Qualify the absolute loss of the frozen PR5687 two-call invoice (no new κ)

docs/ci-verification.md

@@ -3,7 +3,9 @@
 ## Current selected-family replay: #5599
 
 The [selected record](../certificates/selected-result.json) binds conditional
-κ = 0.00310045. `make frontier5599-verify` authenticates the complete source
+κ = 0.00310045 to the [proof review](research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md)
+and [explicit source closure](../certificates/frontier5599-source.json).
+`make frontier5599-verify` authenticates the complete source
 closure and claim derivation, then executes primary, independent caller
 positive-control, adapted caller and optimized primary checks serially. The
 Linux [audit run](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40253906841)

research/uniform-two-call-invoice-qualification/NOTICE

@@ -0,0 +1,3 @@
+Uniform two-call invoice qualification
+
+Prepared with substantial OpenAI Codex assistance. New text is intended for the upstream Apache-2.0 repository. The unchanged PR5687 caller and PR5673 method excerpt retain their authors, source notices and credit. The classical rank-separation method receives no novelty claim. No scientific replay or new kappa is claimed.

research/uniform-two-call-invoice-qualification/PROOF.md

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+# The frozen [PR #5687](https://github.com/CrocSwap/integer-mult-bounds/pull/5687) G5 invoice has negative raw moment for every positive exponent saving
+
+This is a paper qualification of one explicitly supplied invoice. It assumes the complete paid and productive lists stated in [PR #5687](https://github.com/CrocSwap/integer-mult-bounds/pull/5687), Theorem G5, at head `b3b0fc751d3f5ecdd1498a99a4bfcfa97a0a5b97`. It does not independently establish the physical construction or dispute its relative improvement. The method is elementary rank separation after cancellation; repository precedent includes [PR #5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673) section 4 for a different invoice. No new general inequality is claimed.
+
+## Exact input and conventions
+
+The primary text is `research/restricted-gaussian-coframe-improvement/caller.md`, section 7. The authenticated local copy has SHA256 `f5fe6f217a55a684132114522e15c0f3424c10248fb5d0d4b48ccea0a8c08650` and Git blob `2d6cbce0f076a325cf498989eb2ed2268f16d5f3`. Its immutable public source is [the frozen caller](https://github.com/Dugongue/integer-mult-bounds/blob/b3b0fc751d3f5ecdd1498a99a4bfcfa97a0a5b97/research/restricted-gaussian-coframe-improvement/caller.md).
+
+For a real parameter C define H_r(C)=r^(1-C), for positive integer r. Use the source's complete two-call lists:
+
+$$
+P(C)=3732H_1+1046H_2+836H_3+278H_4+24H_6,
+$$
+
+$$
+T(C)=52H_3+4H_4+24H_7+736H_8+224H_{15}.
+$$
+
+The target-minus-paid raw Gaussian moment is M(C)=T(C)-P(C). All original 224 main inputs and the two distinct 408-helper banks have the source's declared independent ownership. No target is added for scratch, no banks are identified, and no helper is credited a second time. These ownership statements bind the invoice being qualified; they are not an independent replay of the physical endpoint.
+
+## Scoped inequality
+
+Cancel exactly the common 52H3 and 4H4 terms. Then
+
+$$
+M(C)=24H_7+736H_8+224H_{15}
+      -3732H_1-1046H_2-784H_3-274H_4-24H_6.
+$$
+
+Each complete original invoice has rank mass 9,588. Canceling the common terms removes 172 from each side. The remaining productive rank mass is
+
+$$
+24\cdot7+736\cdot8+224\cdot15=168+5888+3360=9416.
+$$
+
+The remaining paid rank mass is
+
+$$
+3732+1046\cdot2+784\cdot3+274\cdot4+24\cdot6
+=3732+2092+2352+1096+144=9416.
+$$
+
+For every C>0, the function r^(-C) is strictly decreasing in r. Write each H_r as r times r^(-C). Every remaining productive rank is at least 7, and every remaining paid rank is at most 6. Consequently
+
+$$
+M(C)\le9416\left(7^{-C}-6^{-C}\right)<0
+\qquad(C>0).
+$$
+
+At C=0, M(0)=0. This proof uses no approximated logarithm, numerical power evaluation, monotonicity of a difference of signed moments, or inference from one tested point. Its declared domain includes every positive C, in particular .0033.
+
+## Meaning for the next gate
+
+The frozen G5 invoice cannot supply a positive raw moment. If a proposed admission route requires a strictly positive target-minus-paid raw moment for these same lists, it fails this necessary condition before nonnegative additional recursive charges or a positive reserve. Positive finite ordinary costs remain separate obligations; they do not turn a negative recursive moment into a positive one.
+
+This does not prove failure of an unspecified compiler architecture or outer theorem with a different premise. Any claimed alternative route must name and justify its actual admission inequality. Nor does this rule out a different consumer, target census, helper profile, source invoice or resource contract.
+
+The source's predecessor-relative gain greater than 148C on its stated interval can coexist with M(C)<0: the predecessor can have a still lower absolute moment. The relative gain and operator construction remain source-author contributions. The present result only settles the previously unproved absolute sign for the displayed frozen lists.
+
+Let an actual changed construction supply productive increment DeltaT(C), paid increment DeltaP(C), and a reserve R(C) expressed in the same units only when a compiler theorem justifies doing so. A necessary raw funding condition is
+
+$$
+\Delta T(C)-\Delta P(C)>-M(C)+R(C).
+$$
+
+The displayed upper bound on M implies the weaker necessary threshold
+
+$$
+\Delta T(C)-\Delta P(C)>
+9416\left(6^{-C}-7^{-C}\right)+R(C).
+$$
+
+Neither inequality grants an implementable repair budget or a new target. Actual physical changes must preserve arbitrary inputs, native phases and charts, observations, ownership and restoration, and carry their complete new invoice plus ordinary, precision, stock, tape, compiler and outer obligations.
+
+## Scope and credit
+
+The result applies only to [PR #5687](https://github.com/CrocSwap/integer-mult-bounds/pull/5687)'s stated G5 lists at the frozen head. [PR #5698](https://github.com/CrocSwap/integer-mult-bounds/pull/5698)'s separate actual-C coframe optimum, [PR #5678](https://github.com/CrocSwap/integer-mult-bounds/pull/5678)'s P104/535-helper stream, historical rank17/rank16 templates, and every other consumer have different contracts. Their inventories are not combined here. The source's two-call endpoint contains rank15 jobs; it is not replaced with a rank16 grant.
+
+Credit Dugongue and the retained source authors for the uniform provider, two-call endpoint, complete lists and relative improvement; [PR #5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673) for the repository's earlier use of the elementary rank-separation technique on another invoice; and Cao Yifan, Cinna Davis and the native/helper/code/controller/compiler lineage retained in the source. Prepared with substantial OpenAI Codex assistance. Static mathematical review is separate from independent physical validation, hosted CI and formal proof-kernel acceptance. No construction, numerical experiment, paid saving or new kappa is claimed.

