DONE: actual #4698 centre-chain terminal loses; shared heterogeneous consumer remains open

[Clear1nsight]

Current checkpoint, 2026-10-23T00:35Z: the main-only two-FULL endpoint is closed in the two distinct scopes of #4920, and both fixed many-bank colour variants in #4922 are stopped. The unchanged five-bank free pre/post completion is now excluded in #4856's finite columnwise model, with exact-head CI passing. Statements below identifying those as future experiments are retained history. The present read/derive gate asks whether actual transformed helper resources, exposed during their producer and returned as the same original jobs, can change the complete two-FULL endpoint. This is not a claim that an already transformed arbitrary helper value is a free query or useful dirty scratch. No new κ.

Scoped #4919 native application check, 2026-10-23

DONE / read-and-derive finding; no chart-family search, advance claim or separate PR. #4919 at 15e8433c431aaaf7fa3756407d44f4700fc8377a supplies a valid shared-three-subspace classification. Its native interpretation allows genuine GL9 freedom: the canonical phase-correct K_G works for arbitrary invertible G, with no invented orthogonality restriction. This freedom alone leaves the frozen #4610 paid profile unchanged. All240 existing forward warp occurrences have zero adjacent paid cost in the static source charts, so shortening those zero windows cannot save recursive work.

We therefore checked one actual positive paid boundary, cap973 in #4610 head 20bb93cd990d2ff683e4d0fc56b87a42cd5d3296, literal research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz (SHA256 7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05). Its complete scalar component has roles33/273/548 and six incident paid climbs: 4H1+H2+H3. The three fixed incoming spaces join to973, while the fixed outgoing spaces intersect to973; hence the nested source chart is uniquely forced. Canonical nonnested dimension5/7 replacements each incur an extraH2 mismatch. Their lower invoice excesses are 2*2^theta+4^theta-3^theta-3 and 3*2^theta-4, respectively. Both exceed1.8878 uniformly for theta in[.9966,1], covering C in[.0030217758,.0034]. This is an interval-scoped nonnested comparison, not an all-C claim.

One smallest changed-boundary offer also loses: move973 and its adjacent two-role component1151 to fixed exit chart1147. Counting their shared paid edge15237 once, the complete source bill is7H1+H2+H3. Three literal noncontainments make the offered bill4H3+3H2, with excess at least5.8164 on the same interval. This closes only that literal coupled offer; arbitrary larger coupled charts, changed scalar incidence or transferred transformed resources remain outside it.

Separate normal/optimized source-geometry extraction independently confirms the joins, meets, all nine union edges and the once-counted shared edge. A distinct accounting review confirms the tariff inequalities. These are local read/derive checks, not a newly published replay package or complete native phase/compiler replay. Credit #4453's rank-shape lever, #4610's actual construction, #4919's shared-map theorem, #4841's native chart compiler, and the retained native lineage. The next constructive gate examines actual transformed helper resources with all original jobs preserved; the positive slot count alone is not a physical consumer. No κ improvement follows from these local gates.

Status: bounded centre-carry and boundary/query policies RELEASED after scoped negative tests. Supporting ceilings are published in #4839/#4867/#4888. Priority revision 7. The active changed-operation construction question is now Discussion #4902.

FINDING: the five-FULL/four-main-bank controller with incidence degrees [2,2,3,3] remains unrealized. Its plain optimistic moment is negative for every C>=0.0030217758 through1. Adding one arbitrary retained centre and hypothetically awarding one rank40 job gives only 19.704463096470434..435 at that saving, before wrappers/compiler/outer funding. The stage selfblock CAN be normalized to a fresh-eight-axis Gaussian by actual phase-exact monomial entry/exit maps; the selfblock alone does not force a paid rank8 glue. This positive local identity does not make the centre output autonomous.

The complete #4698 source, including both 379-operation normalization boundaries, has the exact helper224 column helper224 -> helper224 + centre632, with all main and other centre outputs zero in this column. Setting only the last call's fresh helper input nonzero therefore makes bare five-stage retention fail the proposed independent centre job. Including actual helper input normalizers, separate direct returned-helper responses for the last two calls have ranks9 and17. Even granting all other restoration and expenses free, that policy's optimistic margin is [-6.091057701214709,-6.091057701214708] at C4610 and remains negative for every larger C through1. An emitted PARTIAL terminal restores those two helpers at paid profile {9:2,17:2}, four monomial passes and two ADDs; its three-owner noncommuting word and actual raw inverse transpose check exactly. This is not a full five-call controller/raw dual. Bare carry and separate endpoint reads are stopped; shared native corrections or actual earlier paid intermediates remain unexcluded.

