A rule on the number of cells
An arXiv preprint reports a strict arithmetic rule for a class of mathematical partitions: the number of cells, or depth, must be a power of the prime p. The result addresses whether every possible depth K is forced into that prime-power form, without regularity or cell-symmetry assumptions.
The object is a p-ary bent partition into K nonempty cells. Its defining rule is that every balanced assignment of those cells to the field F_p must produce a bent function. A balanced coarsening groups the fine labels so that each coarse value receives the same number of labels. The conclusion is a necessary condition, not a claim that every prime-power depth can actually be built.
What the shifts reveal
The central structural result concerns translations of the underlying finite space. For every nonzero translation, the number of points whose fine label stays the same after the shift is exactly the total domain size divided by K. That identity identifies the fine cells as a partitioned difference family and the label map as zero-difference balanced, with a common parameter.
The proof reaches this result by averaging exactly over balanced coarsenings. For two distinct fine labels, a uniformly chosen coarsening makes them share a coarse value with probability equal to the ratio of m minus one to K minus one, where m is the number of fine labels assigned to each coarse value and K is the total number of fine labels. The same averaging gives an equal count of points whose coarse label does not change under the shift, allowing the argument to recover the exact fine-level count.
A tighter bound from dimension
The arithmetic consequence is restrictive. The depth is a power of p with an integer exponent t running from 1 through n, where n is the dimension of the underlying space. Because the cells are nonempty, the exponent is strictly below n, so the depth is smaller than the full domain size.
A further dimension-sensitive result narrows the range to the floor of half the dimension. The even-dimensional refinement is handled through a cell-size divisibility input, while the only odd-dimensional parameter pair stated in the paper is p equal to 3 and K equal to 3. Those classical dimensional restrictions are inputs to the argument, not results reproved by its Lean formalization.
Further consequences
The fine count also carries information about cell sizes. It yields exact moment identities, including a centered second-moment relation. After the labels are relabelled as a vector space over F_p, every nonzero linear component is bent, and the resulting vector derivatives obey the paper's full uniform-count law.
The manuscript also gives a finite exclusion test. In the displayed certificate, five tests are enough to rule out depth six by parity. The certificate adds a specific exclusion to the broader prime-power and dimension restrictions.
A separate transition-measurement argument reports a symmetric trace-zero subspace of dimension K minus one.
A theorem with a clear boundary
Selected parts of the work were formalized in Lean, including the balanced-fusion lemma, the fine same-label count and its arithmetic consequences, the cell-size moment, the five-test certificate and the transition obstruction. The manuscript says replay instructions, source-level scope notes, axiom audits and cryptographic hashes accompany it.
The paper is explicit about what remains unresolved. It establishes no converse: fine zero-difference-balanced data alone do not recover the universal bent-coarsening axiom, and prime-power depth is only necessary for existence. In other words, the theorem filters the possible parameters but does not decide which permitted depths are realizable.
Paper data and sources
Original title: Fine Difference Structure and Prime-Power Depth of Bent Partitions
Authors: Zhaorui Wu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text