Infinite Dark Dimensions in Geometric Harmony: From Simple Rules Beyond the Standard Model

String theory adds 6 or 7 extra dimensions at a specification cost exceeding 3,000 bits. This paper argues the opposite: an infinite tower of geometric dimensions exists above any manifold with a metric, each at zero additional algorithmic cost. The metric bundle (14D) over a 4D manifold is mathematically necessary, not specified. The jet bundle above the metric bundle is mathematically necessary. The tower continues without bound. No specification. No compactification. No landscape. Zero bits. The argument requires only three assumptions: the fundamental description is discrete, has a simple local update rule, and the rule is applied iteratively. These are the minimal assumptions of any computational substrate. Hypergraph rewriting rules, cellular automata, causal sets, and spin networks all satisfy them. SDP is agnostic about which substrate wins. It predicts only that the winning rule has the lowest Kolmogorov complexity among those producing observer-supporting physics. The physics is not in the dimension count but in the amplitude profile. Like modes on a vibrating string, the fundamental (metric bundle) dominates observable physics. Higher modes contribute progressively smaller corrections. The Standard Model is the timbre. This reframes effective field theory as a consequence of the harmonic structure of necessary geometry, not as an approximation to unknown deeper physics.