What question did this study set out to answer?

The aim is to establish structural properties of the sphenic dyadic-window complex using various mathematical results.

July 9, 2026Open Access

Structural results for the sphenic dyadic-window complex

Key Points

The aim is to establish structural properties of the sphenic dyadic-window complex using various mathematical results.
Proving a kernel-pivot theorem about minimal cells and boundaries.
Establishing a counting identity that connects fitting counts to arithmetic quantities.
Verifying results using machine-generated proofs and maintaining records.
A fitting count's structure reveals a discrepancy with the unsigned boundary by the collision deficiency delta.
The first Betti number's properties are outlined, contributing to the overall Betti profile.
A canonical witness for bulk colliding cells leads to computable results using a local completer rule and finite census.

Abstract

We prove four interlocking structural results for the sphenic dyadic-window complex X (N) (vertices the primes; 2-cells the squarefree products pqr of three distinct primes in (N/2, N]). (1) A kernel-pivot theorem: over every field, the (sum, lex) -minimal cell of anycycle is fitting, and fitting cells carry explicit radius-one witnesses; under the signedboundary b₂ equals the fitting count, while the auxiliary unsigned boundary falls short bythe collision deficiency delta, the two boundaries kept rigorously distinct. (2) A countingidentity equating the fitting count with a purely arithmetic quantity elem (N), making theBetti profile computable by counting. (3) The first Betti number's structure. (4) The striptheorem: for bulk colliding cells the canonical witness is dominance-up, the decidingrecursion confined to a strip that never approaches the near-diagonal; consequentlyfreeness is decided by a single local completer rule and delta (N) is computable byarithmetic plus a finite per-N census. All results are machine-verified with frozen, SHA-pinned scripts; adjudication records and claim identifiers (C-0748, C-0749, withinputs C-0743, C-0745, C-0747) accompany the lab archive. Draft of 7 July 2026.

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