Conjecture 15 of CayleyPy-4 fails at k = 5, L = 4, N = 13

Abstract

The k = 5 clause of Conjecture 15 of CayleyPy-4 is false: without inverses, rotations of five consecutive letters need at least 9 moves to turn 0000111111111 into its reversal, whichever way the rotation acts, while the clause gives 8.

Definition 1.1 (Directed moves).

$$\mathrm{StepDir}\left(left,k,x,y\right)\iff (\exists i\in \mathbb{N},i+k\le \mathrm{length}\left(x\right)\wedge y=(\mathrm{if}left\mathrm{then}\mathrm{rotL}\left(k,i,x\right)\mathrm{else}\mathrm{rotR}\left(k,i,x\right)))$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.StepDir (✓ std3).

Citation. A. Chervov and others (2026). 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.

Commentary.

Without inverses a move applies the cycle on a window of k consecutive positions, which rotates the window one place left or one place right depending on the convention for the action of a permutation on a word; the flag left selects the convention.

Definition 1.2 (Directed reachability within m moves).

$$(\mathrm{ReachDir}\left(left,k,0,x,y\right)\iff (y=x))\wedge (\mathrm{ReachDir}\left(left,k,m+1,x,y\right)\iff ((\mathrm{ReachDir}\left(left,k,m,x,y\right))\vee (\exists z\in \mathrm{List}\left(Bool\right),\mathrm{ReachDir}\left(left,k,m,x,z\right)\wedge \mathrm{StepDir}\left(left,k,z,y\right))))$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.ReachDir (✓ std3).

Source. Repository-derived.

Acknowledgement. A. Chervov and others (2026). 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.

Commentary.

ReachDir(left, k, m, x, y) says that y is reached from x in at most m directed moves.

Definition 1.3 (Largest directed distance from the central state).

$$\mathrm{eccDir}\left(left,k,L,N\right)=\mathrm{sInf}\left(\{m\in \mathbb{N}:\forall y\in \mathrm{List}\left(Bool\right),(\mathrm{IsVertex}\left(L,N,y\right))\Rightarrow (\mathrm{ReachDir}\left(left,k,m,\mathrm{normalWord}\left(L,N-L\right),y\right))\}\right)$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.eccDir (✓ std3).

Source. Repository-derived.

Acknowledgement. A. Chervov and others (2026). 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.

Commentary.

The least m such that every vertex is reached from the central state within m directed moves. A vertex is a word of length N with exactly L zeros (IsVertex(L, N, x)); the central state [0]^L + [1]^(N - L), L zeros followed by N - L ones, is normalWord(L, N - L) of D5/S3/ConceptDynamics/Completion/CommutingCompletionExchange. IsVertex, rotL and rotR are reused from D5/S3/Combinatorics/ShrunkenGrassmannianThreeCycleDiameter.

Definition 1.4 (Directed diameter).

$$\mathrm{diamDir}\left(left,k,L,N\right)=\mathrm{sInf}\left(\{m\in \mathbb{N}:\forall x\in \mathrm{List}\left(Bool\right),\forall y\in \mathrm{List}\left(Bool\right),(\mathrm{IsVertex}\left(L,N,x\right)\wedge \mathrm{IsVertex}\left(L,N,y\right))\Rightarrow (\mathrm{ReachDir}\left(left,k,m,x,y\right))\}\right)$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.diamDir (✓ std3).

Source. Repository-derived.

Acknowledgement. A. Chervov and others (2026). 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.

Commentary.

The least m such that every vertex is reached from every vertex within m directed moves.

Definition 1.5 (The k = 5 clause of Conjecture 15).

$$\mathrm{clauseFive}\left(D\right)\iff (\forall L\in \mathbb{N},\forall N\in \mathbb{N},(4\le L\wedge L+9\le N)\Rightarrow (D\left(5,L,N\right)=L\cdot \lfloor \frac{N-L}{4}\rfloor +(\mathrm{if}(N-L)\mathrm{mod}4=0\mathrm{then}1\mathrm{else}0)))$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.clauseFive (✓ std3).

Citation. A. Chervov and others (2026). 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.

Commentary.

For all L >= 4 and N >= L + 9, with t + 1 = N - L, the diameter is L times the floor of (t + 1)/4, plus 1 when t + 1 is divisible by 4.

Definition 1.6 (The clause for some convention and reading).

$$claim\iff (\exists left\in Bool,(\mathrm{clauseFive}\left(\mathrm{eccDir}\left(left\right)\right))\vee (\mathrm{clauseFive}\left(\mathrm{diamDir}\left(left\right)\right)))$$

Formalization. D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.claim (✓ std3).

Citation. A. Chervov and others (2026). 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.

Commentary.

The clause holds for one of the two conventions and one of the two readings of the diameter.

Theorem 1.7 (Refutation).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.result (✓ std3). ∎

Resolves. Problems/chervov-2026-cayleypy4-conjecture-fifteen-refutation (refuted) by D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.result.

Source. Repository-derived.

Acknowledgement. A. Chervov and others (2026). 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.

Commentary.

Count the inversions of a word, the pairs of positions holding a one before a zero. Rotating five consecutive letters one place carries one letter past the other four, so it changes the count by at most 4, and a directed move is in either convention such a rotation. The central state 0000111111111 has no inversion and its reversal 1111111110000 has 36, so reaching the reversal takes at least 9 moves. If the central eccentricity or the diameter were 8 at L = 4, N = 13 in either convention, the least element of its defining set would be 8, and the reversal, a vertex, would be within 8 moves of the central state, also a vertex. The clause gives 4 times the floor of 9/4, which is 8, since 9 is not divisible by 4.

References

• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.ReachDir
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.StepDir
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.claim
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.clauseFive
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.diamDir
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.eccDir
• Truth anchor: D5/S3/Combinatorics/ShrunkenGrassmannianConjectureFifteenRefutation.result
• Dependency: D5/S3/Combinatorics/ShrunkenGrassmannianThreeCycleDiameter