Hodos Diastema: Comparing Processes as Curves of Distributions — an Information-Geometry Distance, Validated Across Modalities on Real Data

A microphone, a spectrometer, a sonar and a pen all do the same abstract thing: at each instant they report how much of something is where. Normalise that and it is a probability distribution; watch it over time and the observation traces a curve of distributions rather than a number. Most pipelines summarise a signal by an average and compare the averages, which throws away time. This paper keeps the whole curve and studies the distance between two of them: elastic temporal alignment composed with an information-geometry ground metric. The central claim is deliberately narrow, and it is the one thing that survived a deliberately adversarial validation: the representation — not the alignment machinery — is what earns a measurable advantage, wherever identity lives in the relative-space pattern of a process. It is shown on two unrelated real-data modalities: recognising a spoken word across different human voices, and recognising a handwritten character from pen motion alone. What it does not claim: this is a representation and a method, validated narrowly on two modalities, and not a universal oracle. It does not invent a domain's law and does not reach discrete or logical truths. Every component — dynamic time warping, Fisher–Rao geometry, the elastic shape representation — is established prior art and is cited as such; the contribution is the composition, the operators and the validation record. The paper reports every negative result and every correction alongside the positives at equal weight, including a ledger of four seductive findings killed by fair controls before they reached the page, and a 'predicted failure' that turned out to be the authors' own broken input and is kept visible rather than deleted. 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finding, including the negative and corrected ones: xfloukiex-lab.github.io/hodos-studyPreprint, not peer reviewed.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.