The Arithmetic of Spacetime: Collar Covering Monoids, Heat-Kernel Modularity, and the Geometric Origin of Prime Numbers
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Abstract
In spectral geometry and operator-algebraic models of the Riemann zeta function—such as Ray-Singer determinants and Bost-Connes systems—the multiplicative monoid of positive integers is introduced as an external algebraic input. Here, on the compact normal collar orbifold C_{ell_p} = S^1_{2 ell_p}/Z_2 ~ [0, ell_p] supporting the ultraviolet boundary layer of a Z_2-symmetric hypersurface, we prove that smooth metric-scaling endomorphisms preserving the interface and boundary stratum form a commutative covering monoid End_scale(C_{ell_p}) ~ (N_{>0}, x) under composition. By geometric pullback, these endomorphisms generate the complete fluctuating eigenbasis of the self-adjoint Neumann Laplacian Delta_perp = -d^2/dy^2 from a single fundamental mode. The Galois correspondence of covering spaces and the Jordan-Holder theorem identify geometrically irreducible coverings with simple deck transformation groups Deck(Phi_p) ~ Z/pZ of prime degree, causing the normalized Mellin-Haar transform of the collar heat kernel A_perp(tau) = exp(-tau ell_p^2 Delta_perp / pi) to factorize unconditionally into the Euler product of zeta(s) for Re(s) > 1. In the meromorphically continued connected impedance Z_int(s) = -zeta'(s)/zeta(s), composite coverings with coprime factors possess decomposable direct-product deck groups Deck(Phi_{ab}) ~ Deck(Phi_a) x Deck(Phi_b) whose cumulants cancel identically, isolating prime-power coverings without invoking Mobius inversion. Poisson modular inversion of the Schwinger proper-time integral recovers the completed xi-function xi(s) = xi(1-s) and yields the Hadamard constant B = xi'(0)/xi(0) = -1 + (1/2) ln(4 pi) - gamma/2 directly from three collar invariants: vacuum-pole subtraction, Gaussian symplectic volume, and scale-drift entropy. This self-contained construction derives the Euler product from collar topology without claiming a proof of the Riemann hypothesis, and admits an optional braneworld realization as reactive interaction mass.
Keywords: collar orbifold; scale covering monoid; simple deck transformation groups; non-local heat kernel; Poisson summation; Hadamard constant ; Riemann xi-function; connected boundary impedance; interaction mass; net-zero entropy.
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BibTeX
@misc{chung2026arithmetic,
title = {{The Arithmetic of Spacetime: Collar Covering Monoids, Heat-Kernel Modularity, and the Geometric Origin of Prime Numbers}},
author = {Jinhyuk Fred Chung},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23669939},
url = {https://zenodo.org/records/23669939},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Arithmetic of Spacetime: Collar Covering Monoids, Heat-Kernel Modularity, and the Geometric Origin of Prime Numbers
AU - Jinhyuk Fred Chung
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23669939
UR - https://zenodo.org/records/23669939
AB - In spectral geometry and operator-algebraic models of the Riemann zeta function—such as Ray-Singer determinants and Bost-Connes systems—the multiplicative monoid of positive integers is introduced as an external algebraic input. Here, on the compact normal collar orbifold C_{ell_p} = S^1_{2 ell_p}/Z_2 ~ [0, ell_p] supporting the ultraviolet boundary layer of a Z_2-symmetric hypersurface, we prove that smooth metric-scaling endomorphisms preserving the interface and boundary stratum form a commutative covering monoid End_scale(C_{ell_p}) ~ (N_{>0}, x) under composition. By geometric pullback, these endomorphisms generate the complete fluctuating eigenbasis of the self-adjoint Neumann Laplacian Delta_perp = -d^2/dy^2 from a single fundamental mode. The Galois correspondence of covering spaces and the Jordan-Holder theorem identify geometrically irreducible coverings with simple deck transformation groups Deck(Phi_p) ~ Z/pZ of prime degree, causing the normalized Mellin-Haar transform of the collar heat kernel A_perp(tau) = exp(-tau ell_p^2 Delta_perp / pi) to factorize unconditionally into the Euler product of zeta(s) for Re(s) > 1. In the meromorphically continued connected impedance Z_int(s) = -zeta'(s)/zeta(s), composite coverings with coprime factors possess decomposable direct-product deck groups Deck(Phi_{ab}) ~ Deck(Phi_a) x Deck(Phi_b) whose cumulants cancel identically, isolating prime-power coverings without invoking Mobius inversion. Poisson modular inversion of the Schwinger proper-time integral recovers the completed xi-function xi(s) = xi(1-s) and yields the Hadamard constant B = xi'(0)/xi(0) = -1 + (1/2) ln(4 pi) - gamma/2 directly from three collar invariants: vacuum-pole subtraction, Gaussian symplectic volume, and scale-drift entropy. This self-contained construction derives the Euler product from collar topology without claiming a proof of the Riemann hypothesis, and admits an optional braneworld realization as reactive interaction mass. Keywords: collar orbifold; scale covering monoid; simple deck transformation groups; non-local heat kernel; Poisson summation; Hadamard constant ; Riemann xi-function; connected boundary impedance; interaction mass; net-zero entropy.
ER -