Structural results for the sphenic dyadic-window complex

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 b_2 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.