A second scoped test inspected exactly one prefix cut, inclusive event11483 after the final centre0 scatter in the 21733-event normalized word. With only physical helper224=e0 nonzero, all132 helper owners currently in empty chart0 have an identically zero free-word column while centre632 has identity1. The centre, nonempty-chart helpers and main fields are excluded observations. The missing native warp chirp was resolved: the actual logical map is K_G=diag(I+G^-T G^-1) perm(G), not bare perm(G). Fresh exact512-column phase checks for18 underlying operators and signed-tableau/absolute-phase checks for36 frame-bound classes authenticate this single-cut zero-observation certificate. It is a derived physical consequence of the source chart commuting diagram and inherited word lift, not an independent raw-array full-prefix simulation or a universal obstruction over other cuts.

These additional carry/prefix checks and independent accounting are local research outputs, not a public standalone package. They prevent an unsupported gain from being claimed; no scientific admission or new kappa follows. Separate gpt-6.1-sol/high agents supplied source construction/accounting and a phase review. The next experiment changes the central controller operation in #4902 instead of extending this losing endpoint policy.

Mandatory compiler gate: retain the #4610 compiler architecture, but bind its quantities to any changed caller. The 54 strict predicates and four functional guards are source-specific; map reused predicates, changed derivations and additional semantic guards explicitly. The proposed five-call centre target's ambient41 versus maximum productive40 does not meet the source's equality premise automatically. #4544 concerns a particular implication in #4483 and cannot silently transfer. Complete endpoint, profitable caller, funded compiler reserve and outer kappa remain distinct milestones. The new two-call candidate uses ambient17=max proposed productive17 but still owes its own binding.

Sources for the completed test distinguish the consumed #4694 snapshot 039a3e4f8906400cd05908ed86b8f9c6866e6113 from its live 2add42d1edf14a263abe90a3bcde67fc1ba30768 head. #4698 is 9079ab66ba9a1717df478a08eb957c20d84d3e61; #4610 is 20bb93cd990d2ff683e4d0fc56b87a42cd5d3296. Fresh overlap reads include #4711 6093fe10aad90178b3c650fed39862c169919f60 (historical two-replica frame-gap model) and #4712 b75a27bf07f1584b20babfddf887493981cd5949 (different changed-owner/dirty-return/query protocols). Their results are mandatory predecessors, not repeated experiments. Targeted searches found no same five-stage single-centre test in their coverage; Discussion pagination and unpublished work remain incomplete. Two dedicated gpt-6.1-sol/high agents separate exact construction from independent accounting.

Correction history: the proposed free entry shear h_j-=R_X[j]X, with a matching exit shear, was rejected by the literal source and an independent AI auditor. The 112 X inputs begin in distinct rank-one charts 2..113; the empty input charts belong to Y. Common full output chart 1 does not make entry mixing free. The earlier pending proposal is superseded by this correction; no free physical data-observation removal was established and no native witness is claimed for it.

The paid direct-query follow-up is now stopped. For selected centres S, let m=|S| and let u be the union size of their actual R_X supports. The declared policy evaluates each affected X inverse frame once, reuses results freely across observations, and optimistically grants all Y repair, pivot/helper restoration, normalization boundaries and other finite work free. Its invoice is u rank-one queries against at most m new rank-eight centre jobs, with no inherited-helper double credit. Thus DeltaM(C)<=m*8^(1-C)-u<=8m-u<0 for every C in [0,1]. This is a policy invoice, not a universal native lower bound.

Selected centres m	Minimum u over all subsets	Best uniform upper bound
1	56	-48
2	88	-72
3	104	-80
4	104	-72
5	112	-72
6	112	-64
7	112	-56
8	112	-48

All 255 nonempty subsets were checked directly from the retained ABI/native payload and independently recomputed. Four-centre subsets split into 14 with u=104 and 56 with u=112. An elementary chronology rewrite also removed 340 scalar/monomial events without exposing paid-call adjacency or changing the recursive profile. Neither finding excludes different native chronology.

