What question did this study set out to answer?

The paper aims to explore the mathematical structures relating the Four Element system in Geumgangsimron to modern number theory.

April 23, 2026Open Access

A Number-Theoretic Analysis of the Four-Element System in Geumgangsimron: The Adelic Bridge A₇ = ℚ₇ × ℝ from the Womb-Realm to the Vajra-Realm

Key Points

The paper aims to explore the mathematical structures relating the Four Element system in Geumgangsimron to modern number theory.
Constructs a number-theoretic bridge to p-adic mathematics using the adele A₇ = ℚ₇ × ℝ.
Analyzes the Four-Element matrix as a spectral filter on the Bruhat-Tits tree T₇.
Presents multiple theorems that reveal connections between elements and mathematical structures.
Identifies eigenvalues that illustrate transitions between Womb-Realm and Vajra-Realm states.
Establishes that dynamic laws of attraction and repulsion emerge from Fermat's theorem.
Shows a unique number-theoretic embedding ties the quasicrystal to T₇.

Abstract

In 1941, the Korean Zen master Geumta (金陀, 1895–1969) composed "The Essence and Measure of the Universe" after seven years of seated meditation, encoding his cosmological observations as a numerical system built from four numbers assigned to the Four Great Elements: Earth = 1, Water = 3, Fire = 5, Wind = 7. This paper constructs a number-theoretic bridge from the source text's numerical system to modern p-adic mathematics via the adele A₇ = ℚ₇ × ℝ. The Four-Element matrix M = [3, 7, 1, 5] is shown to act as a spectral filter on the Bruhat-Tits tree T₇: its eigenvalues λ₁ = 4+2√2 and λ₂ = 4−2√2 determine which states localize at the boundary (crossing into the manifest Vajra-Realm) and which remain in the tree's interior (staying in the hidden generative Womb-Realm). M is proven to be the unique operator naturally determined by the Four Elements and the local structure of T₇. Principal results: (1) The spectral classification on T₇ realizes the Buddhist cosmological transition from the Womb-Realm (True Emptiness, 眞空) to the Vajra-Realm (Wondrous Existence, 妙有) as boundary localization. The manifest-exclusive ratio equals 2 (√2−1) ≈ 82. 84%, identical to the silver-ratio partition of the Ammann-Beenker quasicrystal tiling's integrated density of states. (2) The source text's dynamical laws — attraction = 3¹⁰ and repulsion = 5⁶ — emerge automatically from Fermat's little theorem in F₇*, with exponents given by the row sums of M. (3) The embedding ℚ (√2) ↪ ℚ₇ exists because √2 ≡ 3 (mod 7) ; among the primes appearing as element values 3, 5, 7, only p = 7 admits this embedding, providing a direct number-theoretic link between the Ammann-Beenker quasicrystal and the Bruhat-Tits tree T₇. The paper presents 61 theorems organized across 15 phases and 9 sections. This work is a number-theoretic analysis, not a physical theory: it makes no empirical predictions and claims no correspondence with modern astronomical measurements. Source text: Geumta Daehwasang, "Shinsetsu Uchu no Honshitsu to Keiryo" (New Theory: The Essence and Measure of the Universe), 1941 (Japanese original) ; Korean translation by Cheongwha Seonsa, 1974; incorporated into Geumgangsimron (金剛心論), 1979.

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