The Hodos Family: Equations Generated by a Relational Premise

The Hodos Hypothesis holds that a thing is the pattern of its connections rather than its substrate, and that what a system does beyond its parts arises from that pattern and is never installed as a component. Until now it had one equation attached to it: a distance comparing processes as curves of distributions — a measuring tool, which takes the distributions as given. A premise about how things are constituted should generate more than a ruler. This paper states the premise as the head of a family and adds a second member: an emergence equation measuring what a relationship creates that is in neither part. At each instant it compares what two processes actually do together against what they would do if they did not interact at all. The construction has a property worth stating plainly: the control is a term in the equation. If the parts do not interact the surplus vanishes by arithmetic rather than by measurement, and relabelling either part's coordinates leaves it unchanged, so it reads the relation rather than the substrate — both provable rather than observed. It clears four criteria fixed before it ran, including one that could have killed it: it reproduces a split measured by an entirely different method before this equation existed. Provenance is stated without softening: the quantity is mutual information in general form and is tagged ASSEMBLED; the contribution is the composition. A third question is named and reported as not cleared, with the numbers that stopped it, rather than omitted. 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finding, including the negative and corrected ones: xfloukiex-lab.github.io/hodos-studyPreprint, not peer reviewed.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.