research: classify the full 4080-owner centre kernel using sheaves and integer fibres

[Clear1nsight]

Summary

This contribution extends the centre-path-sharing investigation in PR #3147 and resolves the fixed-subgroup full-kernel question left open by PR #3656.

Using cellular sheaf theory, finite-group geometry, exact linear algebra and algebraic statistics, it proves that the complete common-observation kernel on the specified 4,080-owner subgroup is exactly one-dimensional, even when all 112 ports are included.

The unique primitive kernel generator is the previously established (1, -1, -1) relation on ports (0, 6, 85), repeated across all 4,080 owners.

The new contribution is the complete classification of the kernel, rather than another example of a valid cancellation.

Status: Completed scoped mathematical result with independently implemented source-bound checks.

Scope: Exact scalar information theorem. No new multiplication exponent, physical supplier or recursive cost reduction is claimed.

1. Mathematical contribution

1.1 Full common-kernel classification

Let (R) be the selected (8 \times 112) rational centre-observation matrix and let (\rho_p) denote the corresponding involutory owner routes.

Consider the subgroup

[
H = \langle \rho_0\rho_6,\rho_0\rho_{85}\rangle,
\qquad |H|=4080.
]

Allow arbitrary rational coefficients on all 112 ports at every owner in (H), with zero input outside (H).

Define the earlier and later observations by

[
Sx(g)=\sum_p R_p x(g,p),
]

[
4Tx(z)=\sum_p R_p x(z\rho_p,p).
]

Every touched later-output owner is retained, including those outside the initial subgroup.

The principal result is

[
\boxed{\ker(S)\cap\ker(T)=\mathbb{Q}v},
]

where (v) has coefficients (1, -1, -1) on ports (0, 6, 85) at every owner in (H), and zero elsewhere.

Consequently, the witness constructed in PR #3656 spans the entire common kernel on this block. No additional independent invisible directions exist under the specified observation contract.

1.2 Cellular sheaf reduction

The proof partitions the 112 ports into outgoing right-coset classes.

The exact classification contains:

• 109 singleton classes, each with an independent nonzero observation column.
• One three-port class (0, 6, 85), whose observation columns have rank two.
• A connected compatibility graph with 4,080 vertices, 8,160 edges and a certified 4,079-edge spanning tree.

The singleton output equations eliminate 444,720 source coordinates.

The remaining three-port relation supplies one scalar amplitude at each owner. Compatibility across the connected graph forces these amplitudes to agree.

In cellular sheaf terminology, the reduced system is a connected constant sheaf with a one-dimensional global-section space.

The reduction replaces an otherwise substantial global kernel calculation with exact local rank conditions and a finite connectivity certificate.

1.3 Sharp information dimension

The resulting dimensions are:

Quantity	Exact value
Initial scalar coordinates	456,960
Rank of the earlier observation map	32,640
Kernel dimension of the earlier observation map	424,320
Dimension of the full common kernel	1
Additional linear observations required	424,319

Therefore,

[
\operatorname{rank}(T|_{\ker S})=424319.
]

At least 424,319 additional scalar linear observations are required to reconstruct the entire later-query output from the earlier observations. This lower bound permits an otherwise nonlinear decoder, but assumes the additional observations are linear.

The bound is achievable as an abstract linear quotient. It is not an efficient physical decoder or a recursive tariff.

For every proper subset (B\subsetneq H), the common kernel is trivial, and the corresponding information requirement is

[
104|B|
]

additional scalar linear observations.

The proper-subblock conclusion also applies to left translates of (H). It does not establish a lower bound for arbitrary owner sets of the same size or resolve the global minimum-support problem.

2. Algebraic statistics and integer fibres

The same result is interpreted through the integer kernel of the combined observation matrix.

Let (A) be the integer matrix obtained by stacking (S) and (4T).

