Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers

research/kappa-architecture-frontier/CLAIMS.md

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+# Claims since κ=.0023: architecture, evidence and limits
+
+Checkpoint: canonical main `32890bc5de034ab8fc93ac88ead7279a6a416d04`,
+2026-10-25. Source heads, blob IDs and SHA256 are in SOURCES.json and ORIGINS.json. These are
+selected milestones and relevant high claims, not an exhaustive claim census.
+No PR, Discussion or intake disposition is changed by this local draft.
+
+| Claim | Architectural change | Status used here |
+|---|---|---|
+| [#4359](https://github.com/CrocSwap/integer-mult-bounds/pull/4359), .0023 | Credit independent productive helpers; mixed-rank recursive compiler and a complete outer bill | Historical selected conditional result, not a cap on descendants |
+| [#4440](https://github.com/CrocSwap/integer-mult-bounds/pull/4440), .00296 | Replace FULL stages by genuine queries, losing their helper credit and specializing their stage factors | Historical conditional proposal in selected ancestry |
+| [#4470](https://github.com/CrocSwap/integer-mult-bounds/pull/4470), .00302015 | Longer optimized productive path and native/helper lineage | Historical selected conditional checkpoint |
+| [#4610](https://github.com/CrocSwap/integer-mult-bounds/pull/4610), .0030217758 | Native paid-profile improvement under retained productive path | Historical selected conditional checkpoint; new arithmetic sharpens its all-path cap to <.00302177587079 |
+| [#5377](https://github.com/CrocSwap/integer-mult-bounds/pull/5377), .00302235; [#5383](https://github.com/CrocSwap/integer-mult-bounds/pull/5383), .00302666 | Owned helper relations change the charged local debt | Retained ancestry; not merely outer rounding |
+| [#5387](https://github.com/CrocSwap/integer-mult-bounds/pull/5387), .003026674765 | Smaller compiler/completion reserve | Funding improvement to the same economic proposal |
+| [#5392](https://github.com/CrocSwap/integer-mult-bounds/pull/5392), .00302725; [#5398](https://github.com/CrocSwap/integer-mult-bounds/pull/5398), .00302728823 | Owned circuit transposition, then a direct backward caller | Stronger fully charged local contract; raw profile later retained by neutral completion |
+| [#5413](https://github.com/CrocSwap/integer-mult-bounds/pull/5413), .003027288235 | Complete the coupled provider, paying and crediting equal extra H8 jobs | Current selected conditional result; signed moment unchanged by that neutral completion |
+| [#5462](https://github.com/CrocSwap/integer-mult-bounds/pull/5462), .0030272882389; [#5468](https://github.com/CrocSwap/integer-mult-bounds/pull/5468), .0030272882489783 | Compiler/precision headroom inside the same raw profile | Open numerical refinements, not new recursive economics; bounded by the same raw root |
+| [#5544](https://github.com/CrocSwap/integer-mult-bounds/pull/5544), .00304057 | Productive P104 provider with 104 ports and 527 helper outputs | Open complete conditional proposal; source receipt independently rebound to path arithmetic here |
+| [#5545](https://github.com/CrocSwap/integer-mult-bounds/pull/5545), .00304117 | Legal paid contact H1+H5 -> H6 | Open proposal; strict raw benefit at every 0<t<1 follows from concavity |
+| [#5551](https://github.com/CrocSwap/integer-mult-bounds/pull/5551), .00304153 | Two coupled paid-path changes, net +2H1-3H2+H3-H4+H5 | Open proposal; new all-length cap here is <.00304153116715 for its declared profile |
+| [#5440](https://github.com/CrocSwap/integer-mult-bounds/pull/5440), about .0033999964 | Centre-free coded reverse stages and inverse-coded queries; complex planning budget .007578535190 | Open, explicitly conditional physical attachment. Missing chain-to-native-event compiler remains recorded; not a selected or newly admitted κ |
+
+## Why the p19 headline does not close its admission task
+
+At source head `c0f52b5b18bba78a6a5e8057edf9571bade88549`, the retained
+`P19_PRECOMPILATION.json` reports `FINDING_PRECOMPILATION_CONTRACT_MISSING`.
+The source README and current PR description distinguish macro/operator checks
+from a serialized chain-to-event compiler. Its paid profile is derived from a
+base program trace and reused for p19; it is not derived from a p19 native event
+trace. The PR explicitly records `physical_attachment: CONDITIONAL` and
+`kappa_attached_to_physical_chain: false`.
+
+This is a concrete remaining proof obligation. It is neither proof that p19 is
+impossible nor permission to treat missing lowering costs as zero. The package
+here authenticates that declaration and inspects the mathematical boundary;
+it does not run p19's large source replay or resolve native attachment.
+
+The productive reason to investigate this branch is structural: coded states
+can avoid an owned-centre restoration that an autonomous-output contract pays.
+Whether they yield an actual improvement depends on the complete consumer,
+all recodings, phase/frame semantics and compiler/precision/outer join. The
+retained .0034 admitted BIT budget separately limits the stated outer claim.
+
+## What the P104 comparison establishes
+
+The new checker independently counts each complete invoice from the retained
+local P/H histograms and the canonical controller. Every source coefficient
+matches, including all thin queries and whole idles. It separately proves the
+all-p ceiling and both contact-delta signs. The source native interpreters,
+ordinary implementation bounds and outer ledgers are not imported or re-run.
+Thus these are source-bound architectural moment theorems conditional on the
+stated local contracts, not fresh complete supplier audits.
+
+The difference between .003027 and .003041 is enough to disprove a profile-only
+argument that the whole rank-eight productive path is stuck at its old number.
+It does not prove the entire native mechanism language has been optimized.
+In particular the exploratory floating LP screens in
+[#5465](https://github.com/CrocSwap/integer-mult-bounds/pull/5465) are not used as
+exact architectural lower bounds here. Its restrictions to one word, commutation
+poset or selected refactorings do not quantify over a redesigned native word.
+
+## Evidence already available and the actual delta
+
+The infinite FULL/QUERY ceiling and convexity reduction were already established
+in [#5405](https://github.com/CrocSwap/integer-mult-bounds/pull/5405) and independently
+checked in [#5409](https://github.com/CrocSwap/integer-mult-bounds/pull/5409).
+The older #4610 all-thinning result also already includes a centre-free relaxation.
+This contribution does not claim those as new discoveries.
+
+The new work is the parametric local-debt frontier, its ratio-minimum certificate,
+the transfer to the actual neutral/P104 bills, the resulting all-length P104
+ceilings, the updated neutral free-completion bound and the explicit composition
+closure/search decisions. Exact source equality and smaller numerical brackets
+alone would not supply that architectural extension.

