Emergence XXXIV: The Spectral Bridge — Chemistry and Mathematics Unified — Electron Shells and Prime Numbers from a Common Origin in the Canvas Model
Fuente:
Zenodo
Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
|
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866902092559417344 |
|---|---|
| author | Ong, Edwin |
| author_facet | Ong, Edwin |
| contents | <p>We demonstrate that the electron shell structure of atoms, the distribution of prime numbers, and the zeros of the Riemann zeta function are all manifestations of a single underlying equation—the Unified Wave Equation of the Canvas Model. This equation, together with a threshold condition and an eigenvalue equation, generates all spectral phenomena across physics and mathematics.</p> <p>What this paper provides:</p> <p>· A periodic table of spectral transforms that classifies 48 known and predicted transforms, 31 Hilbert spaces, and over 450 mathematical concepts across 14 branches of mathematics. The hydrogen atom (Schrödinger spectral transform) occupies row T35. The prime number distribution (TAC spectral transform) occupies row T29. Both are configurations of the same eight primitives on different lattices with different potentials. Both are spectral resonances of the same equation.<br>· A unified spectral framework for chemistry and number theory. The hydrogen atom's Coulomb potential V(r) = -e^2/r (Amplitude configuration B3) and the prime lattice's von Mangoldt potential \Lambda(n) = \ln p (B6) are different potentials in the same Unified Wave Equation. The eigenvalues of the hydrogen atom are E_n = -13.6/n^2 eV with degeneracy 2n^2, giving the electron shell filling: 2, 8, 18, 32, 50, 72, 98... The eigenvalues of the TAC operator are the zeros of the Riemann zeta function, conjectured to lie on \operatorname{Re}(s) = 1/2, giving the prime distribution \pi(x) \sim x/\log x + \text{oscillations}. The periodic table of elements and the distribution of primes are two columns of the same classification.<br>· A physical interpretation of the Riemann Hypothesis. The non-trivial zeros are closed waves formed by matter-antimatter annihilation on the prime lattice. The critical line \operatorname{Re}(s) = 1/2 is the unique equilibrium where matter and antimatter amplitudes balance. The trivial zeros on the negative real axis are pure antimatter states from the Gamma field. The two zero lines are perpendicular, reflecting the perpendicularity of the Gamma and prime fields in the TAC operator.<br>· A reduction of the Riemann Hypothesis to a single lemma. The vanishing of the spectral energy E_0 = \sum_\rho (\beta_\rho - 1/2)^2 = 0 is equivalent to RH. Within Canvas Temporal Mathematics (CTM), this lemma follows from the Local Equilibrium axiom: each zero individually satisfies \mathcal{S}[\rho] = \rho. Combined with the functional equation \mathcal{S}[\rho] = 1-\rho, this gives \rho = 1-\rho and \operatorname{Re}(\rho) = 1/2. The 165-year failure to prove RH in ZFC is explained: ZFC is the static limit of CTM, lacking meta-order dynamics and the Local Equilibrium principle.<br>· The complete classification of mathematical structures by primitive configurations. 26 lattice types (Order), 19 operator types (Acceleration), 12 amplitude configurations, 9 polarity configurations, and 31 Hilbert spaces are systematically classified. Eleven new spectral transforms (T38–T48) are predicted. The full classification space contains 53,352 possible primitive configurations.<br>· A bridge between chemistry and mathematics. The same eight primitives that generate the electron shells of atoms generate the distribution of primes. Physics and mathematics are not separate disciplines. They are the same subject, viewed from different rows of the same table.</p> <p>Why this matters:</p> <p>The periodic table of elements and the distribution of prime numbers have been studied for centuries. They were thought to be completely unrelated—one belongs to chemistry and physics, the other to pure mathematics. The Canvas Model reveals they are the same phenomenon. Electron shells and prime zeros are both eigenvalues of the Unified Wave Equation. The unification of atomic physics and number theory through a common spectral framework is the central result of this paper.