Then

[
\ker_{\mathbb Z}(A)=\mathbb Zv.
]

Every nonempty nonnegative integer fibre has the form

[
{n_0+tv:t\in\mathbb Z,\ L\le t\le U}.
]

Thus a single primitive move, together with its reverse, connects every nonempty fibre under this exact coefficient-observation model.

The associated toric ideal is principal.

However, this move has:

• 4,080 positive entries.
• 8,160 negative entries.
• Total support of 12,240 coordinates.

Its total coefficient change is

[
\sum_i v_i=-4080.
]

Consequently, adding a fixed total-count constraint destroys the kernel freedom.

This distinction is significant: an information-preserving coefficient transformation is not automatically a count-preserving or physically legal scheduling transformation.

The result does not exclude physical programs that temporarily leave the observation fibre and subsequently restore the required state.

3. Task alignment and prior research

Repository task: centre-path-sharing

Priority revision: 6

Scope status: Changed, with a documented justification.

Canonical source base:

20ecf2891d80b5acf18d980b2164fb44ea7faa90

Contribution commit:

37091843c18491abda68b465ad768180c0176a03

The original task investigates whether the selected centre path, which charges eight rank-eight copied totals per call, can be shared or reduced.

The investigation is narrowed to the outstanding information-theoretic question in PR #3656: whether its explicit common-kernel witness spans the full kernel or merely represents one member of a larger family.

This result complements the following work:

• PR research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147: Centre sink duality, route relations and dependent-triple analysis.
• PR Research: route-realized centre common-kernel vector on 4080 owners (block floor range 15..4080; no κ) #3656: Construction of the original 4,080-owner common-kernel witness.
• PR Prove sharp selected-centre repair dimension for owner blocks up to six #3071: Selected centre block-repair observation model.
• PR Repair centre block proof with a sharp fifteen-owner graph floor #3291: Correction and strengthening of the centre block-repair graph bound.
• PR Independently check the seven-owner centre-repair Fano exclusion #3168: Independent route-relation and incidence verification.
• PR test(research): audit centre-circle demands and late responses #3151 and PR Prove native F2 innovation beyond the earlier centre history #3194: Separate chronological and later-observation obstructions to simple centre cancellation.

The contribution does not reintroduce the earlier circle-elision proposal or claim that the resulting scalar information reduction removes a physical recursive call.

4. Verification and reproducibility

The mathematical result was verified against the pinned selected source archive using two separately implemented approaches.

Primary verification

The primary checker uses:

• Authenticated source extraction from the selected archive.
• Exact rational matrix arithmetic.
• Binary column-matrix representations of the physical routes.
• Exhaustive outgoing coset comparisons for all 112 ports.
• Exact class-rank and subgroup-connectivity calculations.

Independent implementation

The second implementation:

• Parses the pinned archive independently.
• Reconstructs the source coefficients through forward propagation.
• Uses binary row matrices and a different group-closure implementation.
• Establishes ranks using nonzero modular minors.
• Checks the complete routed witness residuals directly.
• Independently verifies compatibility-graph connectivity.

Both implementations reproduce the same source-derived geometry and common-kernel classification.

Additional controls include:

• Eight small-group matrix, proper-subblock and integer-fibre tests.
• Normal and optimised Python execution.
• Eight typed-receipt and malformed-JSON rejection controls.
• Exact committed-result comparison.
• Detection of three nonzero later-output failures after deleting one witness owner.
• Verification that committed inputs remain unchanged.

Hosted verification

Upstream GitHub Actions: PASSED

https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39171350386

The complete registered replay also passed on the same scientific source tree in the contributor fork:

https://github.com/Clear1nsight/integer-mult-bounds/actions/runs/39167870503

Environment: Ubuntu 24.04.5, CPython 3.12.15.

Canonical result SHA-256:

9e23c75d59502748f930e3bc029d528c684224963113b7fa1d2f46357ae381b9

Reproduction command

From the repository root:

python3.12 -I -B scripts/research_package.py \
  --package research/centre-sheaf-fibre

The package contains the complete written proof, implementation, independent checker, deterministic result, source bindings and executable controls.

