Physical polarization and the complete spatial current variation

The homogeneous current action preserves an entire spatial eigenspace. Its stability still requires perturbations outside that eigenspace. The calculation below uses actual spatial fields and retains the contribution of omitted angular modes to the positive square.

At depth two, a differential-form parent now produces the same component carrier as the spinor comparison. Its tangent polarization is an actual spatial vector, with scalar and top-form companions retained. This does not establish the complete observed particle spin assignments or a photon.

A bosonic depth-two parent

On the stated physical round two-sphere, let $D_{\mathrm f}=d+\delta$ act on all differential forms, and let $\Gamma=(-1)^{\text{form degree}}$. Their anticommutation gives

$$\boxed{ B_{\mathrm f}=D_{\mathrm f}+\frac{\Gamma}{2R},\qquad B_{\mathrm f}^2=\Delta_{\mathrm H}+\frac1{4R^2}. } \tag{1}$$

For a scalar harmonic $Y_n$, write $L=n(n+1)/R^2$, $E=dY_n/\sqrt L$, $M=*dY_n/\sqrt L$, and $T=*Y_n$. The ordered form basis $(Y_n,E,M,T)$ has

$$D_{\mathrm f}=\sqrt L \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&-1&0 \end{pmatrix},\qquad \Gamma=\operatorname{diag}(1,-1,-1,1). \tag{2}$$

The positive eigenvalue $(n+1/2)/R$ has two families,

$$u_E=aY_n+bE,\qquad u_M=aT-bM,\qquad a^2=\frac{n+1}{2n+1},\quad b^2=\frac n{2n+1}. \tag{3}$$

Insert this actual spatial $B_{\mathrm f}$ into

$$T_2=\begin{pmatrix} cB_{\mathrm f}-c/(2R)&0\\G&cB_{\mathrm f}+c/(2R) \end{pmatrix},\qquad H=\begin{pmatrix}0&T_2^\dagger\\T_2&0\end{pmatrix}. \tag{4}$$

On (3), its two diagonal entries are exactly $cn/R$ and $c(n+1)/R$. The same finite mass carrier therefore has a form realization. The current channel couples the first upper component to the second lower component; reversing that connection changes the law.

For a real standing $Y_n$, the tangent parts $E$ and $M$ are orthogonal and equally long at each nonzero point. Their complex relative phase gives linear, circular or elliptical tangent motion. The full forms transform under ordinary spatial rotations and return after a complete turn. A harmonic index is not, by itself, the observed spin of a particle. The maintained Higgs index is a spectral control; this mixed-form construction does not identify the Higgs with a vector.

At $n=0$, the scalar and top-form modes have no nonzero exact or coexact one-form companion. They do not supply a polarized photon zero. The photon remains a separate massless question.

The positive Hessian retains the full derivative

At a stationary field $F[\phi]=\lambda\phi$, let $L=DF[\phi]$ be the real self-adjoint derivative. Homogeneity gives $D^2F_\phi(h,\phi)=0$. Symmetry of the third variation of the generating functional then removes the extra term from the second variation of the positive potential:

$$\boxed{ \frac12\delta^2\|F\|^2[h,h]=\|Lh\|^2. } \tag{5}$$

The identity also has a directional proof that covers the nodal fields used here. The local nonlinear map $N_0=F-H_0$ is globally Lipschitz for each fixed $\eta>0$. Indeed, $\|K\|=1$, and its normalized current direction obeys $\|(K-kI)\Psi\|^2/\rho\le1$. Equation (12) of the current construction gives the uniform local bound

$$\|DN_0\|\le g(1+\eta+2/\eta). \tag{5a}$$

Continuity at zero extends the Lipschitz estimate along any straight field segment, including one that crosses the origin. If the background field is nonzero almost everywhere and $h$ lies in the domain of $H_0$, dominated convergence then gives $F(\phi+th)=F(\phi)+tLh+o_{L^2}(t)$. The highest-weight zero sets have zero volume, so they satisfy this hypothesis. The generating functional has the directional expansion

$$E_1(\phi+th)=E_1(\phi) +2t\operatorname{Re}\langle F(\phi),h\rangle +t^2\operatorname{Re}\langle Lh,h\rangle+o(t^2). \tag{5b}$$

Homogeneity gives $E_1(\phi+th)=\operatorname{Re}\langle\phi+th,F(\phi+th)\rangle$. Subtracting $t\operatorname{Re}\langle h,F(\phi+th)\rangle$ from (5b) cancels its quadratic term. Consequently the remainder in $F$, paired with $\phi$, is $o(t^2)$. At $F(\phi)=\lambda\phi$, expanding $\|F(\phi+th)\|^2$ now proves (5) directly.

