The Geometry Where the Invariant Lives: Minkowski’s Spacetime, and Not Conflating the “Geometries of Relativity”.
To close Chapter 2 (connecting royal-road nodes, Papers 101-108), the corpus places Hermann Minkowski. In 1908 he translated Einstein’s special relativity into “the geometry of spacetime”: space and time by themselves fade into shadows, and only their union preserves an independent reality. Its core is the INVARIANT INTERVAL ds^2 = c^2 dt^2 – dx^2 – dy^2 – dz^2: time and length change with the observer, but this interval alone is common to all. The Lorentz transformation is the (hyperbolic) rotation preserving it, and ds^2=0 carves the light cone = causal structure. Core: the SEPARATION OF DISTINCT ROOTS (58) — the corpus has several “geometries of relativity”: (a) Minkowski’s flat spacetime ds^2 (this paper, invariant interval, flat, Lorentzian); (b) the hyperbolic geometry of VELOCITY SPACE K=-1/c^2 (Paper 60, negative curvature); (c) PROPER TIME tau=int ds/c (Paper 82, invariant along a worldline). These share “invariant/relativity” but are distinct roots; Minkowski is the geometry of the interval itself (the flat limit). Moreover, where Ignatowski (83) answered “why the Lorentz group” by group theory, Minkowski gives the geometry (invariant) that group preserves — 83 (group root) and 108 (geometry root) are complementary but distinct roots. Minkowski’s flat spacetime is the zero-curvature special case of Riemann’s intrinsic curvature (91) = the flat limit / tangent space of GR, whose algebra is Clifford’s spacetime algebra Cl(1,3) (104). IMPORTANT (scope): a Layer-2 structural interpretation of established relativity; SR is NOT rewritten; no new mathematics and no new prediction; partly overlaps 82/83/60 (as Ch.2 finale it makes explicit their geometric core of the invariant). The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Geometry Where the Invariant Lives [Paper 108]. Zenodo.
DOI: 10.5281/zenodo.21837313
ORCID: 0009-0006-8587-3450