Reading the Interior from the Boundary: the Calderón Problem, Electrical Impedance Tomography (EIT), and the Inverse-Problem Stance.
In Paper 89 (Zero-Tension and the Inverse Problem) the corpus raised the INVERSE-PROBLEM STANCE — reading the unseen “cause” from the observable “effect.” This paper finds its mathematical archetype in Alberto Calderón’s problem. The CALDERON PROBLEM (1980) asks: by applying voltages at the surface (boundary) of a body and measuring outgoing currents, can one determine the interior conductivity gamma(x)? The boundary “voltage -> current” correspondence is the Dirichlet-to-Neumann map (DtN map). This is the mathematical foundation of Electrical Impedance Tomography (EIT), and Calderón reached it from oil prospecting (reading the underground from surface measurements). Skeleton: UNIQUENESS is proven (Sylvester-Uhlmann 1987 for n>=3 via complex geometrical optics; Nachman/Astala-Paivarinta for n=2) — the boundary DtN map determines the interior uniquely. But HONESTLY: the problem is severely ILL-POSED; a tiny boundary error is greatly amplified in the interior (LOGARITHMIC stability), so practice requires REGULARIZATION — inverse problems are powerful but unstable (“unique” is not “stable”). The point is twofold: (1) place the Calderón problem as the mathematical archetype of Paper 89’s stance; (2) sort the corpus’s inverse problems — inverse SCATTERING (Gel’fand-Levitan-Marchenko, 89) and inverse BOUNDARY-VALUE (Calderón) — as the same stance but different mathematics = DISTINCT ROOTS (58/48; the statistical inverse of Mandelbrot 94 is yet another root). Connections: the DtN “boundary determines interior” resonates with n=2 uniqueness (01) and weak-assumptions->Lie-group (90); “read hidden cause from visible effect” is the author’s integral-brain origin (prospecting/fishing/markets). IMPORTANT (scope): a Layer-2 structural interpretation of established inverse-problem theory; no new mathematics and no new prediction. 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). Reading the Interior from the Boundary [Paper 99]. Zenodo.
DOI: 10.5281/zenodo.21758790
ORCID: 0009-0006-8587-3450