This establishes the required directional quadratic form for the stated nodal backgrounds. It does not make the local map differentiable at the origin, prove a uniform Frechet Hessian on the full field space, or establish global nonlinear well-posedness.

For a normalized component ray, write $v=(K-kI)\phi$. The local real current part is

$$Nh=2f''v\operatorname{Re}(v^\dagger h),\qquad \boxed{N^2=2f''\|v\|^2N.} \tag{6}$$

The spatial normalized eigensection is included in the local direction in (6). Its pointwise norm has been divided out, so the component $\|v\|$ is constant on the background.

Let $P$ be an orthogonal angular projection invariant under the base spatial operator $A=H_0+GK+C$, and set $N_P=PNP$. Projecting the square of the full derivative gives the exact identity

$$\boxed{ PL^2P=(A_P+N_P)^2+ \underbrace{2f''\|v\|^2N_P-N_P^2}_{\displaystyle (QNP)^*(QNP)\ \ge0}, \qquad Q=I-P. } \tag{7}$$

Thus the omitted angular square is available without truncating the nonlinear source first. Its positivity does not mean it is negligible. Squaring $A_P+N_P$ alone would be a different calculation.

For a rotating positive-frequency background, the normal and conjugate perturbations combine into $\eta$. Let $\Sigma$ be plus the identity on the normal part and minus the identity on its conjugate. The complete projected squared operator $\mathcal K=PL^2P$ gives

$$\ddot\eta-2i\omega\Sigma\dot\eta+ (\mathcal K-\omega^2I)\eta=0. \tag{8}$$

For $\eta\propto e^{-i\Omega t}$, the actual frequency pencil is $\mathcal K-(\Omega I+\omega\Sigma)^2$. Positivity of $\mathcal K$ alone is insufficient to infer stability of (8). The numerical test retains the gyroscopic term and solves this pencil.

Actual spinor angular channels at every depth

For $b=(d-1)/2$, choose $z=x_1+ix_2$, $q=|z|^2$, and a constant spinor with $\Gamma_u\zeta=0$, $u=e_1+ie_2$. Convenient raw fields are

$$F_l=z^m q^l\zeta,\qquad G_l=z^{m+1}q^l\Gamma_x\Gamma_{\bar u}\zeta. \tag{9}$$

The actual shifted spatial Dirac operator acts as

$$\begin{aligned} BF_l&=(m+b+2l)F_l-lG_{l-1},\\ BG_l&=4(m+1+l)F_l-(m+b+2+2l)G_l. \end{aligned} \tag{10}$$

For $m\ge0$, the finite invariant span contains $F_0,\ldots,F_L$ and $G_0,\ldots,G_{L-1}$. For $m=-M<0$, use $F_M,\ldots,F_{M+L}$ and $G_{M-1},\ldots,G_{M+L-1}$. The latter expressions are regular fields: for example, $z^{-M}q^M=\bar z^M$. The coefficient that would leave the span at its lower end is zero in (10).

The probability-volume Gram entries are

$$\begin{aligned} \langle F_i,F_j\rangle&=b\,\mathrm B(m+i+j+1,b),\\ \langle F_i,G_j\rangle&=2b\,\mathrm B(m+i+j+2,b),\\ \langle G_i,G_j\rangle&=4b\,\mathrm B(m+i+j+2,b). \end{aligned} \tag{11}$$

The normal current projector sends $F_l$ to itself and $G_l$ to $2F_{l+1}$. The anomalous part pairs an azimuth $m$ with $m'=2n-m$ for a background $z^n\zeta$. The three corresponding cross entries are

$$b\,\mathrm B(n+i+j+1,b),\qquad 2b\,\mathrm B(n+i+j+2,b),\qquad 4b\,\mathrm B(n+i+j+3,b). \tag{12}$$

These are actual spatial inner products, not independent copies of a scalar mode. Orthogonalization uses exact high-precision Gram matrices. Stable evaluation uses Jacobi polynomials in a moving two-component frame inside the physical spinor bundle. It does not reduce the sphere's dimension or assert that the complete spinor rank is two.

