Deterministic fractal complexity of solution graphs in multi-delay fractional systems

Fractional differential equations encode long-range memory through weakly singular Volterra kernels, while delay terms reintroduce past states at discrete times, creating a nonlocal interaction between memory smoothing and delay reinjection. Although well-posedness and stability of fractional delay systems are well established, rigorous geometric characterizations of deterministic solution graphs in the multi-delay Caputo setting remain largely unexplored. In particular, it is unclear when multiscale geometric complexity arises as an intrinsic structural consequence of memory–delay interaction rather than from stochastic forcing or classical chaos. In this work, we develop a mechanism-based geometric framework for multi-delay Caputo systems by reformulating the dynamics as a Volterra integral equation. We establish global well-posedness and global Hölder regularity, which impose a universal geometric envelope on scalar solution graphs and yield an unconditional upper bound on their box-counting dimension. We then identify explicit structural conditions — densification of delay-induced injection times together with discrete-scale persistence of cusp-like oscillations — under which this envelope becomes sharp and a matching lower bound on the graph dimension holds. These results provide a rigorous separation between unconditional memory-induced smoothing and delay-activated fractal saturation, thereby clarifying when deterministic multi-delay fractional systems can and cannot exhibit fractal graph complexity.