research/uniform-two-call-invoice-qualification/README.md

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+# Absolute moment qualification of the frozen uniform two-call invoice
+
+## Change and mathematical scope
+
+The exact two-call invoice in [PR #5687](https://github.com/CrocSwap/integer-mult-bounds/pull/5687), Theorem G5, has a negative target-minus-paid Gaussian moment for every positive saving parameter C, under its stated tariff H_r(C)=r^(1-C). The [paper proof](PROOF.md) settles the absolute sign left unestablished by the source's positive predecessor-relative comparison. It does not challenge that relative improvement or claim a physical impossibility theorem.
+
+After cancellation, productive and paid rank masses both equal 9,416; every remaining productive rank is at least seven and every paid rank is at most six. Hence M(C) is at most 9416(7^(-C)-6^(-C)), which is strictly negative for C>0. At zero it is zero. The full original masses are 9,588 each.
+
+An admission route requiring a positive raw moment for these exact lists fails this necessary condition. A different invoice, target census, tariff or compiler premise needs a separate argument. No paid saving, emitted construction, numerical price, scientific replay or new kappa is claimed.
+
+## Task and overlap
+
+This is a source-bound application to head `b3b0fc751d3f5ecdd1498a99a4bfcfa97a0a5b97`, `research/restricted-gaussian-coframe-improvement/caller.md`, section 7. The unchanged [primary caller text](inputs/PR5687-CALLER.md) supplies 224 independent main inputs and two distinct 408-helper banks. The complete paired profiles, cancellation convention and exact source pins are inspectable in this package.
+
+The method is elementary and established. [PR #5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673), section 4, uses rank separation on a different invoice; its [minimum primary excerpt](inputs/PR5673-METHOD-EXCERPT.md) is preserved for credit. The canonical signed-invoice preflight also supplies a generic classical comparator. No new inequality or screening algorithm is claimed. The contribution is the absolute sign of this specific published G5 invoice.
+
+Bounded source and discussion checks found no matching exact G5 result in their declared coverage; this is not exhaustive novelty. [PR #5698](https://github.com/CrocSwap/integer-mult-bounds/pull/5698)'s actual-C coframe optimum and [PR #5678](https://github.com/CrocSwap/integer-mult-bounds/pull/5678)'s P104/535-helper contract are distinct. Their lists are not substituted or combined here.
+
+## Validation
+
+This is an unregistered static proof package. Read the exact histograms and three-line rank-separation argument in PROOF.md. No verifier, TASK, VERIFY or synthetic RESULT is supplied. SOURCE-CUSTODY.json is custom packaging provenance, not generic runner metadata or a scientific execution receipt.
+
+The root argument passed a separate local AI mathematical review at its original exact bytes. The publication proof differs only by linked and spaced PR references and the explicit full-versus-reduced rank-mass sentence; final publication-copy review is recorded separately against the actual package hashes. Source copies retain exact bytes. Acceptance of an earlier argument is not silently transferred to changed package bytes.
+
+No logarithm, power evaluator, source engine, WORD/archive, matrix, solver, numerical screen or scientific replay was executed. The physical producer/caller, native phases and ROW interface, protected reads, ownership/restoration, fixed-odd arithmetic, ordinary costs, stock, precision, tape, compiler and outer hypotheses were not independently verified here. The source author's physical and execution claims remain source evidence.
+
+## Credit
+
+Credit Dugongue and the retained source authors for the uniform provider, two-call endpoint, complete invoice and relative comparison. Credit PR5673 and the classical signed-invoice method for rank separation; Cao Yifan, Cinna Davis and the native/helper/code/controller/compiler lineage remain credited through the unchanged primary text. Source notices and Apache-2.0 terms are retained. Prepared with substantial OpenAI Codex assistance, including separate local AI review; no independent human scientific or proof-kernel acceptance is claimed.
+
+## Open next questions
+
+1. Supply one changed complete endpoint or paid bill with independently owned targets whose legitimate gain exceeds the exact deficit and applicable compiler reserve. State the actual source-bound operations before construction; a relative gain alone is insufficient.
+2. For a proposed joint raw-code caller, retain all entrance-target debits and provide exact all-width boundary operators, residual corrections and native tariffs. A low owner rank or inexpensive residual alone supplies no independent Gaussian reward.
+3. Review this frozen source binding and elementary sign argument. An emitted physical host would not rescue these unchanged lists under an admission route requiring a positive raw moment; a new scientific construction is therefore not the next task for this invoice.