A separate shared-producer review identified the principal prior theorem in Ali Romman's #4720, exact head cb102a57254748651ce16aa4454bb611f6380e53: QUERY-SPAN.md already handles arbitrary sequential noncommuting payload queries and proves the actual 112-source line-observation floor at every positive tensor depth. We do not claim that theorem as new. A scoped extension allows fixed common address monomials for free when applied identically to EVERY owner, including final outputs. Replace all queries by identity; their skeleton input rows form L. For original inputs alpha_i e0 with alpha in ker L, induction makes every real query input zero. Skeleton outputs share one final address w. Each actual inverse-frame column has nonzero origin coefficient (1-i)/2 and a nonzero private coefficient (1+i)/2 at its distinct v_i. Private addresses force every affected alpha_i to vanish, with the origin coefficient handling the possible single v_i=w exception. Therefore q>=u, also at positive tensor depth using repeated private addresses.

This extension counts ALL queries of a separately added linear module starting from initial physical X fields and independent arbitrary work inputs. To infer a Gaussian bill, every counted non-common payload operation must be a newly paid positive-rank query. It does not lower-bound additional calls when a full caller reuses previously funded transformed state. Role-dependent native monomials, output-specific address operations and multi-owner native primitives remain outside this grammar. Under these explicit premises the table is also a complete optimistic screen for shared scalar-owner queries. An independent sparse rational circuit evaluator confirmed all 112 origin/private phases and 130,560 subset/common-support cases. The local seven-file package is frozen at 559f49a29a05a8f3201127fa8827fbb132cc438f; it is not advertised as a public artifact. We retain this short proof as a coordination finding rather than submit the prior 112-source theorem again.

Supporting changed-contract ceiling is now published as PR #4867, exact head 5e4cce12a9851fbb46cd7c32cc8177e983771539. It exposes the 184 centre contacts frozen in #4841/#4690 while retaining all 640 endpoint charts, every incident paid edge and all 408 helper jobs; independent centre productive credit is zero. All 188,048 single-contact and 1,834,488 adjacent paired-transvection cases lose, with attained minimum 1+7^theta-8^theta>0 for 0<theta<1. More broadly, every nonempty subset of the 176 empty-chart contacts retagged toward one common one-transvection alternative loses at least 2-2^theta. All 9,198 integer prefix-tariff cuts have feasible flow equal to an attaining cut; the first-prefix empty minimum is unique, so summation by parts proves the strict all-exponent bound. Separate AI review recomputed all 34,830 baseline edge ranks and checked the proof; parent optimized replay also passes. Hosted run 39864815675 passed at exact head 5e4cce12a9851fbb46cd7c32cc8177e983771539, including shared helpers, the sole selected scientific replay and committed-input mutation gate. At this predecessor head, mixed original groups and independent paired axes were outside the ceiling; the following additive result addresses those declared successor grammars. Changed operations remain outside it. Nick Petalas's concurrent cycle-constrained tuple-enumeration claim is credited and distinct; no general neighborhood theorem is claimed anew.

The complete supporting extension is PR #4888, final head b33e3d8670dc604d430e9cbeb86682f258acd52e. It covers all184 contacts choosing old/Ta across nine original-chart groups for all262143 common axes: 419 rank-energy configurations and3771 exact prefix cuts prove nonimprovement for 0<theta<1, with unchanged choices attaining zero. A separate additive theorem closes arbitrary independent distance-one neighbours at both ends of any one of264 actual internal edges, strictly for0<=theta<1:261632 opposite-neighbour cases/44 profiles plus analytic different-hyperplane and all8 rank8-edge bounds. The nominal8.35million rank8 scan was not run. All640 endpoints/408helper targets remain fixed with zero centre credit. Separate read-only audit and registered/export checks pass; Hosted run39871223220 passed at exact final head b33e3d8670dc604d430e9cbeb86682f258acd52e:346 helper tests with11 expected skips, original normal/optimized common-axis checks, independent pair replay and committed-input mutation gate. This is registered automated replay, not independent mathematical acceptance. Larger independent blocks and changed operations/demand remain outside scope.

HANDOFF: reopen with a concrete role-dependent native routing, changed coupled demand, or reuse of already funded caller state that changes this complete invoice. State its actual entry maps, chronological word, raw inverse transpose, ownership-disjoint outputs and phase obligations before assigning credit. These are unexcluded mechanisms, not established savings. The fixed #4610 compiler route and compiler/outer obligations remain required before any new kappa claim.

