How Far Does Light Determine Geometry? Two Roads That Stop at Conformal Structure (Differential Geometry and Order Theory), and the Distinct Roots of the Residual Scale.
This series has treated the speed of light c repeatedly (63, 66, 83, 95, 108), yet one question was never asked directly: GIVEN LIGHT ALONE, HOW MUCH OF SPACETIME GEOMETRY IS DETERMINED? This paper answers with established theorems, as a LAYER-2 structural interpretation and DISTINCT-ROOTS CENSUS. (i) EHLERS-PIRANI-SCHILD (1972): light rays determine a CONFORMAL STRUCTURE (the null cones, i.e. the metric up to a positive conformal factor, g ~ Omega^2 g); freely falling particles determine a PROJECTIVE STRUCTURE (the straightness of geodesics); their compatibility fixes a WEYL STRUCTURE uniquely. Reaching a Lorentzian metric requires ONE FURTHER STEP: the absence of the SECOND-CLOCK EFFECT. (ii) ZEEMAN (1964): the bijections of Minkowski spacetime (dimension three or higher) that preserve the CAUSAL ORDER are generated by orthochronous Lorentz transformations, translations and dilatations, with NEITHER CONTINUITY NOR LINEARITY ASSUMED (both are conclusions, not hypotheses). (iii) HAWKING-KING-McCARTHY (1976) / MALAMENT (1977): in general (distinguishing) spacetimes as well, the causal structure determines the conformal structure, the topology and the differentiable structure. Starting from entirely different mathematics (differential geometry / order theory), the roads HIT THE SAME WALL: light and causality determine the metric UP TO A CONFORMAL FACTOR, and stop there. Of the ten independent components of a four-dimensional Lorentzian metric, light determines the NINE of the conformal class; the remaining ONE, the scale, it does not. CORE OF THE PAPER: that residual scale is TWO DISTINCT ROOTS carried by ONE WORD. What remains in EPS is a FUNCTION on spacetime (the local conformal factor, removed by CLOCKS, i.e. the absence of the second-clock effect, 96); what remains in Zeeman is a GLOBAL CONSTANT (the dilatation, a choice of unit, i.e. a CONVENTION, 95 and SI 2019). A function is not a constant: the former is filled by physics, the latter agreed by convention. The paper also takes a roll call of THREE DISTINCT ROOTS to the Lorentz group: group axioms (Ignatowski, 83) / the geometry of the invariant interval (Minkowski, 108) / causal order (Zeeman, this paper) – and only the order root lets the dilatation slip through. A further link: the final EPS axiom is THE SAME POINT INVERTED as the failure of Weyl’s 1918 theory recorded in 96 – a headstone in one place, a cornerstone in the other. IMPORTANT (scope): a Layer-2 structural interpretation of established theorems; no new theorem, no new physics, no prediction. IT DOES NOT CLAIM THAT SPACETIME CAN BE CONSTRUCTED FROM LIGHT – what it records is the LIMIT of light, since EPS requires freely falling particles as an indispensable second input. It does not enter the discussion on the status of the EPS final step (adopting only the uncontested point that a further step is required), and it does not prove why 3+1 (Zeeman’s dimension condition is one line for the census, not a derivation; 69/105). Agreement of the three roots is a signature, not a proof (100); no post-hoc numerical adjustment (101). The Layer-3 lens (reading light as building material) is quarantined with a non-scientific label and is not offered as a mechanism.
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). How Far Does Light Determine Geometry? — Two Roads That Stop at Conformal Structure (Differential Geometry and Order Theory), and the Distinct Roots of the Residual “Scale” — [Paper 111]. Zenodo.
DOI: 10.5281/zenodo.21961685
ORCID: 0009-0006-8587-3450