A Clock Made of Relations: Intrinsic duration for a system of interacting parts, and the one-sided nonlinearity that hid it for thirteen attempts

Abstract

Can the duration a process has lived be computed from its own internal relations rather than read off an external clock? Hodos Chronos, the fourth member of the Hodos family, first measured duration as arc length through a statistical manifold. That construction failed a sampling-invariance criterion, and the failure was attributed to the coastline problem. This paper shows that attribution is too broad: it is correct about arc length and is not a verdict on the question. The construction offered here takes duration to be the accumulated null-corrected relational surplus, oriented by the third moment of that surplus's own increments. Over a six-part emergence graph it clears four of five criteria fixed in advance — time-asymmetry, stillness, relation-sensitivity and exactness — including two that a path length cannot clear even in principle. It still fails sampling invariance, which is reported rather than worked around. The route matters more than the result. Thirteen intermediate constructions traced what appeared to be a structural trade-off between registering irreversibility and measuring stillness. It was not one. It was an artifact of a single one-sided nonlinearity applied to the wrong one of two readings, and removing it dissolved the axis. The paper also reports three one-line impossibility results constraining any direction-sensitive quantity, and a false pass caught by seed-checking. Preprint, not peer reviewed. Scored by the authors' own harness, with no external reader. Every figure is re-derived from a run ledger by a checker that refuses to pass on a mismatch.