Massless radial scalar contact

Summary: Regular flat radial scalar Dirac fields have no finite-norm stationary state for cubic or smooth saturating contact; the homogeneous root is also excluded under the stated virial hypotheses.

Equation and exact scope

Consider only the flat local scalar Dirac equation

$$i\partial_t\Psi= \left[-i\alpha\cdot\nabla+g(\Psi^\dagger\beta\Psi)\beta\right]\Psi, \qquad x\in\mathbb R^d,\quad d\ge1,\quad g\in\mathbb R. \tag{1}$$

There is no bare mass, prescribed background or additional kernel. The coordinates are ordinary physical spatial coordinates. This comparison does not replace the proposed curved spatial geometry of an IDWT particle.

Choose a constant unit spinor $\chi$ with $\beta\chi=\chi$ and the standard stationary radial ansatz

$$\Psi(t,x)=e^{-i\omega t}\left[ f(r)\chi+i u(r)\alpha(\widehat x)\chi\right], \qquad r=|x|,\quad \omega\in\mathbb R. \tag{2}$$

The profiles may be complex. Assume they are classical away from zero, regular at the origin with finite $f(0)$ and $u(0)=0$, and have finite positive norm

$$Q=|S^{d-1}|\int_0^\infty r^{d-1} \bigl(|f(r)|^2+|u(r)|^2\bigr)dr<\infty. \tag{3}$$

For $d=1$, the spherical factor accounts for the two half-lines in the even/odd extension of (2). All allowed constant rest polarizations $\chi$ are retained. Higher angular sectors and spatially varying polarization textures are outside this ansatz.

The Clifford relations give $\sigma=\Psi^\dagger\beta\Psi=|f|^2-|u|^2$ and reduce (1) to

$$\boxed{f'=-(\omega+V)u,\qquad u'=(\omega-V)f-\frac{d-1}{r}u, \qquad V=g(|f|^2-|u|^2).} \tag{4}$$

This radial representation follows the spatial Dirac construction; it does not turn the physical manifold into a one-dimensional space. The angular dependence is contained explicitly in (2).

Under precisely these assumptions, the only solution is zero. The proof separates zero and nonzero frequency.

Zero frequency: a monotone scalar density

At $\omega=0$, differentiate the scalar using both equations (4):

$$\boxed{\sigma' =2\operatorname{Re}(\bar f f'-\bar u u') =\frac{2(d-1)}r|u|^2\ge0.} \tag{5}$$

The nonlinear terms cancel for either sign of $g$ and for complex profiles. Regularity gives $\sigma(0)=|f(0)|^2\ge0$. If $\sigma$ were positive at any radius, it would remain bounded below by a positive constant thereafter. Since $|f|^2+|u|^2\ge\sigma$, (3) would diverge. Hence $\sigma\equiv0$ and $V\equiv0$.

For $d>1$, (5) also gives $u\equiv0$, and then $f\equiv0$. For $d=1$, (4) at zero frequency and $V=0$ makes both profiles constant, and finite norm again makes both zero.

Nonzero frequency: the free radial fundamental matrix

Let $\omega\ne0$ and remove the radial measure by defining

$$U(r)=r^{(d-1)/2}\begin{pmatrix}f(r)\\u(r)\end{pmatrix}.$$

Then $Q/|S^{d-1}|=\int\|U(r)\|^2dr$, and (4) becomes

$$U'=\left[ \begin{pmatrix}(d-1)/(2r)&-\omega\\ \omega&-(d-1)/(2r)\end{pmatrix} +\begin{pmatrix}0&-V\\-V&0\end{pmatrix} \right]U. \tag{6}$$

Set $k=|\omega|$ and $\nu=(d-2)/2$. A fundamental matrix for the free part is

$$\boxed{F(r)=\sqrt r\begin{pmatrix} J_\nu(kr)&Y_\nu(kr)\\ \operatorname{sgn}(\omega)J_{\nu+1}(kr) &\operatorname{sgn}(\omega)Y_{\nu+1}(kr) \end{pmatrix},\qquad \det F=-\frac2{\pi\omega}.} \tag{7}$$

The Bessel recurrence verifies the differential equation. The Wronskian identity gives the determinant, and the large-argument expansions show that both $F$ and $F^{-1}$ are bounded on every sufficiently far tail. Its columns have oscillatory, nondecaying amplitudes in the measure-normalized coordinate $U$.