A linear fast gap for the entire angular spectrum

The linear parent admits a bound independent of angular degree. First remove the bounded diagonal geometric shifts and write $T_0=xI+GJ$, for any real spatial eigenvalue $x$. With $e_d$ the last coordinate vector,

$$T_0^\dagger T_0=A_0-G^2e_de_d^\dagger,\qquad A_0=(x^2+G^2)I+xG(J+J^\dagger). \tag{13}$$

The eigenvalues of the tridiagonal matrix $A_0$ are $x^2+G^2+2xG\cos[r\pi/(d+1)]$, for $1\le r\le d$. Their smallest value is at least $G^2\sin^2[\pi/(d+1)]$: complete the square in $|x|-G\cos[\pi/(d+1)]$. Rank-one interlacing in (13) then bounds the second-smallest singular value of $T_0$.

The actual diagonal geometric shift has norm $c(d-1)/(2R)$. Restoring it, and then the scalar current offset $C$, gives

$$\boxed{\operatorname{gap}_{\rm linear,fast}\ge G\sin\!\frac{\pi}{d+1}-\frac{c(d-1)}{2R}-|C|.} \tag{14}$$

When positive, (14) controls all component branches except the lowest signed pair at each angular eigenvalue. It covers the complete infinite angular spectrum of this linear parent, not just the numerical cutoff. The real current derivative $N$ still has to be included. Subtracting its norm $2|f''|\|v\|^2$ gives a crude further bound, which fails to prove a gap in some of the reviewed cases. Such a failed bound is not evidence that the physical spectrum has become unstable.

The current fixed point also gives a component stability criterion

At one fixed spatial eigenvalue, take the normalized real component eigenvector $\phi$, the bare positive frequency $\epsilon$, and $A=H(G)+C$. Let $\lambda=\epsilon+C>0$. The current vector $v=(K-kI)\phi$ is orthogonal both to $\phi$ and to its chiral partner. The current derivative therefore leaves both low component rays unchanged. Its real and imaginary blocks are

$$L_x=A+2f''vv^T,\qquad L_y=A. \tag{15}$$

Assume the remaining bare component eigenvalues lie outside $[-\lambda,\lambda]$. On that fast subspace the rank-one secular function is

$$D(z)=1+2f''\langle v,(A_f-z)^{-1}v\rangle,\qquad D'(z)=2f''\|(A_f-z)^{-1}v\|^2\le0. \tag{16}$$

There are no poles in the specified interval. The exact second Hellmann--Feynman derivative gives

$$\epsilon_{GG}=-2\langle v,(H_f-\epsilon)^{-1}v\rangle, \qquad \boxed{D(\lambda)=1-f''\epsilon_{GG}.} \tag{17}$$

The right-hand side is precisely the derivative of the scalar fixed-point equation $G-f'(\epsilon_G)=0$. If it is positive, monotonicity in (16) prevents a secular zero anywhere in the particle gap. No shifted fast component eigenvalue enters that interval.

The unchanged opposite partner has magnitude $\epsilon-C\ge\lambda$, because this current has $C\le0$. Thus the complete component rotating quadratic stiffness $L^2-\lambda^2$ is nonnegative under these conditions. Its positive kinetic term and skew gyroscopic term exclude an exponentially growing finite-component mode. A zero stiffness can still permit a neutral or secular perturbation; this is not a claim of strict asymptotic stability.

Every maintained index passes these conditions at the illustrative strengths $gR/c=100$ and $1000$, with an independent direct component eigensystem calculation. The result explains that part of the numerical stability without an additional fitted coefficient. It does not replace the spatial test: angularly varying perturbations can mix spatial eigenspaces through the local current direction.

The competing zero-current branch is unstable in the controls

The same current law also permits a branch with $G=0$. In units $c=R=1$, its component vector has equal upper and lower first entries, each $1/\sqrt2$, and all other entries zero. Its current vanishes and its signed frequency is $\lambda_0=n-g\eta$. It is an exact standing branch of the fully varied positive action. Finding the near-saturated branch alone would not remove this competitor.

