bibkey: chervov2026cayleypy4 authors: A. Chervov and others year: 2026 title: ‘CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks’ doi: 10.48550/arXiv.2603.22195 url: https://arxiv.org/abs/2603.22195v1 claim: “The paper reports diameter formulas, found with the CayleyPy library, for Cayley and Schreier graphs of symmetric groups; Conjecture 16 gives the diameters of the inverse-closed Schreier coset graphs S_n / (S_l x S_(n-l)) generated by consecutive k-cycles for k = 3, 4, 5, and its k = 3 clause is the ceiling of L(N - L)/2; Conjecture 15 gives the diameters without inverses as quasipolynomials in n and t = n - (l + 1), with k = 5 clause L floor((t + 1)/4) + [(t + 1) = 0 mod 4] for all L >= 4 and N >= L + 9; Conjecture 11 gives the consecutive-four-cycle inverse-closed Cayley diameter n(n-1)/6 - 1 in residues zero and one modulo three and n(n-1)/6 + 2/3 in residue two, for all n >= 6.” strata_touched:

• D5/S3/Combinatorics/ShrunkenGrassmannianThreeCycleDiameter
• D5/S3/Combinatorics/ShrunkenGrassmannianConjectureSixteenRefutation
• D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation
• D5/S0/CayleyGrowth/ConsecutiveFourCycleDiameterRefutation license: citation-only triage: anchor

CayleyPy-4: AI-Holography

The fourth CayleyPy paper collects computational conjectures on diameters and growth of Cayley and Schreier graphs of symmetric groups. One subsection studies the Schreier coset graph S_n / (S_l × S_{n−l}), called a “k-shrunken” Grassmannian. Its vertices are the binary vectors with exactly l zeros and n − l ones, CayleyPy’s central state is [0]^l + [1]^{n−l}, and the generators are the consecutive k-cycles (i, i + 1, …, i + k − 1) for i = 0, …, n − k, taken for n > k. Conjecture 15 treats generators without their inverses; Conjecture 16 treats the inverse-closed case:

For $k=3$ from $L\ge 2$(?): $D(L,N)=\lceil \frac{L(N-L)}{2}\rceil$

followed by clauses for k = 4 and k = 5. The k = 5 clause of Conjecture 15 reads:

For $k=5$ with period~$4$ (again $t=n-(l+1)$, for all $L\ge 4$, starting from $t+1\ge 9$ that is, $N\ge L+9$): $D(L,N)=L\lfloor \frac{t+1}{4}\rfloor +1_{(t+1)\equiv 0(\mathrm{mod}4)}$.

The source motivates the leading term l(n − l)/(k − 1) by comparison with the neighbour-transposition case k = 2, where the diameter is l(n − l).

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2603.22195
• URL: https://arxiv.org/abs/2603.22195v1
• Version and location: arXiv:2603.22195v1 (2026-03-23), the only version listed by the arXiv API on 2026-09-26; source file main_holography.tex, subsection “Schreier coset graph: S_n / (S_l × S_{n−l}) (“k-shrunken” Grassmannian Gr(l,n,k))“. The environment Conj has its own counter, not reset by section; the inverse-closed conjecture is its sixteenth instance, Conjecture 16, and the conjecture without inverses (line 5650) is its fifteenth, Conjecture 15.

Consecutive four-cycle Cayley diameter

Version 1, section 8.1 defines nonwrapped consecutive cycles (i,i+1,...,i+k-1), 0 <= i <= n-k. Section 8.5 specifies the Cayley graph, not a Schreier graph, with inverse-closed generators. On printed page 57, Conjecture 11 states, for k=4 and every n>=6,

$$D_4(n)=\{\begin{array}{ll}\frac{n(n-1)}{6}-1,& n\equiv 0,1(\mathrm{mod}3),\\ \frac{n(n-1)}{6}+\frac{2}{3},& n\equiv 2(\mathrm{mod}3).\end{array}$$

The original PDF SHA-256 is 3dd686df0fde2af02dbb7936ba2d525a9b7f41edbd21243b4884c859484b76db. Theorem 5 of the same version gives the published bound D_k(n) >= ceil((n(n-1)-2)/(2(k-1))), which at (n,k)=(6,4) gives five against the conjectured four. Its accompanying parity/subgroup sentence has the odd/even cases reversed; the consecutive four-cycles are odd permutations and the full symmetric-group graph is the target.

ConsecutiveFourCycleDiameterRefutation.result refutes the complete printed four-cycle clause by a native finite permutation certificate and a graph-distance bridge. It neither claims that the lower bound or method is new nor gives an exact diameter or a corrected formula. This target does not settle the older CayleyPy Growth Conjecture 14(1), whose prose prints inconsistent generator indices.