Finite norm would make the self-potential integrable

For any fixed $R>0$, the assumed finite norm implies

$$\int_R^\infty |V(r)|dr \le |g|\int_R^\infty(|f|^2+|u|^2)dr \le\frac{|g|}{R^{d-1}} \int_R^\infty r^{d-1}(|f|^2+|u|^2)dr<\infty. \tag{8}$$

Along a hypothetical solution, regard its actual real $V(r)$ as a fixed coefficient and put $W=F^{-1}U$. Equation (6) gives

$$W'=C(r)W,\qquad C(r)=F(r)^{-1}\begin{pmatrix}0&-V(r)\\-V(r)&0\end{pmatrix}F(r), \qquad \int_R^\infty\|C(r)\|dr<\infty. \tag{9}$$

For completeness, let $Y(R)=I$ and $Y'=CY$. Gronwall's inequality bounds $Y$ and its inverse by the exponential of the integral in (9). Their derivatives are integrable, so both have limits at infinity. Since $Y^{-1}Y=I$, the limiting matrix $Y_\infty$ is invertible. Thus every nonzero initial $W(R)$ has a nonzero limit $W_\infty=Y_\infty W(R)$.

The bound on $F^{-1}$ now implies that $\|U(r)\|$ is bounded below by a positive constant on a sufficiently far tail. Consequently $\int\|U(r)\|^2dr$ diverges, contradicting (3). A solution that vanishes at one positive radius is identically zero by the ordinary uniqueness theorem for (4). This proves the nonzero-frequency case.

Smooth self-mass has the same radial obstruction

The proof above uses the cubic law only in the integrability bound (8) and in uniqueness at a zero field. It therefore applies to any real scalar source $V=F(J)$ with $F(0)=0$, a well-defined radial equation, and

$$|F(J)|\le C|J|. \tag{10}$$

Here $J=|f|^2-|u|^2$ is the scalar current, and $C$ is fixed for the specified equation. The zero-frequency identity is independent of $F$. At nonzero frequency, (10) gives exactly the integrable coefficient in (9). Smoothness of the saturating source below supplies ordinary uniqueness.

In particular, the common self-mass action on a constant coherent coefficient ray has

$$F_N(J)=M_N\frac{J}{\sqrt{J^2+\sigma^2}},\qquad M_N=\mu D_N>0,\qquad \sigma>0, \qquad |F_N(J)|\le\frac{M_N}{\sigma}|J|. \tag{11}$$

Every fixed positive smoothing scale therefore has the same radial nonexistence result, however small that scale is. Its large-density mass coefficient does not create a nonzero asymptotic mass as the field vanishes. The conclusion is an exact consequence of (5)–(11), not a claim based on a finite numerical tail or an unsuccessful search.

A useful wider sufficient condition follows directly from Hölder's inequality. If $0<p<1$, $|F(J)|\le C|J|^p$, and $p>1/d$, put $\rho=|f|^2+|u|^2$. For $R>0$,

$$\begin{aligned} \int_R^\infty |F(J)|\,dr &\le C\int_R^\infty (r^{d-1}\rho)^p r^{-(d-1)p}\,dr\\ &\le C\left(\int_R^\infty r^{d-1}\rho\,dr\right)^p \left(\int_R^\infty r^{-\frac{(d-1)p}{1-p}}\,dr\right)^{1-p} <\infty. \end{aligned} \tag{12}$$

The zero-frequency argument still requires $F(0)=0$. At a zero vector, the bound $|F(J)|\le C\rho^p$ gives a linear bound on the radial vector field in a sufficiently small bounded neighborhood, so a solution cannot emerge from zero at a positive radius. This suffices for the last step of the tail argument even when $F$ is not differentiable at scalar-null nonzero vectors. The statement concerns classical radial solutions of the specified source law.

The threshold in (12) is a limit of this sufficient argument. For $d>1$, take the comparison density $\rho(r)=r^{-d}(\log r)^{-a}$ on $r\ge e$, with $1<a\le d$. Its weighted norm is finite, but $\int\rho^{1/d}dr$ diverges. This is not a solution of the field equation and does not establish localization at the threshold.

The homogeneous root requires a different proof

At $\sigma=0$, the potential is $M_N\operatorname{sgn}J$ wherever $J\ne0$. It does not obey (10). A fixed-sign coherent field on a trivial flat coefficient bundle nevertheless obeys a constant-mass Dirac equation:

$$\left(-i\alpha\cdot\nabla\pm M_N\beta\right)\phi =\omega\phi. \tag{13}$$

Squaring in Fourier space gives $(|p|^2+M_N^2-\omega^2)\widehat\phi(p)=0$. For any real $\omega$, the allowed momentum set is empty or a sphere of Lebesgue measure zero. An $L^2$ field supported on that set is zero. This fixed-sign statement includes every spatial polarization and does not require radial symmetry. It assumes the trivial flat connection; it does not apply to a nontrivial curved coefficient bundle.

