When a two-frequency spatial symmetry exists

Summary: A constant antilinear reduction requires a specific involution of the complete spatial Dirac matrices, and its existence, scalar interference and compatibility with nested field projections can be checked directly.

A special solution family need not be a symmetry of arbitrary fields

The radial two-frequency map constructs full solutions from arbitrary complex radial profiles. Extending that map to arbitrary spatial perturbations requires an additional matrix identity. The number of field components alone does not supply it.

Here $r$ counts the ordinary macroscopic spatial derivatives actually retained in the equation. A bound depth $d$ has $r=d$ when normal motion is suppressed, and $r=d+1$ when one existing normal direction is included. These coordinates belong to the one nested manifold. Matrix ranks and conjugation operations describe its field components; they do not introduce new physical dimensions.

Consider

$$H=-i\sum_{j=1}^r\alpha_j\partial_j+\beta\mathfrak m, \qquad \mathfrak m\in\mathbb R. \tag{1}$$

Use constant Clifford matrices in the stated flat spatial model, on a common domain preserved by the proposed conjugation and boundary conditions. A matrix identity alone does not establish a symmetry of a different operator domain or of a curved spatial bundle.

A coordinate-preserving antilinear partner $C=B\mathsf K$, where $\mathsf K$ conjugates the field components, must obey

$$\boxed{B\alpha_j^*=\alpha_jB,\qquad B\beta^*=-\beta B,\qquad B^\dagger B=I.} \tag{2}$$

These identities give $CH=-HC$, including every retained derivative. They also give $(C\psi)^\dagger\beta(C\psi)=-\psi^\dagger\beta\psi$. To mix $\psi$ and $C\psi$ while preserving the scalar source for arbitrary complex fields, their cross bilinear must vanish. This requires

$$\boxed{(\beta B)^T=-\beta B, \qquad C^2=BB^*=I.} \tag{3}$$

Under (2), the two conditions in (3) are equivalent. For example, $BB^*=I$ makes the unitary $B$ symmetric, and then $(\beta B)^T=B\beta^*=-\beta B$.

The related nonlinear Dirac symmetry and its matrix requirements are discussed by Boussaïd and Comech. The calculations below check the actual spatial matrices and the centered moment source used here, rather than assuming a symmetry from a smaller field representation.

Check the complete Clifford representation

The standard complex Clifford representation can be constructed from Pauli matrices:

$$\gamma_{2j-1}=Z^{\otimes(j-1)}\otimes X\otimes I^{\otimes(k-j)}, \qquad \gamma_{2j}=Z^{\otimes(j-1)}\otimes Y\otimes I^{\otimes(k-j)}. \tag{4}$$

For an odd number $2k+1$ of generators, append $\gamma_{2k+1}=Z^{\otimes k}$. Use the first $r$ generators as $\alpha_j$ and the last as $\beta$. This is an irreducible complex representation of the full indicated Clifford algebra.

Each generator has $\gamma_j^*=\epsilon_j\gamma_j$, with $\epsilon_j=\pm1$. For a product $B=\prod_{j\in S}\gamma_j$ of $m=|S|$ distinct generators,

$$B\gamma_jB^{-1}=(-1)^{m-\mathbf1_{j\in S}}\gamma_j, \qquad BB^*=(-1)^{N_{\rm imag}+m(m-1)/2}I. \tag{5}$$

Here $N_{\rm imag}$ counts imaginary generators in the product. The required signs are $\epsilon_j$ on spatial generators and $-\epsilon_\beta$ on the scalar generator. Equations (4)–(5) provide explicit intertwiners and their squares.

For $r=4$ and $r=8$, the product of all $r+1$ generators is a nonzero real scalar times $I$. Equations (2) would make antilinear conjugation change its sign, but a real scalar cannot change under such conjugation. No $B$ satisfying (2) exists in those stated representations.

When a solution exists, any two invertible solutions of (2) differ by a matrix commuting with every generator. Irreducibility makes that matrix a scalar. A phase change of $B$ cannot change $BB^*$, so a computed negative square cannot be replaced by a positive square in the same irreducible representation.

Thus the minimal five- and six-dimensional bound equations do not have the involution (3). Including one normal derivative gives $r=6$ or $r=7$ and still has a negative square. The corresponding ten-dimensional examples have a positive square. These are matrix statements about the specified operators, not assignments of particle charge, stability or interaction strength.

A positive square gives a controlled nonlinear map

When (2)–(3) hold, define

$$\mathcal T\psi=a\psi+bC\psi, \qquad s=|a|^2-|b|^2>0. \tag{6}$$

Pointwise scalar interference vanishes, giving $\sigma[\mathcal T\psi]=s\sigma[\psi]$. In the centered moment source, the scalar center is unchanged and its second moment is multiplied by $s$, provided its scalar normalization is nonzero. Consequently, if $\psi$ solves the complete spatial source equation at coefficient $g_0s$, then $\mathcal T\psi$ solves it at $g_0$ with the same bare scalar term. This follows directly by applying (2) to each derivative and using the transformed scalar moment.

