Uncertainty on the Spiral: The Real and Imaginary Parts of the Quantum Exponent
Abstract
Quantum mechanics carries two uncertainty relations usually taught as one. This paper reads them as the real and imaginary parts of a single complex exponent e^(λ+i)θ. Position–momentum (ΔxΔp ≥ ℏ/2) is an i-phenomenon — Fourier conjugacy, substance-independent, importing ℏ/2 without deriving it; the slogan that uncertainty is "the equation of state of the atomic realm" is named and refused. Energy–time is different: not a conjugate-operator relation (t is not an operator; Pauli) but lifetime–linewidth, ΔE ≈ ℏΓ, the complex energy E₀ − iΓ/2 of Gamow and Weisskopf–Wigner — the inward λ-drain at the atomic floor of the nesting ladder, with Γ = −2ωλ from Entropy on the Spiral. This is the real equation of state the exponent discloses: a relation carrying a measurable rate. The framework is then stress-tested where most exposed — number–phase uncertainty against the absence of a self-adjoint phase operator (Dirac; Susskind–Glogower; Pegg–Barnett) — and the result is logged as a wall, not rescued. One organizing arrangement, one logged obstruction, no new number. The parent cosmology's falsification is reprinted.
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BibTeX
@misc{sanders2026uncertainty,
title = {{Uncertainty on the Spiral: The Real and Imaginary Parts of the Quantum Exponent}},
author = {Nicholas Archer Sanders},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jul,
doi = {10.5281/zenodo.21131949},
url = {https://doi.org/10.5281/zenodo.21131949},
note = {CC BY},
}
RIS
TY - UNPB
TI - Uncertainty on the Spiral: The Real and Imaginary Parts of the Quantum Exponent
AU - Nicholas Archer Sanders
PY - 2026
DA - 2026/07/02
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21131949
UR - https://doi.org/10.5281/zenodo.21131949
AB - Quantum mechanics carries two uncertainty relations usually taught as one. This paper reads them as the real and imaginary parts of a single complex exponent e^(λ+i)θ. Position–momentum (ΔxΔp ≥ ℏ/2) is an i-phenomenon — Fourier conjugacy, substance-independent, importing ℏ/2 without deriving it; the slogan that uncertainty is "the equation of state of the atomic realm" is named and refused. Energy–time is different: not a conjugate-operator relation (t is not an operator; Pauli) but lifetime–linewidth, ΔE ≈ ℏΓ, the complex energy E₀ − iΓ/2 of Gamow and Weisskopf–Wigner — the inward λ-drain at the atomic floor of the nesting ladder, with Γ = −2ωλ from Entropy on the Spiral. This is the real equation of state the exponent discloses: a relation carrying a measurable rate. The framework is then stress-tested where most exposed — number–phase uncertainty against the absence of a self-adjoint phase operator (Dirac; Susskind–Glogower; Pegg–Barnett) — and the result is logged as a wall, not rescued. One organizing arrangement, one logged obstruction, no new number. The parent cosmology's falsification is reprinted.
ER -