[zxv]

We prove all-length caps and a per-FULL price frontier for rank-eight productive-helper paths, identifying the local savings needed for further growth beyond controller-only changes.

Background

#4359 raised κ to .0023 with productive helpers. Independent helper jobs finish with autonomous transforms and earn output credit, and a mixed-rank compiler turns that template into a uniform supplier. A supplier with saving C passes when its paid jobs cost less than its targets in the rank moment M_t(X) = Σ X_r r^t, with t = 1−C.

#4440 and #4470 developed this into a specific construction: a paired path of FULL stages, each adding a fresh rank-eight block, interleaved with rank-one QUERY stages. It reached .00302. Every κ gain since has kept that path and changed only the local helper bill. These gains came from the native profile of #4610, the owned helper relations of #5377 and #5383, and the transposition and direct caller of #5392 and #5398. The last was the neutral completion of #5413, now the selected result at .003027288235. The open P104 proposals #5544, #5545 and #5551 change the bill again and reach .00304153.

That history raises two questions. For a fixed bill, can a longer or rearranged path still raise κ? If not, how much cheaper must the bill become to reach a target κ? This PR answers both for the rank-eight path. It makes no new κ claim.

Significance for researchers

We give a stopping rule for controller changes at fixed local cost and a quantitative target for redesigned providers. A universal ceiling for all native productive-helper mechanisms would additionally need a lower bound on local debt over arbitrary legal factorizations; that bound remains open.

Summary

• Controller-only search on a fixed local bill is finished. For the rank-eight prefix/suffix path, every length, split and FULL/QUERY mask reduces to one strictly convex invoice per FULL count. So each bill has an exact all-length cap on raw saving, and it holds for every larger saving.
• The selected bill has essentially no headroom left. #5413's κ is within 1.5×10⁻¹¹ of its all-length cap, and #5468's refinement is within 10⁻¹². Further compiler or precision work on this bill can gain only digits.
• New local mechanisms get a price test. With net local debt L(t), a favorable path exists exactly when L(t) < v·λ₈(t). Reaching C=.0034 from #5551 needs more than 4.91394927369 moment units of net saving per FULL. That is about 0.28% of its net local debt, and about 30× the reduction in the required per-FULL discount from #5413 to #5551 at that C.
• The ~.003 number comes from local debt, not from the path topology. If the neutral bill's eight H8 completion calls were free, the same path would allow about .0081. So "productive helpers are stuck at .003" does not follow from this work, and this work does not claim it.

Caps on raw saving

The raw-saving ceiling is the supremum of values C for which the supplier moment test M_t(P) < M_t(T), t = 1−C, passes. It comes before compiler, BIT and outer losses, so it bounds κ for the same bill from above.

Local contract	Claimed κ	Sup of raw saving, all lengths/splits/masks	Best FULL count
Selected #5413	.003027288235	(.00302728824993, .00302728824995)	18
P104 #5544	.00304057	(.00304057807630, .00304057807631)	18
P104 #5545	.00304117	(.00304117181649, .00304117181650)	18
P104 #5551	.00304153	(.00304153116714, .00304153116715)	18
Neutral bill, eight H8 completions free	none (relaxation)	(.00807671283977, .00807671283979)	7

The P104 rows extend the submitted fixed-p16 crossings to the whole path family. They take the native circuits as given and do not re-verify them. The last row is an optimistic relaxation, not a construction. Even this relaxation misses .01 by more than 21.85 units per FULL.

The universal exchange/convexity reduction builds on Ali Romman's #5405 and Chris R's independent #5409. This package generalizes it to arbitrary ports v, block rank b > 1 and local debt L. The new parts are the application to P104, the price frontier and the closure result.

Price frontier for changed providers

Write the optimal invoice with n FULLs as C_n = nL + v·c_n(t), where c_n depends only on the path geometry and t. Then λ_b(t) = max_n (−c_n/n). A provider that keeps v, b and the controller contract must cut net local debt per FULL by more than min_n C_n/n. Each new repair request counts against that saving. The minimizing n depends only on b and t, not on the local word. Three rational signs certify it over all n, so no large-controller search is needed.

