From Waveform Inversion to Deep-Earth Anomalies: A Formal Working Paper on Membranous High-Pressure Structures, Observational Limits, and the Epistemic Status of Speculative Geophysical Models

This paper formalizes a speculative but non-arbitrary line of reasoning concerning the epistemic limits of human knowledge about the deep Earth. The central claim is not that a literal hollow Earth or inhabitable underground world exists. Rather, the paper asks whether complex, high-pressure, multi-layered, shell-like or membrane-like anomalous regions could be under-resolved or over-smoothed by present geophysical inverse models. The argument begins from an observation: human knowledge of the mantle, core-mantle boundary, outer core, and inner core is overwhelmingly indirect, mediated by seismic waves, gravity fields, magnetic fields, normal modes, heat-flow constraints, mineral physics, and model-based inversion. This makes planetary interior science powerful but not equivalent to direct inspection. We introduce the Membranous Deep-Earth Anomaly Hypothesis (MDEAH): under billions of years of high pressure, high temperature, convection, phase transitions, slab subduction, partial melting, chemical segregation, and core-mantle coupling, the interior may contain multi-scale, layered, shell-like, high-pressure anomalies whose true geometry may be more complex than current smoothed velocity or density models suggest. The hypothesis is distinguished from a literal hollow-Earth claim. It is framed as a testable residual hypothesis: if such structures exist, they should appear as systematic residuals in seismic travel times, waveform scattering, normal-mode splitting, gravity-density inversions, geomagnetic coupling, and eventually neutrino tomography. Mathematically, the paper uses the language of forward modeling and inverse problems, showing that geophysical observations satisfy equations of the form d=F(m)+\epsilon, where m is an internal Earth model, F is a forward operator, and \epsilon is noise and modeling error. Because F^{-1} is generally non-unique and resolution-limited, the absence of a feature in a model is not equivalent to a proof that no such structure exists. At the same time, large low-density voids are strongly constrained by mass, moment of inertia, seismic-wave propagation, and pressure stability. Thus, the scientifically defensible version of the speculation is not an empty cavity, but a high-P anomalous domain: a complex, possibly shell-like, partially molten, dense, low-velocity, anisotropic, or otherwise exotic region that may be simplified by current inversions. The conclusion is deliberately balanced. Present evidence strongly disfavors literal large empty caverns below it, but it does not eliminate the possibility that deep-Earth anomalies are structurally richer than current models can resolve. The value of the hypothesis lies not in claiming discovery, but in converting intuition into falsifiable questions, residual tests, and multi-observable predictions. Copyright (c) 2026 Chen, Ho Yiing (norika). Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).