Finite-flux normal kernels and their spatial projections

A finite connection flux can select a complete finite set of normalizable Dirac modes on an ordinary Euclidean plane. This gives an explicit spatial realization of a finite polynomial family. The kernel is not separated from the continuum, however, and bounded local multiplication generally forces its complement. These facts matter when using such a family in a mass-frequency operator.

The connection below is a prescribed coefficient of the same-field spatial calculation. Its physical source in IDWT is not derived. No exchanged particle or Yukawa interaction is part of this test. The plane is unbounded ordinary space, without compactification.

An explicit full-plane kernel

Use z=x+iy, r=|z| and phi=(q+2) log(1+r^2)/2, with q a nonnegative integer. Set A=(-partial_y phi,partial_x phi). The normal Dirac operator and its chiral derivative are

$$D_A=\begin{pmatrix}0&Q^\dagger\\Q&0\end{pmatrix},\qquad Q=-2i(\partial_{\bar z}+\partial_{\bar z}\phi). \tag{1}$$

Its curvature and total flux are

$$B=\Delta\phi=\frac{2(q+2)}{(1+r^2)^2},\qquad \int B\,dx\,dy=2\pi(q+2). \tag{2}$$

The general regular upper-chirality zero mode is e^(-phi) f(z), with f entire. The lower chirality would contain e^(+phi) times an anti-holomorphic entire function and has no nonzero L2 mode. Angular orthogonality in the power series of f shows that its nonzero powers must have degree k<q+1. Thus the complete kernel has dimension q+1, with canonical basis

$$\varphi_k(z)=\sqrt{\frac{q+1}{\pi}\binom qk} \frac{z^k}{(1+|z|^2)^{(q+2)/2}},\qquad k=0,\ldots,q. \tag{3}$$

The count follows from the full spatial integrability condition; it is not a truncation of otherwise normalizable zero modes. This is an explicit case of the Aharonov–Casher zero-mode calculation. The distinction between integer flux and a marginal, non-normalizable next mode on the plane is also explained in the direct infinite-plane treatment.

The complete projection kernel has diagonal

$$\sum_{k=0}^q|\varphi_k(z)|^2 =\frac{q+1}{\pi(1+r^2)^2}. \tag{4}$$

This is a count of available modes. It does not specify that one particle occupies all of them, or turn their count into a frequency.

Bounded spatial coordinates give the angular matrices

Define the three bounded real functions

$$(n_x,n_y,n_z)= \frac{(2\operatorname{Re}z,2\operatorname{Im}z,r^2-1)}{1+r^2}. \tag{5}$$

Their unit-vector parameterization describes functions of the two ordinary coordinates. It introduces neither another physical space nor compactification of the plane.

Let P project onto (3). In the basis k=0,...,q, define J_z=k-q/2 and J_+ with entries sqrt((k+1)(q-k)) from k to k+1. Direct beta integrals on the plane give

$$Pn_iP=\frac{2J_i}{q+2},\qquad Pn_in_jP=\frac{\delta_{ij}}{q+3}I +\frac{2\{J_i,J_j\}}{(q+2)(q+3)}. \tag{6}$$

Consequently the bounded local multiplier T(z)=beta+(q+2)[n_z+gamma(n_x+i n_y)]/2 projects to beta+J_z+gamma J_+. Its complete Hermitian pair therefore has the same projected rising-product frequency as the finite Schwinger normal band. All factors in (6) come from canonical plane integrals, not separately adjusted mode amplitudes.

The same multiplication forces other normal modes

Projection does not make the bounded coordinate functions preserve ker D_A. Their complete leakage is fixed by (6):

$$\sum_i Pn_i(I-P)n_iP=\frac{2}{q+2}I. \tag{7}$$

More generally, subtracting (Pn_iP)(Pn_jP) from the second identity in (6) gives every cross term of the omitted source. Both components of the Hermitian pair must be included.

For its small positive branch at large gamma, the complete source outside P obeys

$$\|(I-P)V\Psi\| \sim\gamma\sqrt{\frac{(q+1)(q+2)}{q+3}}. \tag{8}$$

For q=5, beta=15.5 and gamma=10000, the projected frequency is 1.11383945e-15, while the actual complementary multiplier source has norm about 22912.877. A finite projected frequency is therefore not by itself a solution of the full spatial equation.

There is no complementary normal gap

The connection decays as 1/r. A normalized smooth spinor supported in a disk centered at distance R, with disk width sqrt(R), has Dirac residual bounded by O(R^(-1/2))+O(R^(-1)). Its support escapes to infinity. Removing its projection on the finite kernel changes its norm by a quantity tending to zero, and does not change its Dirac residual. This is a Weyl sequence in the complete complement. Thus zero lies in the essential spectrum, and no positive bound of the form ||D_A Q psi|| >= gap ||Q psi|| holds there.

A finite numerical box can manufacture a gap that the actual plane does not possess. Controlling a perturbed finite-flux kernel instead requires an appropriate full weighted resolvent argument or additional derived supporting dynamics. The finite count alone is insufficient.

Spatial width also distinguishes this band from an oscillator

The highest mode has density proportional to r^(-4) at large radius. It is square integrable, but its second position moment diverges logarithmically. More explicitly, with t=R^2/(1+R^2),

$$\int_{r<R}r^2|\varphi_q|^2\,d^2x =(q+1)\left[\log(1+R^2)- \sum_{j=1}^{q+1}\frac{t^j}{j}\right] \sim 2(q+1)\log R. \tag{9}$$

This has a simple general explanation. If the highest polynomial zero mode had finite second moment, multiplying it by z would give an additional square-integrable holomorphic zero mode. A nonzero finite-dimensional kernel of this form cannot have finite second moment for every vector.

A finite characteristic core size or a radius containing a fixed fraction of the norm still exists. Equation (9) concerns the second moment specifically; it does not by itself refute every possible notion of a particle's cavity size.

finite-flux-normal-20260914T050943_681569Z.json passes 328 checks. It verifies actual Cartesian zero-mode equations, canonical plane norms, every first and second coordinate projection, the full local multiplier source, projected frequencies and the complete highest-mode tail. The separate gapped construction in A gapped finite normal kernel for the simplex parent addresses the complementary-gap requirement using a different prescribed spatial operator.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.