research: bound causal centre sharing and audit selected family coverage

[Clear1nsight]

Summary

This contribution addresses two research priorities in the selected integer multiplication construction:

1. Centre-path sharing: Establishes an exact lower bound against uniform causal linear scalar-oracle sharing.
2. Coverage-law change: Verifies the selected physical-family coverage ledger and identifies the additional endpoint work that must change to realise two unpaid coordinates per family.

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: 20ecf2891d80b5acf18d980b2164fb44ea7faa90

Priority revision: 6

Contribution head: fe75e9f5a2f9b99f02e7cae42ac2123dd57c0313

---

1. Mathematical contribution: Causal centre sharing

Task: centre-path-sharing

Package: 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 recursive-call lower bound for a single known native Fourier operator.
• An impossibility result for operator-specific algebraic identities.
• A lower bound against nonlinear or adaptive algorithms.
• A complete physical centre-sharing construction.
• A changed paid recursive-child histogram.
• An improvement in the selected multiplication exponent.

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-change

Package: 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:

Quantity	Exact value
Physical families	14,336
Unpaid coordinates per family	1
Ordinary paid mass	1,032,192
Private-centre paid mass	9,216
Total selected paid mass	1,041,408
Hypothetical two-unpaid paid mass	1,027,072
Additional ordinary advances requiring removal	14,336

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:

1. Whether the aggregate resource histogram admits a favourable integer partition.
2. Whether the existing physical-family endpoints satisfy the proposed allocation.
3. Whether a changed construction can realise the allocation with a valid execution programme.

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:

• Reducing the significant centre-path information and resource cost.
• Changing the family coverage law to increase available unpaid coordinates.

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:

• Exact source and response rank certification over two primes.
• Complete causal feedback reconstruction.
• Source-dependent derivative and discrete secant calculations.
• Independent literal scalar execution comparisons.
• Full checks of 1,152 innovation cuts and 14,336 output coordinates.
• Tests detecting omitted causal feedback.
• Rejection of invalid causal cycles.
• Generic model-discrimination and corruption controls.
• Normal and optimised Python execution.

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:

python3.12 -I -B scripts/research_package.py \
  --package research/centre-causal-tangent

Selected family coverage

Verification includes:

• Exact selected-source archive authentication.
• Reconstruction of the complete family ledger.
• Checks of every physical-family endpoint.
• Verification of ordinary and private-centre paid mass.
• Independent counting from family capacity and actual advances.
• Pooling and aggregate-partition controls.
• Failure controls for transferred, deleted and double-counted resources.
• Normal and optimised Python execution.
• Immutable input and result-record checks.

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:

python3.12 -I -B scripts/research_package.py \
  --package research/selected-family-coverage-boundary

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 questions

Both 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:

• PR Certify composed scalar centre-source rank for a selected call pair #1918 and Negative result: calls-only centre-source rank census across all 144 selected calls (PR2048 Q2; no new kappa) #2199: Earlier centre and cost investigations.
• PR Certify complete projected centre-error endpoint response #2622 and Audit centre-source coefficients with an independent adjoint and sign control #2763: Source/response ranks and bridge-source auditing.
• PR Research: certify terminal X/Y readout kernels for selected centre errors #2799 and Research: transport terminal centre-error kernels through causal feedback #2814: Chronological structure and causal feedback.
• PR research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147: Centre-sharing investigation.
• PR Research: route-realized centre common-kernel vector on 4080 owners (block floor range 15..4080; no κ) #3656 and Prove the exact repair minimum for one centre block #3797: Fixed-block common-kernel investigations.
• PR research: classify the full 4080-owner centre kernel using sheaves and integer fibres #3828: Complete fixed-subgroup common-kernel classification.

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:

• PR Research note: restoration deferral is mass-neutral in the hosted T37 recurrence (no new kappa) #1012: Original family coverage identity.
• PR Research: near-ceiling idle packing stability and exact integer optimum #1685: Earlier integer optimisation.
• PR Audit idle integrality before per-call normalization #3747: Relevant resource-accounting distinctions.
• PR research: pool the replica idle block; the two-unpaid premise is per-call-forbidden only #3754: Two-unpaid pooled counterfactual.
• PR Cap line-port replicas at one unpaid coordinate per family #3561 and Account for both native payloads in paired-plane identity blocks #2203: Distinct restricted endpoint and payload obstructions.

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.

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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.