research/uniform-two-call-invoice-qualification/SOURCE-CUSTODY.json

@@ -0,0 +1,60 @@
+{
+  "schema": "uniform-two-call-invoice-custom-custody/v1",
+  "generic_runner_registration": false,
+  "scientific_execution": false,
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+  "primary_head": "b3b0fc751d3f5ecdd1498a99a4bfcfa97a0a5b97",
+  "primary_caller_git_blob": "2d6cbce0f076a325cf498989eb2ed2268f16d5f3",
+  "primary_caller_sha256": "f5fe6f217a55a684132114522e15c0f3424c10248fb5d0d4b48ccea0a8c08650",
+  "method_source": {
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+    "source_path": "research/crossed-connector-caller/PROOF.md",
+    "url": "https://github.com/aliromman/integer-mult-bounds/blob/c10f75f09e41803ae8efe478331a7509b7379b2f/research/crossed-connector-caller/PROOF.md",
+    "whole_git_blob": "e3800fbcf1c5f5633e14d9856ca1400985292a0d",
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+    "meaning": "Priorrank-separatedmomentboundonDIFFERENT8fieldescapeinvoice;creditmethodonly,noP/Torphysicalclaimtransfer",
+    "current_guidance_main": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+    "guidance_basis": "Freshmaincheck06:37:48UTCsameimmutablecommit;existingguidance/template/catalogpinsunchanged",
+    "original_capture_excerpt_file": "PR5673-SECTION4-METHOD-EXCERPT.md",
+    "packaged_excerpt_path": "inputs/PR5673-METHOD-EXCERPT.md"
+  },
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+  "original_review_sha256": "1cec509bff3c7910e5fced45176b40783e1ff0a84c9e31b4f6bb81e47da82d09",
+  "publication_changes": "Only spaced/linked PR references and explicit full9588 versus reduced9416 mass sentence; final publication-copy review recorded separately against the actual package hashes",
+  "source_urls": [
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+    "https://github.com/aliromman/integer-mult-bounds/blob/c10f75f09e41803ae8efe478331a7509b7379b2f/research/crossed-connector-caller/PROOF.md"
+  ],
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+}

research/uniform-two-call-invoice-qualification/inputs/PR5673-METHOD-EXCERPT.md

@@ -0,0 +1,22 @@
+## 4. Constructive escape and its price
+
+`ESCAPE-WORD.json` contains a complete eight-field binary word. It has three
+paired stages, six full calls with rank-one queries, and eight rank-one thin calls.
+Every local projector, query rank and compatibility relation is checked.
+The verifier retains every exponent of each independent D variable and
+checks every coefficient of the full word. It also checks that the thin
+maps distinguish all three leaves in both directions.
+
+Thus the changed connector contract has a real positive operator. It does
+not give a profitable supplier. At the pinned PR5599 tariff, 104 copies have
+complete optimistic moment coefficients
+
+```text
+208 H25 + 624 H9 + 6 helpers - 6 paid - 832 H1.
+```
+
+Its linear margin is -336. After common ranks are cancelled, all positive
+terms have rank at least eight and all negative terms have rank at most
+seven. For every `0<=t<=1`, the margin is at most `8^(t-1)*(-336)<0`.
+Additional physical adapters cannot repair that price.
+