A node rule for information-theoretic nonlinear extensions of the Schrödinger equation: the deformed uncertainty relation and the harmonic oscillator

Parwani and Tabia established in 2006 that in information-theoretic nonlinear extensions of the Schrödinger equation the energy shift of an excited state is carried predominantly by the nodes of the unperturbed wavefunction, while nodeless ground states receive only subleading shifts. This paper shows that for the rotation-invariant branch of that family, regulated by the small components of the Dirac spinor as proposed by Liew and Parwani, the node contribution closes in a single expression: each node of the density contributes exactly −4πC²/ε to the shift integral, where C is the slope of the unnormalised wavefunction at the node and ε is the ratio of the reduced Compton wavelength to the characteristic length of the system. The rule contains no free parameter and covers every bound state of every smooth potential. Two applications follow. For hydrogen it reproduces in three lines the closed coefficient obtained previously by radial integration. For the harmonic oscillator it gives a finite sum over the roots of the Hermite polynomials; the first two evaluate in closed form. The oscillator spectrum ceases to be equidistant, with two parameter-free level-spacing ratios, 1.4261 and 1.7514, which separate this model from the commutator branch, where the corresponding ratios are integers. The relative shift scales as the square root of the trap frequency. Two questions left open in the companion paper are settled: the momentum entering the deformed uncertainty relation is the one a time-of-flight measurement records, and the leading constant of that relation is 4, so that the Gaussian is its minimum-uncertainty state to first order. The same constant governs the entropy-power form, which is uniformly stronger for every non-Gaussian state; the corresponding statement for the product of entropy powers is formulated as a conjecture, with the obstruction to proving it identified. Throughout, other deformations are discussed under an explicit rule: parameters of different frameworks are never compared, only predictions for one and the same measured observable. The paper claims neither the family of equations nor the node mechanism as new, and reports no evidence for a nonzero deformation. Companion paper: https://doi.org/10.5281/zenodo.22072059A working log documenting all calculations and twenty-one self-corrections is included in the files.