Within the regular radial ansatz (2), sign changes can also be tested. Assume in addition that the stationary field admits integration by parts and the norm-preserving spatial dilation used in local scalar normalization. Write

$$E=T+M_N\int|J|\,dx,\qquad T=\int\phi^\dagger(-i\alpha\cdot\nabla)\phi\,dx, \qquad Q=\int\phi^\dagger\phi\,dx. \tag{14}$$

The dilation $\phi_\lambda(x)=\lambda^{d/2}\phi(\lambda x)$ leaves both $Q$ and the root interaction unchanged and multiplies $T$ by $\lambda$. Stationarity and the integrated field equation give

$$T=0,\qquad \omega Q=M_N\int|J|\,dx,\qquad \boxed{0\le\omega\le M_N.} \tag{15}$$

This argument remains valid if the source at $J=0$ is interpreted by a real subgradient $V\in[-M_N,M_N]$, since $VJ=M_N|J|$ there and elsewhere. No choice at scalar zero is used in the following interval, where the current remains strictly positive.

A nonzero regular radial solution must have $f(0)\ne0$; the bounded source and the integral form of (4) otherwise force both profiles to vanish near zero and hence everywhere. Remove the common phase of $f(0)$. Until the first scalar zero, the radial equation has the real constant coefficient $V=M_N$. For $0\le\omega<M_N$, set $k=\sqrt{M_N^2-\omega^2}$ and $\nu=(d-2)/2$. Its unique regular solution is

$$\begin{aligned} f(r)&=f(0)\,{}_0F_1\!\left(;\frac d2;\frac{k^2r^2}{4}\right),\\ u(r)&=-f(0)\frac{(M_N-\omega)r}{d}\, {}_0F_1\!\left(;\frac d2+1;\frac{k^2r^2}{4}\right),\\ \left|\frac{u}{f}\right| &=\sqrt{\frac{M_N-\omega}{M_N+\omega}} \frac{I_{\nu+1}(kr)}{I_\nu(kr)}<1. \end{aligned} \tag{16}$$

The positive series for ${}_0F_1$ makes $f(r)\ge f(0)>0$. For a direct proof of the ratio bound, let $R(x)=I_{\nu+1}(x)/I_\nu(x)$. Its recurrence gives $R'=1-R^2-(d-1)R/x$, with $R(x)\sim x/d$ at zero. For $d>1$, the derivative is strictly negative at $R=1$, so a solution starting below one cannot cross it. At $d=1$, the regular ratio is $\tanh x<1$. Thus (16) never reaches a scalar zero. It remains the actual root solution and has infinite norm. At $\omega=M_N$, the limiting solution is simply $f=f(0)$, $u=0$, again with infinite norm.

Consequently no nonzero regular radial root solution satisfying the stated virial hypotheses can evade localization failure by changing the sign of its scalar current. The argument does not classify nonradial sign-changing fields, singular solutions, other current-matrix structures, or other spatial geometries.

Consequence for the contact construction

Replacing $g$ by the graph coefficient $\kappa D_N$ cannot change this radial existence result. The graph pairing alone therefore does not supply a normalized stationary profile of form (2) when reduced to (1) on flat infinite space.

This conclusion identifies the limits of a specific spatial reduction. The cubic and smooth proofs use the flat infinite tail, zero bare mass, an integrable scalar self-potential and the stated radial form. The root case uses the additional virial hypotheses just stated. The fixed-sign Fourier argument (13) has its separate nonradial scope. The radial tail proof also does not apply to the nonlocal scalar moment model, whose self-generated quadratic mass profile grows at infinity.

The exact massive one-dimensional solution in local scalar normalization provides a positive control: adding $m>0$ creates a frequency gap and admits decaying profiles. Its fixed-norm frequency is nevertheless a nonlinear function of the contact coefficient.

The positive-current localization theorem gives a separate full-field restriction for the convex matrix-current hierarchy. Its virial and scalar identity retain every coefficient and spinor component, and identify a spatially changing mass-sign projector as a possible contribution outside a fixed positive-current decomposition.

Verification

local_scalar_profile_checks.py checks (4) against independent Cartesian differentiation with full Clifford matrices and complex rest polarizations, (5) for complex profiles, and (7) by independent Bessel differentiation and determinants. Representative massless radial initial-value solutions have nonzero tail norm density in the $U$ variable. Those finite integrations illustrate the mechanism; the nonexistence result is the argument (5)–(9), not a conclusion drawn from unsuccessful numerical shooting.

local_self_mass_localization_checks.py extends the verification to the smooth source bound, complex zero-frequency identity, independent Hölder integrals and illustrative smooth tails. It checks the root profiles by both spatial differences and integration of the actual sign-dependent equation in dimensions $1,2,3,5,6,10$. Independent full-profile dilations include scalar sign changes and verify the norm, root energy and general smooth virial. These checks support the displayed analytic proofs with their stated scope.