the optimal pairing theorem. Seven invalid input controls
+reject, and a strict crossed-pair example detects reversing the pairing rule.
+The source hash, mass preservation, job-count reduction and explicit pair
+multiplicities are executable invariants. Every run prints a generated receipt
+binding the package files by SHA256.
+
+## Reproduce
+
+From the repository root, use CPython 3.12, standard library only:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/joint-cross-merge-threshold/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/joint-cross-merge-threshold/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/joint-cross-merge-threshold
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/joint-cross-merge-threshold --check-source-history
+```
+
+Use an available CPython 3.12 path on another machine. The expected marker is
+`PASS_JOINT_CROSS_MERGE_THRESHOLD` with `minimum_merges: 406`.
+Verification writes no files. See `VERIFY.json` for both registered commands.
+
+## Scope and limitations
+
+This is an exact threshold in an optimistic, finite accounting model of two
+historical invoices. It supplies no physical merge, joint native word, compiler,
+controller, dirty-state restoration, tape bill or new kappa. The target `7/2000`
+is an explicit benchmark above the frozen side roots, not a selected admission
+threshold. The current selected record remains unchanged.
+
+Additional fees can consume the positive margin. Restrictions on eligible roles
+can raise the minimum. Repeated merges, splitting, altered targets, extra credits
+or a different invoice are outside the theorem's threshold application. Native
+and all-size multiplication obligations were not run. Independent human and
+formal proof-kernel review are unclaimed.
+
+The runtime is managed Python. The verifier's largest supplied side has fewer
+than 100,000 jobs and its exhaustive oracle has fixed finite loops. Dynamic
+integer, rational and collection allocations are bounded by this fixture;
+Python cannot enforce allocation-free execution. Memoization is limited to 512
+power bounds. Functions remain below 60 physical lines. These are the applicable
+Power of Ten deviations, not changes to shared repository checks. The bounded
+cache decorator and seven-entry control dispatch are standard-Python deviations
+from the restrictions on metaprogramming and indirect calls.
+
+## Sources and credit
+
+The exact source is PR #5298 head
+`e63b0a32ed2c021a8f4344e5aeb7e9123ff2eaef`,
+`research/joint-supplier-average-ceiling/RESULT.json`.
+`inputs/PR5298-RESULT.json` retains its bytes verbatim because this pending
+source is not available from the package base history. The verifier authenticates
+that retained file locally; no remote Git object or network access is required.
+`SOURCES.json` binds the original card and selected record at the package base.
+`TASK.json` records the external source URL, head and digest explicitly.
+
+Mark / Matarener supplied the joint invoice, whole-job boundary and merge
+examples. James Reed / jwr404 identified the scope correction. zxv and the
+retained #4470 lineage supplied the complex invoice; ai-nanachi and #4814 on
+#4760 supplied the BIT invoice. Ali Romman's #5341 supplies separate crossing
+results, which are not required or replayed here. The retained Zhihao Chen,
+RaD/hipotures, icekylinx and eumemic controller and moment lineage keeps its
+credit. This package adds the optimal disjoint-pair rule and exact 406 threshold.
+
+## Open next questions
+
+Can a native child legally realize any of the threshold witness's three rank
+pairs while retaining both arbitrary payloads and their returned state? This
+decides whether the relaxed accounting witness has a physical interpretation.
+Start with one `(72,63)` pair and write its complete producer/consumer and fee
+contract. No such joint native contract is supplied here.
+
+Can an explicit role-eligibility graph still admit 406 effective merges? First
+derive its legal edges from one actual joint word, then solve the finite weighted
+matching problem with all fees. The missing physical edge relation is the blocker.

research/joint-cross-merge-threshold/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ]
+}

research/joint-cross-merge-threshold/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "joint-query-controller",
+  "priority_revision": 8,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "original_card": {
+    "id": "joint-query-controller",
+    "question": "Can a changed query demand or caller remove real work while satisfying native chronology and restoration?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Choose an exact caller and a new demand/interface hypothesis. Read the scoped #5617/#5631/#5649/#5747 and #5720/#5745 objections before rebuilding an excluded schedule. Compare the complete matched source/consumer protocols including all retained helper outputs. For an active-mode injection, first check #5745: its canonical delivery claim repays the eight removed centre jobs. Audit that exact claim or change its channel/feedback hypothesis; do not repeat the same free-injection search. Stop if the full favorable budget fails or the required restored output is unobservable.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "query demand",
+      "active mode",
+      "5720",
+      "5745",
+      "5747",
+      "caller"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4166: author-checked installed producer/service/return/bridge connection; native-component comparison loses.",
+        "evidence": "research/qq16-caller-install/README.md",
+        "remaining": "Complete matched physical protocol bill, or changed source-compatible demand/circuit; no affordable supplier yet.",
+        "status": "author-checked",
+        "source_commit": "0b082d22a0e9a442d264ab0633e35ef26c790630",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Sharp minimum number of disjoint cross-side paid-job merges at C=7/2000 in the exact PR5298 frozen joint invoice.",
+  "scope_change_reason": "The current card asks for a physically completed changed caller. This side result resolves a necessary finite accounting threshold at PR5298's explicit whole-job-premise boundary, without claiming a native realization.",
+  "bottleneck_link": "Quantifies the optimistic cross-side merge budget needed to escape the frozen whole-job bound; physical eligibility and all fees remain open.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first exact threshold certificate, or report a bounded failure. No larger saving or additional construction search.",
+  "outcome": "completed",
+  "evidence": "Optimal-pairing proof; exact rational signed-moment signs at 405 and 406, independently certified by 145 integer-power comparisons; exhaustive 13200 partial matchings; eight controls.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a joint native child realize a witness rank pair?",
+      "why_it_matters": "The accounting relaxation has no physical construction.",
+      "first_step": "Specify one (72,63) pair with both arbitrary inputs and exact returned states, including all fees.",
+      "blocked_by": "No joint native producer/consumer contract."
+    },
+    {
+      "question": "Can a physically derived eligibility graph support an effective 406-pair matching?",
+      "why_it_matters": "Restricted roles or extra fees may invalidate the relaxed witness.",
+      "first_step": "Derive legal cross-side edges from one actual joint word and solve its weighted matching.",
+      "blocked_by": "No physical edge relation or complete fee bill."
+    }
+  ],
+  "external_sources": [
+    {
+      "pull_request": 5298,
+      "author": "Matarener",
+      "head": "e63b0a32ed2c021a8f4344e5aeb7e9123ff2eaef",
+      "original_path": "research/joint-supplier-average-ceiling/RESULT.json",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/e63b0a32ed2c021a8f4344e5aeb7e9123ff2eaef/research/joint-supplier-average-ceiling/RESULT.json",
+      "retained_path": "research/joint-cross-merge-threshold/inputs/PR5298-RESULT.json",
+      "sha256": "e712e5373ecbfea0c1b7c115b52eea4fb533c6c7c2ebae3aa98a2dd1f058721c"
+    }
+  ],
+  "environment": {
+    "python_implementation": "CPython",
+    "python_version": "3.12.15",
+    "platform": "macOS-27.0.1-arm64-arm-64bit",
+    "dependencies": "Python standard library only"
+  }
+}

research/joint-cross-merge-threshold/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact 405/406 moment signs, optimal-pairing oracle, source bytes, mass/job invariants and rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Identical full verification under optimized Python."
+    }
+  ]
+}