A common geometric and mass-carrier differential operator

The differential operator obtained from the linked geometric Hessian can also serve as a candidate normal carrier for the mass construction. It has exactly the required number of localized modes, equally spaced projected polarization, positive adjacent hopping integrals, and a controlled normal complement. This comparison is made at depths 3, 4, 5, 6 and 10 with the maintained particle indices.

There is also a simplification that applies to both the old and new carriers: a local anticommutator proves that their entire first projected polarization variation vanishes at the radial configuration. The full first shape source comes from one hopping matrix.

The geometric carrier changes the spatial integrals. It is a candidate consolidation of two differential expressions, rather than an assertion that all previous numerical predictions are unchanged. Its geometric fluctuation module and an ordinary spinor module have different full-turn rotation laws. Their physical identification, the occupied same-field geometry and its total inertia remain to be constructed. Mass remains frequency, in equivalent units.

Later calculations replace the independent hopping profile by the curvature moment, include depth two, and derive the leading complementary kinetic correction. The bounded angular completion specifies a full extension that preserves the leading simplex factors after radial dressing. Its finite-scale bound controls the remaining spectral error. These developments do not identify the physical rotation modules or solve the occupied supporting geometry.

A family that exposes the common algebra

On the ordinary unbounded normal plane, use the linked radial profile $W=w(r)e^{id\theta}$, $A=a(r)e_\theta/r$. With the matrices of the geometric factorization, write

$$D_{t,e}=-i\alpha\cdot\nabla+ \alpha\cdot A\left(\frac{\tau_3}{2}+eI\right)+t\Gamma\cdot W, \qquad t>0,\quad |e|\le\frac12. \tag{1}$$

The former carrier is $(t,e)=(1,0)$. The geometric Hessian fixes $(t,e)=(2,-1/2)$; neither coefficient is adjusted for a particle. Other values below are mathematical controls of the identity, not proposed extra fitting freedom.

A negative-chiral zero mode has

$$\Phi_l=(0,e^{il\theta}U_l, i e^{i(l-d+1)\theta}V_l,0)^T,$$

with radial equations

$$\begin{aligned} U_l'&=\frac{l+(e-1/2)a}{r}U_l-twV_l,\\ V_l'&=\frac{d-1-l-(e+1/2)a}{r}V_l-twU_l. \end{aligned} \tag{2}$$

At the geometric coefficients these are exactly the horizontal geometric mode equations with $k=d-l$. Normalize each by $2\pi\int r(U_l^2+V_l^2)dr=1$. Both radial components are positive in the phase convention used here.

The positive-chiral square is

$$\begin{pmatrix} [-i\nabla+(e+1/2)A]^2+(e+1/2)B+t^2w^2&0\\ 0&[-i\nabla+(e-1/2)A]^2+(1/2-e)B+t^2w^2 \end{pmatrix}. \tag{3}$$

The linked first-order profile cancels its off-diagonal coefficient. In the geometric case the rigorous normal gap is $\delta_d=2/\sqrt{d+1}$, and the complete kernel has dimension $d$, as derived in the factorization note.

The bounded polarization is fixed by an identity

Put $S=\sigma_3\otimes I$ and

$$\boxed{K_e=(d+1-a)S+2eaI.} \tag{4}$$

For $|e|\le1/2$, $\|K_e\|\le d+1$. Integrating the derivative of $r^2(U_l^2-V_l^2)$ gives

$$\boxed{P K_eP=-2J_z,\qquad (J_z)_{ll}=l-\frac{d-1}{2}.} \tag{5}$$

The result is independent of $t$. In the geometric case, $K_{-1/2}=(d+1-a)S-aI$; its two negative-chiral component values are $d+1-2a$ and $-(d+1)$. The expectation (5) still gives all the equally spaced values $d-1,d-3,\ldots,1-d$.

