Entropy on the Spiral: A Reservoir Reading of the Euler Universe — U = √(S/S₀)·e^(iθ)
Abstract
We present a reservoir reading of the Euler Universe in which one principle — a system disperses energy and information into the largest accessible reservoir of microstates until its entropy is maximal, then rests — organizes both the cosmic and the atomic behavior of the spiral U(θ) = e^((λ+i)θ). The principle selects the sign of the growth parameter λ in each domain: outward (λ > 0) in the cosmos, whose reservoir is space and its de Sitter horizon, and inward (λ < 0) at the atomic scale, whose reservoir is the continuum of electromagnetic field modes. At the atomic scale we derive λ explicitly: identifying the spiral's modulus with the survival amplitude of a decaying state gives λ = −Γ/(2ω), the inward pitch fixed by, and only by, the positivity of the decay rate. Combined with the spiral's entropy relation λ = ½ d(ln S)/dθ, this reproduces the decay as d(ln S_atom)/dt = −Γ — the population of the excited sector falling at exactly the decay rate, with the second law preserved not within the atom but by the field reservoir that carries the dispersed energy away. The square in the master expression has two standard faces: the area law S ∝ R² cosmically and the Born rule P ∝ |c|² atomically — what the modulus² measures is a genuine entropy in the cosmos and a population in the atom, unified in form and by the second law acting on the total, not as one quantity. The two domains terminate at opposite bounds of the same kind: an upper ceiling, the Gibbons–Hawking horizon entropy S = A/4, approached once and slowly; and a lower floor, the quantum ground state, reached locally and fast. The result is a nesting — countless fast local floors inside one slow global ceiling — that explains why a universe of ever-increasing entropy is nowhere near maximal: maximality is local and reservoir-relative. The reading is a lens, not a law.
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BibTeX
@misc{sanders2026entropy,
title = {{Entropy on the Spiral: A Reservoir Reading of the Euler Universe — U = √(S/S₀)·e^(iθ)}},
author = {Nicholas Archer Sanders},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20931935},
url = {https://doi.org/10.5281/zenodo.20931935},
note = {CC BY},
}
RIS
TY - UNPB
TI - Entropy on the Spiral: A Reservoir Reading of the Euler Universe — U = √(S/S₀)·e^(iθ)
AU - Nicholas Archer Sanders
PY - 2026
DA - 2026/06/26
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20931935
UR - https://doi.org/10.5281/zenodo.20931935
AB - We present a reservoir reading of the Euler Universe in which one principle — a system disperses energy and information into the largest accessible reservoir of microstates until its entropy is maximal, then rests — organizes both the cosmic and the atomic behavior of the spiral U(θ) = e^((λ+i)θ). The principle selects the sign of the growth parameter λ in each domain: outward (λ > 0) in the cosmos, whose reservoir is space and its de Sitter horizon, and inward (λ < 0) at the atomic scale, whose reservoir is the continuum of electromagnetic field modes. At the atomic scale we derive λ explicitly: identifying the spiral's modulus with the survival amplitude of a decaying state gives λ = −Γ/(2ω), the inward pitch fixed by, and only by, the positivity of the decay rate. Combined with the spiral's entropy relation λ = ½ d(ln S)/dθ, this reproduces the decay as d(ln S_atom)/dt = −Γ — the population of the excited sector falling at exactly the decay rate, with the second law preserved not within the atom but by the field reservoir that carries the dispersed energy away. The square in the master expression has two standard faces: the area law S ∝ R² cosmically and the Born rule P ∝ |c|² atomically — what the modulus² measures is a genuine entropy in the cosmos and a population in the atom, unified in form and by the second law acting on the total, not as one quantity. The two domains terminate at opposite bounds of the same kind: an upper ceiling, the Gibbons–Hawking horizon entropy S = A/4, approached once and slowly; and a lower floor, the quantum ground state, reached locally and fast. The result is a nesting — countless fast local floors inside one slow global ceiling — that explains why a universe of ever-increasing entropy is nowhere near maximal: maximality is local and reservoir-relative. The reading is a lens, not a law.
ER -