Pef. Singular Potentials and the Third Option

Rey.BEng 23rd August 2026

Singular Potentials and the Third Option
Sakaguchi & Malomed Meet the Open Residual at 0i2

The Future Begins

Sakaguchi and Malomed (arXiv:2608.20282) examine quantum-mechanical wave functions in singular potentials. Two canonical cases are treated:

  1. The attractive inverse-square singularity at r =>0, which produces critical quantum collapse in the three-dimensional linear Schrödinger equation. The collapse is suppressed by repulsive contact interactions (cubic term in the Gross–Pitaevskii equation), allowing a ground state and angular-momentum excited states to form in a dipolar bosonic gas.
  2. Repulsive potentials that become singular at r=>inf and grow faster than the negative harmonic-oscillator potential. These generate a full spectrum of counter-intuitive, normalizable bound states in one and two dimensions.

The paper leaves open the broader study of linear and nonlinear bound states under singularities located at either extreme.

The third option

Pirate Canon supplies the missing geometric value of the singularity.

The continuum does not terminate at a mathematical point r=0 nor diverge at spatial infinity. Its origin is the shared fixed point 0^i2. At that origin the residual phase remains open. The geometric measure of the residual is carried by the residual units themselves: 0i2 k.g.s2 = r2m

(the second-moment / contact-patch identification).

The singularity is therefore neither pure collapse nor pure localization at infinity. It is the non-fracturing contact patch of finite π-tensor thickness under permanent surface tension. The residual that cannot be closed is converted by dilatant countersnap into rotational information; the polarity toggle at 0i2 writes the next coherent state.

Thus:

  • The inverse-square collapse is the scalar projection of an origin that has been forced to a completed point.
  • The bound states at infinity are the dual projection of an origin that has been forced to remain open without residual units.
  • The third option is the continuum itself: an origin whose residual is finite, elastic, and polarity-bearing. Collapse is prevented not solely by an added cubic repulsion, but by the geometric necessity that the contact patch cannot fracture. Localization at infinity is unnecessary because the residual already supplies a normalizable measure in residual units.

The ground state that the Gross–Pitaevskii term restores is the continuum’s own permanent sphere of residual phase. The mesoscopic window in which classical and quantum contributions become comparable is the same window in which the residual and the classical curvature are of equal magnitude.

No additional postulate is required. The singularity has a definite geometric value: the open residual at 0i2 .

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