Its stability is decided by one exact two-component block. Set $h=g\eta$, $M=g/\eta$, $a=n+1$, and $\omega_0=n-h>0$. The real and imaginary derivatives on the second upper/lower component pair are

$$X=\begin{pmatrix}-h&a\\a&-h-M\end{pmatrix},\qquad Y=\begin{pmatrix}-h&a\\a&-h\end{pmatrix}. \tag{18}$$

Their full rotating perturbation equation is

$$\begin{aligned} \ddot x+2\omega_0\dot y+(X^2-\omega_0^2I)x&=0,\\ \ddot y-2\omega_0\dot x+(Y^2-\omega_0^2I)y&=0. \end{aligned} \tag{19}$$

There is an exact sufficient instability criterion:

$$\boxed{g\eta\le\frac n2,\qquad \frac g\eta>\frac{2n+1}{n}.} \tag{20}$$

To see this, the smaller-magnitude eigenvalue of $X$ is $-h+\mu$, where $\mu=\sqrt{a^2+M^2/4}-M/2>0$. The second inequality in (20) is equivalent to $\mu<n$; the first ensures $-\omega_0<-h+\mu<\omega_0$. The other eigenvalue of $X$ has magnitude larger than $\omega_0$. Thus $X^2-\omega_0^2I$ has exactly one negative eigenvalue, while $Y^2-\omega_0^2I$ is positive definite.

The determinant of the complete real stiffness in (19) is negative. For a real growth exponent $s$, the characteristic determinant $\det(s^2I+s\mathcal G+\mathcal K)$ is therefore negative at zero and positive for sufficiently large positive $s$. It has a positive real root. This establishes an exponentially growing mode with the gyroscopic matrix $\mathcal G$ included; it does not infer instability from a negative curvature alone.

All sixteen maintained index controls satisfy (20) at the illustrative common strength $gR/c=100$ and the same within-depth regularizer used for the near-saturated states. Direct high-precision solutions of the complete block (19) give growth rates from about 0.09 to 0.68 times the uncoupled branch frequency. The calculation applies equally to the actual form realization at depth two because the affected component block acts as the identity on its spatial form indices.

This distinguishes two branches of one specified action: the uncoupled branch is unstable in these controls, while the near-saturated branch passes the component criterion above. It does not prove attraction to that branch, select the occupied particle indices, or establish full spatial stability and geometric support.

What the numerical stability tests establish

The depth-two form calculation tests the maintained indices 76, 81 and 95, two complex polarization choices, azimuth shifts one and three, and three angular cutoffs. At $gR/c=1000$, all 36 cases have growth rates indistinguishable from numerical zero in the tested frequency pencil. Independent finer quadrature agrees with the complete squared operator. All four form types are retained.

The spinor calculation uses the same current law at depths two, three, four, five, six and ten, with the maintained indices and three axial cutoffs. Its illustration uses $gR/c=100$ to resolve the small frequencies more reliably. No resolved growing mode appears in those channels. A conservative squared-matrix resolution estimate reaches roughly one percent of the selected frequency for the depth-ten control; that numerical limitation is reported separately.

These are finite spatial perturbation tests, including their exact positive angular-square contribution. They do not prove stability in all azimuthal channels, nonlinear stability, supporting geometry, or complete particle inertia. Increasing the spatial basis is also required when a two-mode encounter drives new angular components.

Verification

depth_two_form_parent_checks.py passes 65 checks of the actual form parent and polarization. form_spatial_stability_checks.py passes 228 checks of the full form perturbation calculation. spinor_spatial_stability_checks.py passes 576 checks across the maintained spinor controls. complete_axial_spinor_checks.py passes 276 independent checks of direct spatial Clifford derivatives, negative azimuths, physical Gram matrices, and stable Jacobi functions. The reports retain the original source hashes and numerical frequency arrays where used. uniform_component_gap_checks.py passes 262 checks of the rank-one identity, interlacing bound, and geometric shifts. It records the unsuccessful nonlinear norm bounds. current_fixed_point_stability_checks.py passes 320 checks of the component secular criterion, its exact fixed-point derivative, and the independently diagonalized real derivative. zero_current_branch_checks.py passes 96 checks of the competing exact branch, its full real and imaginary derivative blocks, and the high-precision gyroscopic eigenproblem. The instability conditions (20) hold in every recorded case.