The Geometry of Dimensional Lift: Why the Power of pi Rises with Dimension, a Census of Royal-Road Topology.
In physics and mathematics the way pi appears differs qualitatively with dimension: 3D shows 4pi or 8pi (Einstein’s G=8pi G T), 4D shows pi^2 coefficients (QED one-loop 1/16pi^2, 4D Gauss-Bonnet 1/32pi^2, instanton number 1/8pi^2). This paper organizes the observation as a LAYER-2 CENSUS. The core is NOT the specific numbers (8pi -> 16pi^2) but one vertical thread: “the power of pi rises with dimension.” Its geometric necessity is the sphere-volume recursion Vol(S^n) = (2pi/(n-1)) Vol(S^{n-2}) — every two dimensions up adds one pi (S^2:4pi -> S^3:2pi^2 -> S^5:pi^3). This single structure is illuminated by established theorems: CHERN-WEIL / TRANSGRESSION (3D-boundary Chern-Simons and 4D-bulk c2 joined by d CS = tr(F^F) = an instance of Paper 92’s generalized Stokes), the APS INDEX THEOREM (4D-bulk index and 3D-boundary eta-invariant = a child of Paper 70), and the HOPF fibration (S^3 -> S^2 — but NOT a simple product: S^3 is a non-trivially twisted S^1-bundle over S^2, Vol does NOT factor as 4pi x 2pi = 8pi^2 != 2pi^2, so it is a “twisted auxiliary line,” not a multiplicative lift). CHIEF DISCIPLINE: do NOT say 8pi and 16pi^2 are “unified” — DISTINCT ROOTS (86). The roots producing pi^2 split into analysis (loop integrals), topology (characteristic classes), geometry (fiber-bundle volume); sharing the number is RESONANCE (a signature), NOT a proof (100). The coefficient varies with convention (8pi^2/16pi^2/32pi^2) — the essence is the power of pi, not the number; no numerology / post-hoc adjustment (101). IMPORTANT (scope): a Layer-2 structural interpretation/census of established geometry and topology; no new mathematics and no new prediction; it does NOT solve “why 3+1 dimensions” (deferred to 69/105, admire); the “substance/information zone” metaphor is Layer-3. 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 of Dimensional Lift [Paper 109]. Zenodo.
DOI: 10.5281/zenodo.21837375
ORCID: 0009-0006-8587-3450