Local scalar contact normalization

Summary: Spatial normalization turns a scalar contact coefficient into a profile-dependent frequency, with exact virial, scaling and massive one-dimensional benchmarks.

A spatial action after the graph pairing

The defined graph pairing reduces its single-source interaction to $g_N(\bar\Psi\Psi)^2/2$, with $g_N=\kappa D_N$ and $D_N=\binom{N+d}{d}$. This identifies a scalar nonlinear coefficient. Extracting a frequency still requires the spatial field equation.

Consider a specified flat comparison on the physical spatial space $\mathbb R^d$, with Hermitian Clifford matrices $\alpha_j,\beta$, $\hbar=c=1$, and fixed bare scalar coefficient $m\ge0$:

$$\mathcal E_g[\Psi]=\int\left[ \Psi^\dagger(-i\alpha\cdot\nabla+m\beta)\Psi +\frac g2(\Psi^\dagger\beta\Psi)^2\right]dx, \qquad i\partial_t\Psi= \left[-i\alpha\cdot\nabla+(m+g\sigma)\beta\right]\Psi, \tag{1}$$

where $\sigma=\Psi^\dagger\beta\Psi$ and $g$ is real. The spatial measure and time-kinetic norm are fixed. The spinor components do not add physical dimensions or independent fields. The complete nonlinear source and its conjugate response are given in Dirac source linearization.

The present comparison supplies no confining background or additional interaction. It must not replace the proposed global spatial geometry of a particle.

Frequency, scalar overlap and total energy

Suppose $\Psi=e^{-i\omega t}\phi$ is an admissible stationary solution, with finite integrals and sufficient decay for the variations below. Define

$$Q=\int\phi^\dagger\phi,\qquad S=\int\phi^\dagger\beta\phi,\qquad I=\int(\phi^\dagger\beta\phi)^2,\qquad T=\int\phi^\dagger(-i\alpha\cdot\nabla)\phi.$$

$Q$ is the positive phase-symmetry norm, not an assigned ordinary electric charge. $S$ is signed; $I$ is nonnegative. Multiplying the stationary equation by $\phi^\dagger$ and integrating gives

$$\boxed{\omega Q=T+mS+gI, \qquad \mathcal E=T+mS+\frac g2I =\omega Q-\frac g2I.} \tag{2}$$

Thus the nonlinear spectral frequency and the total energy per norm are different quantities. A physical rest-energy identification must specify which quantity and normalization it uses.

The norm-preserving dilation $\phi_\lambda(x)=\lambda^{d/2}\phi(\lambda x)$ leaves $Q,S$ fixed, while $T\mapsto\lambda T$ and $I\mapsto\lambda^dI$. Stationarity of $\mathcal E-\omega Q$ gives

$$\boxed{T+\frac{dg}{2}I=0,\qquad \omega Q=mS+\left(1-\frac d2\right)gI.} \tag{3}$$

This is the local quartic instance of the nonlinear Dirac virial identity; see Comech, Lemma 3.2. It uses an admissible dilation on flat infinite space. A boundary, curvature, a nonlocal kernel or a spatially varying background adds terms or changes the allowed variation.

At $m=0$ and $d>2$, (3) implies

$$\omega Q=-\frac{d-2}{2}gI,\qquad \mathcal E=-\frac{d-1}{2}gI =\frac{d-1}{d-2}\omega Q. \tag{4}$$

For example, the positive-$g$ branch cannot have positive frequency and positive norm under these hypotheses. This sign condition is only necessary; it does not prove that the other sign supports a bound state.

Removing the coupling changes the normalization

Write $g=\varepsilon h$, where $h=|g|>0$ and $\varepsilon=\operatorname{sgn}g$. For any fixed spatial operator, the substitution $\phi=h^{-1/2}\varphi$ turns (1) into the same equation at unit coupling $\varepsilon$. The frequency is unchanged by this amplitude rescaling, but

$$Q_g(\omega)=\frac{Q_\varepsilon(\omega)}h. \tag{5}$$

Consequently, a fixed physical norm $Q_*$ requires

$$\boxed{Q_\varepsilon(\omega)=hQ_*.} \tag{6}$$

The frequency is obtained from the branch's norm-frequency curve; it is not the bare coupling multiplied by an unchanged normalized profile. Several branches or turning points require their own selection.

For the massless flat equation there is an additional homogeneity. If a unit-coupling solution $\phi_0$ of frequency $\omega_0$ and norm $Q_0$ exists, then

$$\phi_{h,s}(x)=h^{-1/2}s^{1/2}\phi_0(sx),\qquad \omega_{h,s}=s\omega_0,\qquad Q_{h,s}=\frac{Q_0}{h}s^{1-d}. \tag{7}$$

The Dirac derivative and the cubic source both acquire the factor $h^{-1/2}s^{3/2}$, proving the field equation directly. For $d>1$, fixed norm therefore selects

$$\boxed{\omega_h=\omega_0 \left(\frac{Q_0}{hQ_*}\right)^{1/(d-1)}.} \tag{8}$$

This is a conditional scaling relation, not an existence theorem. The massless radial analysis shows that the standard regular radial ansatz has no such finite-norm solution in this particular flat local model.