No full selected cold/native replay, formal proof-assistant verification or independent human acceptance is claimed.

5. Scope, limitations and research decision

Decision: NARROW / COMPLETE for the specified fixed-block information question.

The result proves that the 4,080-owner subgroup contains exactly one independent common-kernel direction under the full selected calls-0/4 scalar observation contract.

This closes the search for additional invisible directions inside that specific block.

However, it does not establish:

• A physical centre-sharing circuit.
• A reduction in the paid recursive-child histogram.
• A legal native acquisition or return program.
• Physical routing, ownership or hosting feasibility.
• Arbitrary dirty-state restoration.
• Finite precision and storage admission.
• A complete recursive supplier invoice.
• A new multiplication exponent.

The selected frontier remains unchanged.

The distinction between abstract information dimensions and actual recursive costs is maintained throughout the proof and implementation.

6. Open next questions

A. Larger or differently structured owner blocks

Can a specified union of distinct subgroup cosets create additional dependent outgoing classes and a materially larger common kernel?

The first step is to define the expanded owner set, construct its exact outgoing coset-incidence structure and compute the resulting compatibility ranks.

Any promising construction must then satisfy the existing family expansion and complete-cost requirements.

B. A weaker observation contract tied to an actual consumer

Does an identifiable downstream consumer require less information than the entire later-query field?

A useful next contribution must specify the exact required outputs, allowed source access, arbitrary dirty inputs and restoration requirements before deriving a reduced observation map.

C. Resource-preserving integer transformations

Can an information-preserving primitive move also preserve the actual operation-count, ownership and chronological resource constraints of a changed caller?

The first step is to specify those resource equations and test whether the proposed integer-kernel moves preserve them.

The current fixed-count constraint does not permit the known primitive move.

7. Attribution and references

This contribution preserves the following predecessor credit:

• Matarener, PR Research: route-realized centre common-kernel vector on 4080 owners (block floor range 15..4080; no κ) #3656: Original 4,080-owner subgroup and explicit common-kernel witness.
• David Charbonneau, PR research: centre sink duality, dependent-triple braid sparsity (|B| <= 12), and circle-cancellation sharing ceilings #3147: Centre-sharing investigation, route structure and dependent-triple analysis.
• Ross Richardson, PR Prove sharp selected-centre repair dimension for owner blocks up to six #3071: Centre block-repair observation model and source extraction.
• Sagar Thakrar, PR Repair centre block proof with a sharp fifteen-owner graph floor #3291: Corrected graph argument and strengthened information lower bound.
• Sudarsh Chaturvedi, PR Independently check the seven-owner centre-repair Fano exclusion #3168: Independent route and incidence checks.
• Gonzalo Mier, PR Research: rule out seven-owner centre information compression #3097: Earlier owner-support and incidence work.
• Aaron Wang/navinor and ywz65: Selected centre observation and sharing lineage.
• PR test(research): audit centre-circle demands and late responses #3151 and PR Prove native F2 innovation beyond the earlier centre history #3194: Distinct chronological and centre-history obstructions.

The retained selected-construction, native-geometry, compiler and supplier contributors retain their existing attribution and notices.

Mathematical foundations

Hansen, J., and Ghrist, R. (2019). Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3, 315–358.

Diaconis, P., and Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions. Annals of Statistics, 26(1), 363–397.

These are established mathematical theories used to formulate the result; no claim is made to have invented cellular sheaf theory, toric ideals or Markov bases.

Development disclosure

Prepared by Clear1nsight with substantial OpenAI assistance in mathematical development, implementation, testing and documentation.

The separately implemented verification approaches share the same assisted author and mathematical source. They are not represented as independent human or formal proof-kernel review.

License: Apache-2.0.

All changes are isolated to research/centre-sheaf-fibre/. Frozen predecessor inputs, source notices and the selected frontier remain unchanged.