There is a stronger local identity. For arbitrary, possibly nonradial, coefficients $W,A$, set

$$a(y)=y_1A_2-y_2A_1,\quad L=-i(y_1\partial_2-y_2\partial_1),\quad \mathcal R=-iS\alpha\cdot y,\quad Q_A=\tau_3/2+eI.$$

Direct Clifford multiplication gives

$$\boxed{\{D_{t,e},\mathcal R\}=2L+2S+2aQ_A.} \tag{6}$$

The texture term anticommutes with $\mathcal R$, regardless of its spatial form. On negative chirality $\tau_3=-S$; thus, with $\mathcal J=L-(d-1)S/2$,

$$K_e=\{D_{t,e},\mathcal R\}-2\mathcal J. \tag{7}$$

Compression to an isolated kernel gives $PK_eP=-2P\mathcal JP$. At the radial configuration each $\Phi_l$ is an eigenvector of $\mathcal J$ with eigenvalue $l-(d-1)/2$, proving (5). Also

$$D_{t,e}(\mathcal R\Phi_l) =[K_e-(d-1-2l)]\Phi_l, \qquad \mathcal R\Phi_l=r(e^{i(l-d)\theta}V_l,0,0, i e^{i(l+1)\theta}U_l)^T. \tag{8}$$

This is the exact complementary polarization inverse. The modes decay sufficiently fast for all the coordinate-weighted functions in these identities to lie in their stated operator domains.

The full first polarization source cancels

Let the coefficients deform smoothly while the isolated chiral kernel persists. Transport its orthonormal basis with $\langle\Phi_i,\delta\Phi_j\rangle=0$. Differentiating the compressed identity (7) at the radial configuration gives

$$\begin{aligned} \delta\langle\Phi_i,K_e\Phi_j\rangle &=-2\left[\langle\delta\Phi_i,\mathcal J\Phi_j\rangle +\langle\mathcal J\Phi_i,\delta\Phi_j\rangle\right]\\ &=0. \end{aligned} \tag{9}$$

The explicit $\delta K_e$ is included in the left side. Equation (9) uses invariance of the original kernel under $\mathcal J$, together with the orthogonal basis derivative; it does not assume an unchanged profile. The source cancellation holds for all polynomial deformation orders, their complex phases, and both special carriers.

This is a first-variation result. At a nonradial configuration the kernel need not be invariant under $\mathcal J$, so a higher-order change of the polarization spectrum is not excluded.

The same simplex numerator survives

Keep the bounded spatial multiplier $h=z/\sqrt{1+r^2}$. Its adjacent matrix entries are

$$h_l=2\pi\int_0^\infty\frac{r^2}{\sqrt{1+r^2}} (U_{l+1}U_l+V_{l+1}V_l)dr>0. \tag{10}$$

They are common to all occupied surface branches at one depth. The full projected canonical pair remains

$$H_P=\begin{pmatrix}0&T^\dagger\\T&0\end{pmatrix},\qquad T_{ll}=\beta+l-\frac{d-1}{2},\quad T_{l+1,l}=\gamma h_l. \tag{11}$$

For the maintained particle index $n$, $\beta=n+(d-1)/2$, and therefore

$$\prod_{l=0}^{d-1}\left(\beta+l-\frac{d-1}{2}\right) =\prod_{l=0}^{d-1}(n+l) =d!\binom{n+d-1}{d}. \tag{12}$$

The complete triangular inverse gives

$$\boxed{\omega_{n,d}= \frac{d!}{\gamma^{d-1}\prod_lh_l} \binom{n+d-1}{d}\,[1+O(\gamma^{-2})].} \tag{13}$$

Only the common per-depth factor changes with the geometric carrier. This does not derive the shared charged-lepton frequency unit between depths six and ten. Their different hopping products, normal scales and physical length factors must produce that relation under a common rule. The preserved simplex numerator alone cannot establish it. The small singular value has a nonzero finite-hopping correction; the calculation does not produce an exact simplex law at finite $\gamma$. Both surface signs and the two bottom component controls are retained; a physical bottom geometric-mean pairing is not derived here.