Target C	Best FULL count	Net saving needed per FULL from #5551	From #5413
.0031	18	.80638594481	.99667194384
.0033	17	3.54923453123	3.72139618536
.0034	16	4.91394927369	5.07657009697
.004	14	12.96623724304	13.06167562796
.006	9	38.56153416704	38.34368776890
.01	5	85.66448729816	84.53539335470

Values are rounded down; the exact outward endpoints are in RESULT.json. Units are rank-power moments, not ADD counts. For scale, #5551's net local debt is L(t) ≈ 1736 at C=.0034. At C=.006 and C=.01, the 112-port neutral bill needs less saving than the 104-port #5551. P104's advantage near the current crossing does not persist at every tested target; a changed provider's port count must be included in the comparison.

Why .0034: it is the selected outer contract's admitted BIT budget. With that budget fixed, a change to COMPLEX alone cannot promote κ ≥ .0034, whatever the frontier says. This limit belongs to the outer interface. It is not a BIT impossibility result.

Why the margins are small: every completed native profile here has L(1) = 17v − 48. At t = 1 the optimal invoice therefore has surplus exactly 48n − v: 752 for the selected n=18 invoice and 760 for P104. The positive-C crossing still depends on how paid and target ranks are distributed.

To price a new autonomous bill (ports, paid, helper_targets):

python3.12 -B research/kappa-architecture-frontier/verify.py --price PROFILE.json --C 0.0034

The command reports exact discount bounds and the all-n certificate. If an interval sign is ambiguous, or the minimizer lies beyond n=4096, it fails as unknown. It does not inspect a native word or admit a supplier.

Closure: what cannot amplify a failed bill

Fix t and take bills that each fail the moment test. None of the following produces a passing bill:

• disjoint combinations;
• positive scaling;
• declared tensor products;
• exact acyclic typed cancellation;
• charged splitting or completion.

So more tensor depth, batching, repetition or a finer funding grid cannot rescue a failed bill. A mixture of compatible autonomous providers cannot beat its best member. None of this covers:

• a new cross-boundary consumer;
• correlated or coded outputs;
• a changed idle law or controller graph;
• an arbitrary native refactorization.

Relation to other claims

The frontier assumes autonomous outputs, so it neither refutes nor supports the coded, centre-free p19 proposal #5440 (.0033999964). That PR is outside this contract. Its source at c0f52b5b reports FINDING_PRECOMPILATION_CONTRACT_MISSING and kappa_attached_to_physical_chain: false. The claim map lists it separately from admitted results for that reason. The floating-point LP screens in #5465 are not used as lower bounds. They cover one word or commutation poset, not a redesigned native word.

Evidence and scope

The selected construction, frontier, allocation and queue dispositions remain unchanged. Checkpoint: canonical main 32890bc5; source bills bound at #5544 e2a6cce2, #5545 18968423, #5551 1123635a and #5440 c0f52b5b (full hashes in SOURCES.json and ORIGINS.json).

See the proof, claim map, exact receipt, source origins and validation. The checks use an independent rational engine written with only the Python standard library: interval arithmetic on a 2²²⁴ grid, with no floating-point acceptance. The engine:

• reconstructs every coefficient of four complete source invoices;
• proves six all-length gates that also cover every larger saving;
• proves thirteen all-length discount gates;
• checks 912 small-mask comparisons.

Fourteen rejection controls reject changed source bytes, forged Git trees, omitted query/idle/helper bills and false admission receipts. They pass in both normal and optimized Python.

Validation used CPython 3.12.8, registered serial replays and a Git-free, network-free read-only export (archive SHA256 c741d2d2…). The caps are raw moment theorems under the stated controller and local contracts. Native words and the five framework assumptions are inherited. This package does not claim a whole selected replay, a proof-kernel theorem, hosted CI or independent human acceptance.

Open next questions

• Native lower bound at .0034. Can any legal rank-eight native factorization beat #5551 by more than 4.91394927369 units per FULL? A yes needs a complete charged word. A no must cover arbitrary factorizations, not one poset.
• Lower #5440's coded consumer. Price its actual native event chain, keeping recodings, phases, ownership and all later calls.
• Change an invariant. Change one query, idle, state or geometry invariant and derive its full optimistic invoice before extending a search.
• Check the ratio-frontier proof. Independently check the written proof, especially the n=2/3 convexity junction and the J_(n+1)−J_n sign gate. The finite controls do not replace this check.

We retain the productive/compiler lineage of Cinna Davis, Cao Yifan, Dillon Amburgey, Jacob Alan Sussman, Ali Romman, Chris R, Gareth Davidson, David Arias, thu-nmrc, zxv and all notices in the unchanged source extracts. Prepared with substantial OpenAI Codex assistance; Apache-2.0.