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research: bound causal centre sharing and audit selected family coverage - #3873
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Separate parameterised scalar oracle rank from fixed native operator identities. Preserve selected inputs and frontend; draft source-dependent replay pending.
…erage Audit the actual selected frame/owner/idle ledger. Separate legitimate pooling from a changed endpoint and distinguish private centre mass from ordinary unpaid credits. Source-dependent draft checks pending.
Freeze the verified derivative receipt and add a strict causal feedback and secant certificate. The stronger gate concerns a programme uniform at scalar oracles +1 and -1; a single fixed native transform remains outside scope.
Retain the actual selected per-family ledger, distinguish pooled arithmetic from a second unpaid ordinary coordinate, and add real Git-free and command-line corruption controls. All original inputs remain unchanged.
Record both source-bound rank certificates and the exact causal secant receipt. Add finite two-point controls and Git-free command-line corruption checks. The uniform scalar-oracle conclusion does not claim a fixed native Fourier lower bound.
…e-boundary-20ec research: bind the two-unpaid premise to selected family coverage
…ngent-20ec research: test a first-variation gate for causal centre sharing
Clear1nsight
changed the title
Resolve RUNNER_TEMP at step scope and retain the existing manual trigger, repository gate, replay commands and pass criteria. Prepared with substantial OpenAI assistance.
This was referenced Oct 19, 2026
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Summary
This contribution addresses two research priorities in the selected integer multiplication construction:
Both results are source-bound investigations of existing construction bottlenecks. They distinguish mathematically valid relaxations from physically admissible improvements.
Status: Two completed scoped research results, with remaining native-construction obligations explicitly identified.
Scope: Mathematical analysis, exact verification and reproducible audits. No new multiplication exponent, physical supplier, recursive tariff improvement or selected-frontier change is claimed.
Canonical base:
20ecf2891d80b5acf18d980b2164fb44ea7faa90Priority revision: 6
Contribution head:
fe75e9f5a2f9b99f02e7cae42ac2123dd57c03131. Mathematical contribution: Causal centre sharing
Task:
centre-path-sharingPackage:
research/centre-causal-tangent/Research question
Can the selected centre path be shared across calls through causal recoding, reducing the required centre operations without losing information needed by the existing decoder?
Result
The investigation replaces centre operations in the complete selected scalar projection with a common scalar parameter.
For the resulting boundary-transfer operator F(z), the exact source-derived ranks are:
[
\operatorname{rank} F'(1) = 1152
]
and
[
\operatorname{rank}(F(-1)-F(1)) = 1152.
]
The derivative result constrains infinitesimal variation near the identity.
The discrete secant result establishes the stronger conclusion that a finite, fixed causal linear circuit required to work for both scalar-oracle choices +1 and -1 must invoke the rank-one oracle at least 1,152 times.
Consequently, a uniform 576-query replacement cannot reproduce the required boundary response within this model.
The result permits arbitrary fixed linear recoding, finite workspace and singular clearing operations.
Mathematical significance
The analysis establishes that generic causal linear recoding cannot halve the centre-oracle count under the specified uniform scalar contract.
The argument uses a time-expanded causal representation, exact source and response ranks, and a discrete operator-difference factorisation.
The source and response maps each have rank 1,152, while the chronological feedback operator is strictly lower triangular.
This distinguishes information-theoretic restrictions on generic sharing from opportunities that depend on the particular algebraic structure of the selected native operators.
Scope and limitations
The bound applies to the specified scalar-oracle model.
It does not establish:
A fixed native operator may admit identities unavailable to a circuit required to operate uniformly over distinct scalar choices.
The native sharing problem therefore remains open.
2. Mathematical contribution: Selected family coverage
Task:
coverage-law-changePackage:
research/selected-family-coverage-boundary/Research question
Can the existing selected construction support two unpaid coordinates per live family through pooling, changed accounting or a modified coverage allocation?
Result
An exact audit of the selected source establishes that every one of the 14,336 physical families has exactly one unpaid coordinate.
The audited resource quantities are:
The selected physical-family identity is:
[
8k_f+h_f+i_f+1=73.
]
This identity is inherited from earlier work and is independently rebound to the selected original-source ledger.
The audit demonstrates that regrouping or pooling existing endpoint rows does not change the number of unpaid coordinates associated with each physical family.
A hypothetical two-unpaid allocation can be arithmetically consistent at the aggregate level while failing the actual local endpoint obligations.
Mathematical significance
The result separates three distinct questions:
The first does not establish the second or third.
Under the unchanged selected endpoint contract, obtaining a second unpaid coordinate per family requires removing or replacing 14,336 ordinary coordinate advances.
The private-centre charge is accounted for separately to prevent double-counting potential savings.
Scope and limitations
This is a source-bound conservation and coverage audit of the existing selected construction.
It does not prove that an alternative decoder, changed endpoint contract or new native primitive cannot provide two unpaid coordinates.
A successful future construction must identify the changed physical operations, preserve the required endpoint semantics and charge all associated costs.