For example, at depth six the new hopping entries are approximately $(0.62068735,0.75332397,0.81296850,0.83299374,0.79325947)$. The new vector connection breaks their old endpoint-reflection symmetry. No individual electron or muon coefficient is selected independently.

In the full normal parent, replace its polarization term by $-K_e\chi_1/2$. With perturbation norm $v=\gamma+(d+1)/2$, fixed surface eigenvalue $\beta$, and normal scale $\Lambda$, the same chiral cancellation gives the conservative complement estimate

$$\|H_{\rm eff}(\omega)-H_P\| \le\frac{v^2(|\omega|+|\beta|+v)} {\Lambda\delta_d[\Lambda\delta_d-|\omega|-|\beta|-v]}. \tag{14}$$

The denominator must be positive, and the bound must resolve the small cluster before using that reduction. The anticommutator $\{D,\alpha_i\}=2\Pi_i$ also gives $P\Pi_iP=0$, where $\Pi=-i\nabla+Q_AA$. This is covariant momentum of the retained wave, not the canonical momentum or inertia of the complete supporting field.

Response grouping remains, with different spatial integrals

Equation (9) reduces the first shape derivative to $\delta T=\gamma\delta h$. For the constant polynomial direction, angular selection still gives $\delta T=\gamma\kappa_d\alpha_d|0\rangle\langle d-1|$. Thus

$$\partial_{\alpha_d}\omega =2\gamma\kappa_d\operatorname{Re}(v_0^*u_{d-1}). \tag{15}$$

For a fixed surface sign the last factor approaches a common value, independent of the assigned particle index. This preserves the sought structural pattern of different masses and similar responses within a depth. It is a response of the candidate spatial operator, not a measured interaction rate.

The new coefficients, divided by the actual geometric mode metric, are:

The former carrier's nearly constant value around $-0.312$ is not inherited by this different operator. In particular, neither comparison derives special neutrino suppression. The complete source test still finds a nonzero surviving even-order deformation at the symmetric profile.

A common expression does not determine its physical rotation module

For an ordinary spinor on the normal plane, $U_s(\theta)=\exp(-i\theta S/2)$, and $U_s(2\pi)=-I$. For the geometric fluctuation used to derive (5) of the factorization note, $\delta W$ is a scalar and $\delta A_1+i\delta A_2$ is a vector component. Its four-component transformation is instead

$$U_g(\theta)=e^{i\theta/2}U_s(\theta),\qquad U_g(2\pi)=I. \tag{16}$$

Both give local covariance of the differential expression when $W$ and $A$ are transformed with their spatial fields. Their full-turn laws differ. A change of notation cannot identify those two physical representations. Any proposed common master-field realization must supply its actual bundle and rotation law, rather than declaring the geometric fluctuations themselves to be ordinary spinors.

The calculation therefore reduces the required differential structures, and derives useful common identities, while leaving a specific physical identification to test. It does not establish that nature realizes this geometry or that the complete mass operator has been obtained.

Verification

geometric_normal_carrier.py solves the full radial carrier independently of the geometric zero-mode solver, including both components and every normal label. geometric_carrier_checks.py compares those solutions, applies the full Cartesian operator and square, checks the exact complementary inverse, and compares the complete finite spectrum with 85-digit diagonalization at every maintained control.

Its 319-check report includes the rigorous gap comparison, all geometric polynomial sources, physical rotations with both modules, and general-coefficient controls. The independent solvers use matching radial domains; their boundary and normalization conventions are retained in the report.

normal_polarization_identity_checks.py independently applies (6) to arbitrary radial and nonradial coefficients. Its 139-check report checks every polynomial first derivative and the original carrier as a separate implementation. Raw radial fields and deformation matrices are retained with the exact executed source hashes.