If the only band dependence in a proposed massless spatial reduction were $g_N=\kappa D_N$, (8) would give $\omega_N\propto D_N^{-1/(d-1)}$ on the same scaling branch. The linear simplex frequency would require additional structure beyond that reduction. A band-dependent spatial operator, changing branch, physical geometric scale or other action term changes the hypotheses; none is silently excluded from IDWT.

An exact massive benchmark with fixed norm

A constructive check is available on one ordinary spatial dimension. Take $m>0$, $g=-h<0$, $\beta=\operatorname{diag}(1,-1)$ and $-i\alpha\partial_x=\left(\begin{smallmatrix}0&\partial_x\\ -\partial_x&0\end{smallmatrix}\right)$. A real stationary profile $\phi=(f,u)^{\mathsf T}$ satisfies

$$f'=-(\omega+m-h\sigma)u, \qquad u'=(\omega-m+h\sigma)f, \qquad \sigma=f^2-u^2. \tag{9}$$

For $0<\omega<m$, set $k=\sqrt{m^2-\omega^2}$ and $A(x)=m+\omega\cosh(2kx)$. The exact solution is

$$\boxed{ f(x)=\sqrt{\frac{2(m+\omega)}h}\, \frac{k\cosh(kx)}{A(x)},\qquad u(x)=\sqrt{\frac{2(m-\omega)}h}\, \frac{k\sinh(kx)}{A(x)}.} \tag{10}$$

The unit-coupling formula and its power-law extension are presented by Cuevas-Maraver and collaborators, Section 3.1. Direct differentiation checks both equations (9), including the sign and the factor $h$. Both components are retained.

Here

$$\sigma(x)=\frac{2k^2}{hA(x)},\qquad \rho(x)=\frac{2k^2[m\cosh(2kx)+\omega]}{hA(x)^2}.$$

Substitute $t=\tanh(kx)$ into the norm integral. Its integrand becomes

$$\rho(x)dx=\frac{2k}h\, \frac{(m+\omega)+(m-\omega)t^2} {[(m+\omega)-(m-\omega)t^2]^2}dt =\frac{2k}h\,d\left[ \frac{t}{(m+\omega)-(m-\omega)t^2}\right].$$

Evaluation from $t=-1$ to $t=1$, and the corresponding scalar integral, give

$$\boxed{Q=\frac{2k}{h\omega},\qquad S=\frac2h\operatorname{arcosh}\frac m\omega,\qquad I=\frac4{h^2}\left[m\operatorname{arcosh}\frac m\omega-k\right].} \tag{11}$$

The virial and total energy become $T=hI/2$ and $\mathcal E=mS$, in agreement with (2)–(3). At the same fixed norm $Q_*$ for every coupling, (11) gives the explicit frequency

$$\boxed{\omega(h;Q_*)= \frac{m}{\sqrt{1+(hQ_*/2)^2}}.} \tag{12}$$

Increasing the nonlinear coefficient changes the normalized profile and lowers this branch's frequency. Equation (12) supplies a complete counterexample to inferring a linear frequency law from the absence of a band index in a bare coupling. It is a one-dimensional comparison, not an electron or neutrino model.

Relation to the spatial moment model

The existing scalar Dirac moment branch uses a nonlocal centered quadratic spatial moment. Its interaction scales as $\lambda^{-4}$ under norm-preserving dilation, giving a different virial and a massless frequency proportional to $g_0^{1/5}$. Equations (3) and (8) instead follow from local scalar contact. The physical kernel determines which normalization law applies.

Canonical coupling normalization must also precede comparison of different retained mode families or spatial sectors. An oscillator dimension count cannot replace these profile and kinetic calculations.

It restores the full radial Dirac components and computes the normalized frequency through cubic order. Its different global hypotheses are explicit; the flat scaling and tail arguments cannot be transferred to it.

The nonlinear spatial reduction eliminates the complementary Dirac modes together with their norm. It derives the effective stationary action, distinguishes its normalized frequency and energy corrections, and gives a source-preservation condition for an exact linear contact branch under explicit positivity hypotheses.

Verification

local_scalar_profile_checks.py checks the exact massive equations by independent spatial differences, integrates all quantities in (11), verifies the virial and fixed-norm frequency, and checks massless scaling directly in the complete spatial Dirac operator with arbitrary complex rest polarizations.