No replacement decoder or physically executable two-unpaid construction is supplied.
3. Task alignment and research scope
Both contributions address priority revision 6 of the repository's research guidance.
The original tasks investigate:
The research scopes have been changed and narrowed to exact questions that can be answered against the selected source.
For centre sharing, the result establishes a uniform scalar-oracle obstruction rather than constructing a cheaper native circuit.
For family coverage, the result establishes the limitations of the unchanged selected endpoint ledger rather than proposing a replacement decoder.
These scope changes are intentional and recorded in each package's
TASK.json.Both investigations terminate at their respective mathematical and source-bound conclusions rather than extending speculative constructions without the necessary physical and resource evidence.
The selected frontier, frozen inputs and existing construction are unchanged.
4. Verification and reproducibility
Both packages provide mathematical arguments, source bindings, exact result records, executable verification and failure-mode controls.
Centre causal tangent
Verification includes:
The selected scalar execution contains 144 calls, 1,152 centre occurrences and 14,336 ordinary outputs.
The source-dependent literal executions each process 2,835,792 scalar instructions.
Earlier hosted source-dependent checks passed on Ubuntu 24.04.5 with CPython 3.12.15.
https://github.com/Clear1nsight/integer-mult-bounds/actions/runs/39177162360
Reproduction:
Selected family coverage
Verification includes:
Earlier hosted source-dependent checks passed on Ubuntu 24.04.5 with CPython 3.12.15.
https://github.com/Clear1nsight/integer-mult-bounds/actions/runs/39176163645
Reproduction:
Current upstream validation status
The upstream workflow for this PR has completed with a failure:
https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39181557809
The planning job passed, but both registered package replays stopped during package validation because the README files lack the exact required heading:
## Open next questionsBoth README files contain corresponding next-question material under differently worded headings.
The heading discrepancy requires correction and a new exact-head replay before upstream validation can be marked successful.
The earlier successful fork runs support the source-dependent results at their recorded commits. They are not represented as successful verification of the present PR head.
No independent human review, formal proof-assistant verification, complete selected native cold replay or new physical supplier validation is claimed.
5. Relationship to existing research
Centre-path sharing
This work builds on the source and response geometry and causal scheduling established in earlier contributions.
Relevant predecessors include:
The present contribution is distinct from PR #3828.
PR #3828 studies the exact common kernel of specified earlier and later observation maps over a fixed owner subgroup.
The centre-causal-tangent package instead studies the boundary response of a complete selected scalar projection under causal oracle replacement.
These are different mathematical contracts and neither result is presented as superseding the other.
Family coverage
This work builds on:
The contribution does not claim to have discovered the original coverage identity or the pooled integer allocation.
Its additional value is the selected-source verification of the unchanged physical-family endpoint contract and the identification of the precise ordinary-work deficit.
Predecessor mathematical results, source contributions and attribution are retained in the package documentation.
6. Open next questions
A. Native operator-specific centre sharing
Can an identity specific to the actual fixed native operators permit cheaper centre sharing than the uniform scalar-oracle model?
A useful first step is to identify one explicit operator identity and specify its required source states, output states, frame interval and dirty-input restoration conditions.
The candidate must then be evaluated against the complete recursive resource budget, including acquisition, conversion, restoration, ownership, hosting and precision costs.
The present work supplies no qualifying native identity.
B. Alternative family endpoint contracts
Can a changed decoder, endpoint obligation or native primitive legitimately remove an additional ordinary coordinate advance per family?
A useful first step is to construct one complete small instance with explicit frame/owner rows and verify that the changed endpoint semantics hold for arbitrary dirty inputs.
Any favourable construction must be extended to the selected family structure and evaluated using a fully charged replacement budget.
The present work supplies no physically admissible replacement endpoint.
C. Integration with the selected resource model
Can a validated native change in either research stream produce a favourable complete recursive cost improvement?
Future work must connect local mathematical savings to the actual selected recursive-child histogram and relevant construction constraints.
A favourable abstract relaxation alone is insufficient.
7. Research decision
Centre sharing: NARROW / COMPLETE for the uniform causal linear scalar-oracle question.
Family coverage: NARROW / COMPLETE for the unchanged selected physical-family ledger.
The contributions establish useful restrictions on two proposed improvement pathways.
Neither proves that the corresponding native construction problem is impossible.
Further exploration is justified only when a concrete changed native operation, endpoint contract or operator-specific identity is available for verification and complete cost evaluation.
Selected frontier: Unchanged.
New multiplication exponent: None claimed.
Physical recursive-cost improvement: None claimed.
8. Attribution and development disclosure
Prepared by Clear1nsight with substantial OpenAI assistance in mathematical development, implementation, verification and documentation.
The separately implemented checks share the same underlying selected source and assisted development context. They are not represented as independent human verification or formal proof-kernel acceptance.
All new contributions are provided under Apache-2.0, with predecessor attribution and applicable source notices retained.
The contribution preserves frozen source archives, existing construction records, selected-frontier metadata and unrelated research packages.