Time as a Computational Substrate: The Eligibility Trace and the First Challenge to the von Neumann Architecture

<title>Abstract</title> <p>Every computer built since 1945 inherits an unexamined assumption from the von Neumann architecture: that time is merely a sequence number -- the program counter advances, one instruction after another, and the physical arrow of time carries no computational meaning. This assumption has survived six decades and four generations of neural networks because no working physical structure had ever challenged it. We report the first such structure. The eligibility trace -- a one-line equation, e(t+1) = lambda*e(t) + delta_w -- embeds the causal arrow of time directly into the computational substrate. We demonstrate that this structure (i) is the unique necessary condition for emergent differentiation in spiking neural networks (five-condition ablation, N = 800 to 80,000, delta-Gini = -0.006 without it); (ii) is scale-invariant across two orders of magnitude; and (iii) exhibits an isomorphic causal structure in stochastic gradient descent, where gradient direction consistency is a necessary-but-not-sufficient condition for the long-range retrieval phase transition. The von Neumann architecture is the degenerate limit in which the eligibility window goes to zero. The general case is open.</p>