Black Holes on the Spiral: A phase-horizon reading of collapse, the Euler identity, and the "universe as black hole."
Abstract
This paper asks three questions of the Euler Universe apparatusโthe phasor ๐๐๐, its logarithmic-spiral extension ๐(๐+๐)๐, and the relational redshift law ๐ = (๐/๐ป0) ln(1 + ๐ง)โturned now toward black holes. (i) Does the lens clarify what a black hole is and how it forms? (ii) Is there something in Eulerโs identity that could bring about a black hole? (iii) Is the universe itself a black hole? The honest result is one recurring shape. The lens re-describes the exterior of a black hole beautifullyโthe event horizon becomes a phase horizon where accumulated phase diverges, exactly as the cosmological redshift law diverges as ๐ง โ โ; light near the photon sphere winds as a logarithmic spiral; and the ๐โ๐ demagnification of successive photon rings, with the 2๐ periodicity that fixes the Hawking temperature, are genuinely Euler-flavored facts of standard physics. But at every turn the lens reconstructs general relativity and quantum field theory rather than extending them, and it inherits, unchanged, the four falsifications that already retired the no-expansion cosmology. The โuniverse is a black holeโ claim reduces to the flatness condition ๐๐ = ๐/๐ป0โa true and pretty identity that is evidence of critical density, not of literal black-hole-hood. As with the parent project, the spiral is an excellent intuition pump and not a replacement physics.
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BibTeX
@misc{sanders2026black,
title = {{Black Holes on the Spiral: A phase-horizon reading of collapse, the Euler identity, and the "universe as black hole."}},
author = {Nicholas Archer Sanders},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20802418},
url = {https://doi.org/10.5281/zenodo.20802418},
note = {CC BY},
}
RIS
TY - UNPB
TI - Black Holes on the Spiral: A phase-horizon reading of collapse, the Euler identity, and the "universe as black hole."
AU - Nicholas Archer Sanders
PY - 2026
DA - 2026/06/22
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20802418
UR - https://doi.org/10.5281/zenodo.20802418
AB - This paper asks three questions of the Euler Universe apparatusโthe phasor ๐๐๐, its logarithmic-spiral extension ๐(๐+๐)๐, and the relational redshift law ๐ = (๐/๐ป0) ln(1 + ๐ง)โturned now toward black holes. (i) Does the lens clarify what a black hole is and how it forms? (ii) Is there something in Eulerโs identity that could bring about a black hole? (iii) Is the universe itself a black hole? The honest result is one recurring shape. The lens re-describes the exterior of a black hole beautifullyโthe event horizon becomes a phase horizon where accumulated phase diverges, exactly as the cosmological redshift law diverges as ๐ง โ โ; light near the photon sphere winds as a logarithmic spiral; and the ๐โ๐ demagnification of successive photon rings, with the 2๐ periodicity that fixes the Hawking temperature, are genuinely Euler-flavored facts of standard physics. But at every turn the lens reconstructs general relativity and quantum field theory rather than extending them, and it inherits, unchanged, the four falsifications that already retired the no-expansion cosmology. The โuniverse is a black holeโ claim reduces to the flatness condition ๐๐ = ๐/๐ป0โa true and pretty identity that is evidence of critical density, not of literal black-hole-hood. As with the parent project, the spiral is an excellent intuition pump and not a replacement physics.
ER -