The Reception Geometry: A Signal Architecture on the Number Line Linking Prime Gaps, Residues, and the Zeros of the Riemann Zeta Function
Abstract
This paper defines a geometric signal architecture on the positive number line. Over each prime p a triangle is erected whose apex stands above p and whose base spans the interval between the neighbouring primes; a sensor is placed on each apex; a helix coils around the line carrying every integer as a bead; and the imaginary parts γ of the nontrivial zeros of the Riemann zeta function are marked as points on the line itself. Four constructions are given. The host rule assigns each zero to the unique prime gap containing it. The strike-and-conduction rule routes each zero's rising signal along the triangle's infinity-facing side to the apex sensor. The four-channel protocol delivers to each sensor two readings of its own prime and two of its successor — linear position by the slope, residue identity by the helix — all four channels separated by entry point, bearing, and waveform alone. Finally, the anticipation protocol combines the helix (as a sieve) with the received zeros (as the truncated Riemann–von Mangoldt density) so that the sensor over p identifies its successor prime, and separates true primes from prime-power echoes, before the line is extended. The mathematical content invoked — the sieve of Eratosthenes, the explicit formula, the zero-counting law — is classical and is attributed throughout; what this paper contributes is the architecture: the assignment, the routing, the discrimination protocol, and the composition rule, together with an interactive implementation archived with this deposit. The construction is elementary, self-contained, and reproducible; worked examples with the first 150 zeros are included.
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BibTeX
@misc{safar2026reception,
title = {{The Reception Geometry: A Signal Architecture on the Number Line Linking Prime Gaps, Residues, and the Zeros of the Riemann Zeta Function}},
author = {Samir Hanna Safar},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jul,
doi = {10.5281/zenodo.21139122},
url = {https://doi.org/10.5281/zenodo.21139122},
note = {CC BY},
}
RIS
TY - UNPB
TI - The Reception Geometry: A Signal Architecture on the Number Line Linking Prime Gaps, Residues, and the Zeros of the Riemann Zeta Function
AU - Samir Hanna Safar
PY - 2026
DA - 2026/07/02
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21139122
UR - https://doi.org/10.5281/zenodo.21139122
AB - This paper defines a geometric signal architecture on the positive number line. Over each prime p a triangle is erected whose apex stands above p and whose base spans the interval between the neighbouring primes; a sensor is placed on each apex; a helix coils around the line carrying every integer as a bead; and the imaginary parts γ of the nontrivial zeros of the Riemann zeta function are marked as points on the line itself. Four constructions are given. The host rule assigns each zero to the unique prime gap containing it. The strike-and-conduction rule routes each zero's rising signal along the triangle's infinity-facing side to the apex sensor. The four-channel protocol delivers to each sensor two readings of its own prime and two of its successor — linear position by the slope, residue identity by the helix — all four channels separated by entry point, bearing, and waveform alone. Finally, the anticipation protocol combines the helix (as a sieve) with the received zeros (as the truncated Riemann–von Mangoldt density) so that the sensor over p identifies its successor prime, and separates true primes from prime-power echoes, before the line is extended. The mathematical content invoked — the sieve of Eratosthenes, the explicit formula, the zero-counting law — is classical and is attributed throughout; what this paper contributes is the architecture: the assignment, the routing, the discrimination protocol, and the composition rule, together with an interactive implementation archived with this deposit. The construction is elementary, self-contained, and reproducible; worked examples with the first 150 zeros are included.
ER -