Temporal trace semantics and a bounded-repetition special case

Definitions and mathematical statement

Fix positive integers $n,t$ and nonnegative integers $r_1,\ldots,r_t$ with $\sum_{s=1}^t r_s\geq n$. For each $s$, let $A_s$ be an $n\times n$ zero--one matrix with all diagonal entries one, no pair of opposite off-diagonal entries both one, and every row sum at least $r_s+1$. Explicitly, for $i\ne j$, $(A_s)_{ij}(A_s)_{ji}=0$. The bound $r_s$ is common to all rows of its matrix but may vary with time. No restriction identifies the edge sets encoded by different matrices.

The conjecture asks whether \[ \operatorname{tr}(A_1A_2\cdots A_t)>n \] for every admissible sequence, with the product interpreted as an ordinary integer matrix product in the displayed order. Equivalently, do the row-sum budgets totaling at least $n$ force vertices $i_0,\ldots,i_t=i_0$ such that $(A_s)_{i_{s-1},i_s}=1$ for every $s$, and at least one step changes vertex? Every vertex contributes one stationary walk, so exactly these nonstationary closed walks account for the desired excess trace. Repeated intermediate vertices and stationary steps are allowed. Taking all $r_s$ equal gives the preceding fixed-row-bound problem once the number of factors is large enough. Allowing zero budgets merely permits additional matrices with diagonal ones; it does not impose a hidden positive-degree assumption on every time step. The task is a universal assertion about temporally ordered matrices, rather than the cycle structure of their union alone.

Short English statement

Does a total row-degree budget of at least the matrix size force a nonstationary closed walk through a sequence of oriented matrices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001061 (AIM-COMBINATORICS-0186), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, The Caccetta-Haggkvist conjecture, item 6.5. Original workshop question underlying AIM-COMBINATORICS-0186. https://aimath.org/WWN/caccetta/caccetta.pdf.