Which polarization channels the normal geometry supplies

The generalized cancellation bound permits complete beat cancellation when enough identical polarization channels exist. The current normal geometry does not supply those extra copies automatically. Its modes already form the chain used to generate the simplex numerator.

This is a classification of the specified normal carrier. It does not classify every component or surface harmonic of the master field. A larger physical polarization space remains possible, but its actual spatial modes and dynamics must establish its multiplicity.

Count shapes and repeated copies separately

The horizontal geometric modes have positive norm

$$M_k=2\pi\int_0^\infty r(U_k^2+V_k^2)dr>0, \qquad k=1,\ldots,d. \tag{1}$$

The real and imaginary parts of each polynomial coefficient form a two-dimensional real plane. A spatial rotation, with the harmless common basis phase restoring the monic polynomial convention, acts by

$$P(z)\mapsto e^{id\theta}P(e^{-i\theta}z),\qquad \alpha_k\mapsto e^{ik\theta}\alpha_k. \tag{2}$$

The scalar and vector parts of each actual spatial variation both obey this law. The $k=1$ pair translates the profile; the higher weights change its centered shape. They are distinct representations of rotation in this normal plane. These weights are not, by themselves, an assignment of observed particle spin.

Their positive metric prevents cancellation of a nonzero shape velocity's kinetic norm. A null traveling wave instead uses equality between its time and propagation derivatives, as demonstrated by the complete stress calculation.

The actual normal carrier has no independent repeated factor

In its orthonormal basis, the projected rotation and hopping operators are

$$J_{ll}=l-\frac{d-1}{2},\qquad h=\sum_{l=0}^{d-2}h_l|l+1\rangle\langle l|, \qquad h_l>0. \tag{3}$$

Suppose a complex matrix $X$ commutes with both. Since every eigenvalue of $J$ is distinct, $[X,J]=0$ forces $X$ to be diagonal. Then

$$[X,h]_{l+1,l}=h_l(X_{l+1,l+1}-X_{ll})=0$$

forces all its diagonal entries to agree. Therefore

$$\boxed{[X,J]=[X,h]=0\quad\Longrightarrow\quad X=xI.} \tag{4}$$

For $r$ identical independent copies, the operators would instead be $J\otimes I_r$ and $h\otimes I_r$. Every $I\otimes X_r$ would commute with them, giving at least $r^2$ complex commuting directions. Equation (4) excludes such a hidden repeated factor in this existing $d$-mode normal space.

The two canonical blocks of the retained squared mass operator are still present. The two-block source test already includes their entire complex polarization. Equation (4) addresses the additional proposed copies; it does not remove those blocks or identify them with photon helicities.

The earlier sufficient condition $r\ge d$ for cancelling all occupied band beats is consequently conditional on a genuine enlargement supplied by physical modes. The existing normal labels cannot simultaneously be the mass-chain labels and $d$ independent copies of that whole chain. Integrated orthogonality of extra surface harmonics would also need a local source test before it could establish cancellation of the complete spatial field response.

Reserving the translation mode changes the massive spectrum

A direct trial is to leave the $k=1$ geometric direction massless and uncoupled, and build the massive chain from the remaining normal modes. Since $l=d-k$, this removes the last basis vector $l=d-1$ from

$$T_{ll}=\beta+l-\frac{d-1}{2},\qquad T_{l+1,l}=\gamma h_l. \tag{5}$$

On the positive surface branch $\beta=n+(d-1)/2$, the original numerator and the truncated one are respectively

$$\begin{aligned} \det T&=\prod_{l=0}^{d-1}(n+l)=d!S(n,d),\\ \det T_{\rm cut}&=\prod_{l=0}^{d-2}(n+l)=(d-1)!S(n,d-1). \end{aligned} \tag{6}$$

The removed hopping factor changes a common scale, while the missing diagonal factor $n+d-1$ changes ratios between particles. At large hopping, calibrating one reference index $n_0$ gives

$$\boxed{ \frac{\omega_{\rm cut}(n)/\omega_{\rm cut}(n_0)} {\omega(n)/\omega(n_0)} =\frac{n_0+d-1}{n+d-1}[1+O(\gamma^{-2})].} \tag{7}$$

For the electron and muon this factor is $18/40=0.45$. One common frequency unit cannot repair that change. At depth two the trial leaves only the linear factor $n$ instead of $n(n+1)$. The negative surface branch loses a different endpoint factor; the complete finite test retains both signs.

This calculation does not establish that a photon and massive particles cannot arise from one field. It shows that an apparently simple reuse of the existing translation mode does not preserve the current mass law. The protected geometric translations and the formal $n=0$ zero of the massive matrix are different constructions whose connection still needs a physical derivation.

Verification

normal_polarization_multiplicity_checks.py passes 146 checks. It computes the commuting algebra using the actual curvature hopping integrals, verifies the extra commuting operators of an explicitly repeated comparison, and evaluates the full and truncated singular spectra at every maintained particle index and both surface signs. The independent spatial rotation and metric checks are included in the 942-condition null-stress report. The 146-check multiplicity report records the complete spectral comparison and its input curvature integrals.