The real-linear map has

$$\boxed{\mathcal T^{-1}=\frac{\bar a-bC}{s},\qquad \operatorname{Im}\langle\mathcal T u,\mathcal T v\rangle =s\operatorname{Im}\langle u,v\rangle.} \tag{7}$$

At fixed nonzero $s$ it is invertible and transfers the growth exponents of the corresponding full real field linearizations. The map becomes poorly conditioned when its two weights approach equality:

$$\operatorname{cond}_{\mathbb R}\mathcal T =\frac{|a|+|b|}{|a|-|b|}. \tag{8}$$

The number norm is generally different from the radial isometry:

$$\boxed{\|\mathcal T\psi\|^2 =(|a|^2+|b|^2)\|\psi\|^2 +2\operatorname{Re}\big[\bar a b\langle\psi,C\psi\rangle\big].} \tag{9}$$

The cross term need not vanish on arbitrary fields. A stationary nonzero-frequency eigenfunction is orthogonal to its opposite-frequency partner, but that fact must not be applied to every perturbation.

For $C^2=-I$, equation (2) instead makes $\beta B$ symmetric. Generic complex fields then have nonzero $\psi^\dagger\beta C\psi$, so the single scalar scaling used in (6) fails. Exact radial and real anisotropic two-frequency families can still exist through their own spatial identities. They do not require a global transformation of every field perturbation.

Include the existing normal direction before using the symmetry

A symmetry of the bound equation must also satisfy $BY^*=YB$ to include its normal derivative. In the tested three-dimensional bound representation, the involution instead has $BY^*=-YB$. At normal momentum $q$, it therefore obeys

$$B H_{-q}^*+H_qB=2qYB, \qquad H_q=H_b+qY. \tag{10}$$

Its bound-space reduction fails as soon as that normal propagation is retained. In the tested five-dimensional case, the negative-square partner respects the normal derivative but still fails the arbitrary-field scalar-interference condition. These are distinct requirements, checked separately in the full spatial source.

The occupied polarizations also matter. A background generated by (6) must have $Y\phi_2=C\phi_1$, or $\phi_2=YC\phi_1$. That is a specific correlation between the two occupied profiles. It cannot be presumed for independent polarizations.

For the deformed three-dimensional profile in a $2\subset3$ pair, this correlated preparation cancels all scalar rest sidebands. The full normal response still opens them. The analogous orthogonal preparation in the deformed five-dimensional profile of a $3\subset5$ pair has nonzero scalar sidebands even at rest. Orthogonality alone therefore does not substitute for the full spatial symmetry calculation. The reciprocal periodic response retains these channels directly.

A containing-space involution need not preserve a projected sector

Suppose a higher representation has $C^2=I$, and $P$ selects a lower field copy while retaining its lower spatial Clifford operators and $\beta$. If $CP=PC$, restricting $C$ to that copy would give a positive-square solution of (2) there. That is impossible when the selected minimal lower representation requires $C^2=-I$.

The containing involution must then leave that selected copy. This can be checked explicitly in the $64$-component representation with ten bound derivatives and one common normal derivative. Commuting rotation projectors on the other containing directions select a lower minimal copy of rank $8$ for bound depth five or rank $16$ for bound depth six. For the checked projectors,

$$\boxed{PBP^*=0,\qquad \|(I-P)BP^*\|_F^2=\operatorname{rank}P.} \tag{11}$$

The transformation's full norm goes into complementary field components. Those components still describe the same nested physical manifold. This is a compatibility condition on a symmetry and a chosen field projection, not an obstruction to the nesting of the spatial sectors. The spatial reflection analysis provides related constraints on their polarization multiplicities.

Verification and scope

antilinear_dirac_checks.py constructs (4)–(5), checks every spatial matrix relation, and independently solves the complete complex linear intertwiner equations at the smaller ranks. It checks scalar interference, the real kinetic form, the inverse and number norm on arbitrary complex field vectors.

Complete centered nonlinear source evaluations test the map beyond a stationary ansatz. Independent prepared spatial resolvents test the correlated-polarization sidebands. Explicit containing-space projectors verify (11). Reports are timestamped and checks remain active with Python optimization.

The obstruction concerns constant coordinate-preserving antilinear maps in the specified representations. Additional field multiplicities, spatial reflections, nonlocal transformations and special solution families require their own analysis. The full periodic source equation remains available without assuming this reduction.