Question: can an actual #4698 centre/helper chain be consumed affordably while retaining every arbitrary input, data observation, centre-to-Y coupling and helper return? The completed subtask is the explicit strip/correct/restore terminal in PR #4839, exact head cd6ce17acc1eedb9843c6c346b6aa4327673639b. It is author-checked in the inherited native primitive model, with all 640 input columns and the raw inverse transpose verified. Its complete incremental recursive moment is strictly negative throughout C in [0,1]. This is a scoped construction and policy ceiling, not a universal obstruction or new kappa. The completed strip/correct/restore policy remains released.

The next active task is the explicitly changed central-function experiment in #4902. Other meaningful successors must change the shared heterogeneous observation producer or demonstrate an actual chronological chart cancellation. Begin with the physical entry/call/exit word and ownership-disjoint target census, then screen its complete recursive bill before implementation. Unknown routing, precision, stock and compiler costs remain obligations. Do not repeat a larger search of the losing terminal policy.

The single terminal adds 112 H8 + 114 H9 and 113 ADDs for at most one extra H8 job. The full 379+379 helper-basis transformation is charged. All eight singleton screens and 28 pair screens lose under this declared policy; only centre 0 has a fresh complete native-word replay. The direct candidate Q is nonsymmetric, so the homogeneous two-call template cannot simply be assigned to it. A different Q or fully paid heterogeneous consumer remains open.

Package and proof include exact pins, separate old/live #4694 snapshots, OBLIGATIONS.json, the ownership inventory and rejection controls. #4712 dirty-helper/query rejections and #4703 audit are mandatory predecessors; #4711 supplies neighbouring historical-frame work. Selected conditional kappa stays 0.00302015 (#4470); #4610 remains the stronger developed comparison.

Reproduce: python3 -B scripts/research_package.py --package research/center-chain-terminal-feasibility; or python3 -I -O -B research/center-chain-terminal-feasibility/verify.py. The optimized Git-free replay took about 31 seconds on CPython 3.12.3 Linux x86_64. Source-byte and type-sensitive receipt mutations reject. Hosted CI run 39858851124 passed plan and scientific replay on that exact head; full original phase suite, Lean, complete caller, compiler and outer admission were not freshly replayed.

Credit Cinna Davis, Cao Yifan/intqwq, zxv, Jacob Alan Sussman, Dugongue, Ali Romman and the retained physical/native/Clifford/precision/outer contributors. Chris R / Clear1nsight with substantial OpenAI Codex assistance; the initial terminal study had no subagents; the follow-ups below received separate AI-agent checks, with no independent human review. New work Apache-2.0. Live overlap reading was bounded; gh-based pagination was unavailable, and no exhaustive novelty claim is made.

{"schema":"integer-mult-coordination/v1","kind":"RELEASED","task":"complex-new-helper-inventory","scope":"Bare five-stage arbitrary-centre carry, separate endpoint helper reads and one helper-only prefix cut are stopped by scoped exact failures; supporting chart ceilings in PR4888; changed-operation successor in Discussion4902","base":"624de9dc66858af0945e5f8a0942488d62280d43","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/4888"}

[1337isnot1337]

Status update / 2026-10-23: #5137 was archived by maintainers at f2d8399. The new source-bound native repeated-helper budget gate is PR #5147, head8d1a267b708bd30bdd84a3ab47d5a67de7aeb745: even granting full112-prefix capacity and a free H17 third output, the stated repeated-H9-owner policy loses at least16 moment units for all0<=C<1. It does not close distinct-owner or changed-provider callers. Follow-up source/registered/export checks and all11 targeted onboarding tests pass on updated canonicala5df; hosted follow-up checks pending. No new kappa.

DONE / conditional operator result / 2026-10-23 / priority revision 7. PR #5137, exact head f2d83999edccdad625ea686ff3f0d03b7d208965. Claim released; physical acquisition remains open.

For every chain length n>=2, two calls to U_n=I+QJ_n and its actual raw inverse transpose attain exactly r=2 floor((n-1)/2) independent Q jobs with all other arbitrary inputs restored. The package proves the all-odd factorization, a separate all-even weighted cyclic construction, explicit linear-size reversible boundaries, and a matching rank/parity upper bound with arbitrary restored dirty workspace in the declared scalar/global-P ring. The bound excludes broader physical address/monomial operations and isolated nonfaithful small widths; the identities hold at every positive width. The three-field identity remains credited to #4698.

The complete conditional moment is 2H_t+r(2b+1)^t+(n-r)(2b)^t-2P_t. Direct unary acquisition loses at least 2(n-1)-r units and is stopped. The actual native homogeneous producer, source charts, helper autonomy/restoration, physical tariffs/precision, completion, compiler, BIT and outer obligations remain unresolved. No new kappa or native supplier.

