The Shared Alphabet: Cartan’s Differential Forms and the Language Beneath the Corpus’s “Grammar”.
Paper 87 held that the shared GRAMMAR of physical law is least action. In what ALPHABET is that grammar written? This paper finds it in the exterior calculus perfected by Élie Cartan (differential forms, the exterior derivative d, the wedge product, integration) — the shared language flowing one level beneath the corpus’s grammar. Forms capture “things that get integrated” by degree, and d (with d²=0) unifies gradient, curl, and divergence into one operation. At its center is the generalized Stokes theorem (integral over the boundary of Omega of omega = integral over Omega of d-omega), which contains as special cases the fundamental theorem of calculus, Green’s theorem, the Gauss divergence theorem (the solid angle 4pi in the flux of a point source; Paper 02), and the classical Stokes theorem. Cartan’s structure equations express curvature as a 2-form, and the same alphabet writes the GR curvature (Paper 91), the symplectic 2-form omega=dp wedge dx (Paper 76), the electromagnetic field F=dA (Paper 87), and the Gauss-Bonnet curvature integral (Paper 67). KEY LINE (distinct roots): forms are a shared ALPHABET (a language), while the symplectic form, the curvature 2-form, the EM 2-form, and the action are different words = distinct roots (Papers 58/48). What is shared is the language; what must not be conflated is each word — one level below Paper 87 (the shared grammar). IMPORTANT (scope): a Layer-2 structural interpretation of standard exterior calculus; no new mathematics and no new prediction; NOT a unified field theory (a language, not a theory, the same line as Paper 87). 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 Shared Alphabet [Paper 92]. Zenodo.
DOI: 10.5281/zenodo.21758486
ORCID: 0009-0006-8587-3450