</p> <p>Keywords: Canvas Model, spectral bridge, chemistry, number theory, electron shells, prime numbers, Riemann Hypothesis, unified wave equation, periodic table of spectral transforms, TAC operator, hydrogen atom, von Mangoldt potential</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20233920 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Emergence XXXIV: The Spectral Bridge — Chemistry and Mathematics Unified — Electron Shells and Prime Numbers from a Common Origin in the Canvas Model Ong, Edwin <p>We demonstrate that the electron shell structure of atoms, the distribution of prime numbers, and the zeros of the Riemann zeta function are all manifestations of a single underlying equation—the Unified Wave Equation of the Canvas Model. This equation, together with a threshold condition and an eigenvalue equation, generates all spectral phenomena across physics and mathematics.</p> <p>What this paper provides:</p> <p>· A periodic table of spectral transforms that classifies 48 known and predicted transforms, 31 Hilbert spaces, and over 450 mathematical concepts across 14 branches of mathematics. The hydrogen atom (Schrödinger spectral transform) occupies row T35. The prime number distribution (TAC spectral transform) occupies row T29. Both are configurations of the same eight primitives on different lattices with different potentials. Both are spectral resonances of the same equation.<br>· A unified spectral framework for chemistry and number theory. The hydrogen atom's Coulomb potential V(r) = -e^2/r (Amplitude configuration B3) and the prime lattice's von Mangoldt potential \Lambda(n) = \ln p (B6) are different potentials in the same Unified Wave Equation. The eigenvalues of the hydrogen atom are E_n = -13.6/n^2 eV with degeneracy 2n^2, giving the electron shell filling: 2, 8, 18, 32, 50, 72, 98... The eigenvalues of the TAC operator are the zeros of the Riemann zeta function, conjectured to lie on \operatorname{Re}(s) = 1/2, giving the prime distribution \pi(x) \sim x/\log x + \text{oscillations}. The periodic table of elements and the distribution of primes are two columns of the same classification.<br>· A physical interpretation of the Riemann Hypothesis. The non-trivial zeros are closed waves formed by matter-antimatter annihilation on the prime lattice. The critical line \operatorname{Re}(s) = 1/2 is the unique equilibrium where matter and antimatter amplitudes balance. The trivial zeros on the negative real axis are pure antimatter states from the Gamma field. The two zero lines are perpendicular, reflecting the perpendicularity of the Gamma and prime fields in the TAC operator.<br>· A reduction of the Riemann Hypothesis to a single lemma. The vanishing of the spectral energy E_0 = \sum_\rho (\beta_\rho - 1/2)^2 = 0 is equivalent to RH. Within Canvas Temporal Mathematics (CTM), this lemma follows from the Local Equilibrium axiom: each zero individually satisfies \mathcal{S}[\rho] = \rho. Combined with the functional equation \mathcal{S}[\rho] = 1-\rho, this gives \rho = 1-\rho and \operatorname{Re}(\rho) = 1/2. The 165-year failure to prove RH in ZFC is explained: ZFC is the static limit of CTM, lacking meta-order dynamics and the Local Equilibrium principle.<br>· The complete classification of mathematical structures by primitive configurations. 26 lattice types (Order), 19 operator types (Acceleration), 12 amplitude configurations, 9 polarity configurations, and 31 Hilbert spaces are systematically classified. Eleven new spectral transforms (T38–T48) are predicted. The full classification space contains 53,352 possible primitive configurations.<br>· A bridge between chemistry and mathematics. The same eight primitives that generate the electron shells of atoms generate the distribution of primes. Physics and mathematics are not separate disciplines. They are the same subject, viewed from different rows of the same table.</p> <p>Why this matters:</p> <p>The periodic table of elements and the distribution of prime numbers have been studied for centuries. They were thought to be completely unrelated—one belongs to chemistry and physics, the other to pure mathematics. The Canvas Model reveals they are the same phenomenon. Electron shells and prime zeros are both eigenvalues of the Unified Wave Equation. The unification of atomic physics and number theory through a common spectral framework is the central result of this paper.</p> <p>Keywords: Canvas Model, spectral bridge, chemistry, number theory, electron shells, prime numbers, Riemann Hypothesis, unified wave equation, periodic table of spectral transforms, TAC operator, hydrogen atom, von Mangoldt potential</p> |
| title | Emergence XXXIV: The Spectral Bridge — Chemistry and Mathematics Unified — Electron Shells and Prime Numbers from a Common Origin in the Canvas Model |
| url | https://doi.org/10.5281/zenodo.20233920 |