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

Can the duration a process has lived be computed from its own internal relations? Every clock is already made of relations — a second is the Earth turning relative to the Sun, and a caesium second is one oscillation counted against another. So the question is never relation versus no relation. It is whose relations are counted: a wall clock borrows the Earth–Sun relation, and this construction counts the system's own. 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. This paper takes duration to be the accumulated null-corrected relational surplus instead, 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. 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. At the registered configuration it still does not clear sampling invariance, which is reported rather than worked around. 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. Nothing in it is a cosmological claim. Part of the Vektorgeist Method. Research index: xfloukiex-lab.github.io/vektorgeist-researchEvery finding, including the negative and corrected ones: xfloukiex-lab.github.io/hodos-studyPreprint, not peer reviewed. No result here has been reproduced outside the project.