research/kappa-architecture-frontier/EXAMPLE-PROFILE.json

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+{
+  "ports": 104,
+  "paid": {
+    "1": 2430,
+    "2": 554,
+    "3": 390,
+    "4": 111,
+    "5": 108,
+    "6": 17,
+    "7": 13,
+    "9": 1
+  },
+  "helper_targets": {
+    "4": 51,
+    "5": 36,
+    "7": 24,
+    "8": 122,
+    "9": 294
+  }
+}

research/kappa-architecture-frontier/LITERATURE.md

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+# Direct predecessors and a targeted literature check
+
+The proofs here use elementary concavity, positive moment potentials and exact
+intervals. No external theorem is needed to justify the numerical exclusions.
+This is a targeted primary-source check, not a comprehensive literature survey.
+
+## Direct research sources
+
+[Jacob Alan Sussman's Theorem 1G, retained by PR4359](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/proofs/theorem-1g.md)
+uses rank-power potentials for mixed target jobs and grouped paid blocks.
+Its full all-column/causality/separation hypotheses matter. We use its job-accounting
+view to state a closure obstruction; we do not infer that a favorable histogram
+alone supplies its finite physical compiler. The native and outer bridge belongs
+to the separately credited productive-helper construction.
+
+[PR5405](https://github.com/CrocSwap/integer-mult-bounds/pull/5405), by
+[Ali Romman](https://github.com/aliromman), proves an unbounded retained
+FULL/QUERY prefix/suffix ceiling. [PR5409](https://github.com/CrocSwap/integer-mult-bounds/pull/5409),
+by [Chris R](https://github.com/Clear1nsight), supplies the exchange/convexity
+independent check. Their underlying controller is credited to
+[Cinna Davis](https://github.com/1337isnot1337). Frozen proofs in this package
+retain source wording and attribution. They are the direct mathematical
+predecessors, rather than newly discovered literature barriers.
+
+## Related potential methodology, with a precise limit on its use
+
+[Christandl, Vrana and Zuiddam, *Universal points in the asymptotic spectrum of tensors*](https://arxiv.org/abs/1709.07851v3)
+studies parameters that are additive on direct sums, multiplicative on tensor
+products, and monotone under tensor restriction. It provides a broad context
+for additive/multiplicative obstructions to construction closures. This does
+**not** prove that our rank-power price is monotone for arbitrary native
+operators, coded interfaces or tensor restrictions; that is exactly the missing
+hypothesis for an unrestricted architectural barrier.
+
+[Wigderson and Zuiddam, *Asymptotic spectra: Theory, applications and extensions*](https://www.math.ias.edu/~avi/PUBLICATIONS/Wigderson_Zuiddam_2025.pdf),
+August 5, 2025, especially sections 3.4, 4.2 and 4.5, develops the relation
+between monotone homomorphisms, asymptotic algorithms and barriers. Its lesson
+for this investigation is methodological: specify a construction preorder and
+prove the potential inequalities for **every** allowed generating operation.
+The local closure proof here does so only for its explicitly listed invoice
+operations; no general spectral duality theorem is applied to multiplication.
+
+The elementary exchanges and the ratio certificate are proved directly in
+PROOF.md. No finite mask enumeration, assignment weight, old fixed-word ceiling,
+or external tensor theorem supplies the absent universal lower bound over all
+legal native factorizations.