CPython3.12.15 normal/optimized and registered replay pass: all527 independent input columns for n=2..32, thirteen active mutations,256 finite dirty-workspace fixtures and faithful-width checks. Exact committed-byte/source-history preflight, official seven-file export/integrity and fresh Git-free optimized replay pass at the published head. Canonical main is 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, selected conditional kappa0.00302015. Final overlap refresh covered318 open PRs and4290 Discussion records with complete pagination; it is not a novelty guarantee. Hosted planning failed eight pre-existing onboarding assertions; the same eight failures reproduce in eleven targeted tests on exact canonical base20a1e31. The hosted package replay was skipped and remains unrun. CI run; the PR records the baseline reproduction.

Next useful task: a named actual five-field offer with E<2 for the positive-rank screen and a positive complete target/paid moment, retaining every independent owner. No larger direct-query construction is justified by this module. Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0. No independent human or kernel acceptance.

Original prospective claim; retained for provenance

CLAIM — optimal two-call extraction from longer homogeneous query chains

Status: prospective algebraic investigation, LOCAL-ONLY; no native supplier or new kappa. Base 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, priority revision 7, selected conditional kappa 0.00302015. Claim expires 2026-10-23T12:40:00Z.

Question: for odd n=2m+1, can two calls to U_n=I+Q J_n and its actual raw inverse transpose extract n-1 independent Q jobs while restoring one arbitrary spectator? I will construct scalar/global-P boundaries (P=Q^2) and prove a matching upper bound in that expressly restricted homogeneous ring model, including arbitrary restored dirty workspace. This extends the conditional three-field identity of Cinna Davis's #4698; it does not reopen the actual heterogeneous native terminal excluded here. Broader physical monomial/address operations remain outside the proposed lower bound.

The first gate is an all-n written factorization and exact cyclic-ring replay at bounded lengths. The second is a complete conditional target/paid ledger: with n transverse rank-b main fields and local excess E, the rank surplus must be (n-1)-2E before compiler/outer funding. An unpriced native embedding will remain unknown, not free. Stop before physical synthesis unless a named provider meets that complete allowance. Expected artifact: a small registered proof/replay package with source pins, an explicit reversible boundary circuit, negative controls and the unresolved physical offer.

Nearest predecessors: #4694/#4698 (coupled centers and homogeneous three-chain), #4712/#4839/#4867/#4888 and this thread (actual helper contamination/heterogeneous terminal restrictions). Center deletion and physical homogeneity are not claimed. Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0.

[1337isnot1337]

FINDING / exact complete budget gate — PR #5147, head 8d1a267b708bd30bdd84a3ab47d5a67de7aeb745. This follows the all-length controller theorem archived in #5137; no new kappa.

The actual one-port H3 prefix uses an original H9 helper as the third chain owner. For the stated policy of reusing112 such owners through two FULLs, two independent H9 rewards per owner must be replaced by the single composed target. Even granting full112-prefix capacity, all homogeneous charts, free boundaries/precision/cleanup, unchanged other helpers, and zero split cost, the complete bare target 224H17+112H16+2(H-112H9) loses at least128 moment units throughout0<=C<1. Granting an unconstructed free H17 third output at every port still leaves a loss of at least16.

The package authenticates the frozen3772-byte native paid/helper receipt, gives all exact min(r,z) cut residuals, and proves the every-exponent conclusion by increasing concavity. Three controls reject an excess margin, double-counted aliased helper rewards and omitted paid centers. Source counts are reused explicitly; this is not a fresh native replay or an actual112-prefix construction.

Decision: stop expanding prefix capacity for this full repeated-H9-owner policy. Distinct-helper retention, changed productive census/provider, other stage/axis counts, and broader routing remain different open tasks. The source H3-prefix offer and phase-bound normalizers retain Chris R/Clear1nsight's credit; native/source and the earlier conditional controller retain Cao Yifan/intqwq, Cinna Davis, zxv and the existing lineage.

Registered/direct normal/O, committed-byte/source-history checks, eight-file export/integrity and fresh Git-free optimized replay pass. All11 targeted onboarding tests now pass on refreshed canonicala5df48c after its fixture fixes; older #5137 CI failure and its baseline reproduction remain preserved. Hosted #5147 checks are pending. Selected kappa0.00302015; priority7. Author-checked with substantial OpenAI Codex assistance, Apache-2.0; no human/kernel acceptance or frontier promotion.