Exact Orientation Currents of the Inferred Excited Shells

Measured masses determine candidate depths and indices through the shared spectral inference. This calculation tests the geometry of those inferred shells using the same conditional ordinary-sphere realization at every occupied depth. It constructs actual currents rather than inferring their accessibility from the signs of an abstract response kernel.

1. A Disturbance on the Fixed Matrix-Charge Level

Use normalized sphere measure and the positive shell

$$E_l(X;p)=\frac{[A(t)I-iB(t)\alpha(v)]K(X)}{\sqrt N}, \quad t=X\cdot p,\quad v=\nabla t,\quad N=\binom{l+d-1}{d-1},$$ $$A(t)=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad B(t)=C_{l-1}^{(d+1)/2}(t),\qquad l=n-1. \tag{1}$$

Put $B=0$ at $l=0$. The matrix Killing frame has $KK^\dagger=I/r$. Let $a\perp p$ be a unit direction in which to move the center, $w=X\cdot a$, and $z=\nabla w$. Differentiation of (1) gives another exact positive-shell field:

$$\partial_a E_l=\frac{[A'wI-iB'w\alpha(v)-iB\alpha(z)]K}{\sqrt N}. \tag{2}$$

Its component along the original right-column space is nonzero. To remove it, define the Hermitian endomorphism $Q_a$ by

$$Q_a K=K\alpha_a^{\rm right},\qquad Q_a=\alpha(tz-wv)-\frac i2[\alpha(v),\alpha(z)]. \tag{3}$$

Here $\alpha_a^{\rm right}$ is the tangent Clifford matrix at the center transported through the Killing frame. Equation (3) is the actual constant right generator already obtained from the first coexact Killing harmonic; it satisfies $\nabla_iQ_a=(i/2)[\alpha_i,Q_a]$.

The positive-shell reproducing kernel gives

$$\int E_l^\dagger E_l=I/r,\qquad \int E_l^\dagger\partial_aE_l =-\frac{il}{dr}\alpha_a^{\rm right}. \tag{4}$$

For clarity, its kernel is $\Pi_l(X,p)=r[A(t)I-iB(t)\alpha(v)]K(X)K(p)^\dagger$. The isometry-invariant positive-shell projector has this zonal form: the full Dirac equation fixes the two regular Gegenbauer coefficients, and $\Pi_l(p,p)=N I$ fixes the normalization. Projector composition then proves (4); differentiate at coincident centers using $B(1)/N=l/d$.

The resulting horizontal derivative is

$$\boxed{W_a=\partial_aE_l+\frac{il}{d}E_l\alpha_a^{\rm right}, \qquad \int E_l^\dagger W_a=0.} \tag{5}$$

Both $W_a$ and $iW_a$ preserve the full particle matrix charge to first order. Indeed the selected massive-shell map $Z_l=a_lI+b_l\beta$ preserves the integrated canonical inner product on the positive bare shell, so the same orthogonality holds after applying $Z_l$. These variations do not rely on fixing only a scalar norm.

Their common squared canonical norm is exactly

$$\boxed{G_l=\int\operatorname{tr}W_a^\dagger W_a =\frac{l(d-1)(l+d)}{d^2}.} \tag{6}$$

The center derivative has norm $l(l+d-1)/d$; subtracting its right-column projection gives (6). The checker independently integrates the actual Gegenbauer/Clifford contraction for the coefficient in (4), the original norm, and the horizontal norm. At $l=0$, the background is center independent and this entire direction vanishes.

2. The Actual Current, with Its Phase Retained

The real orientation derivative $W_a$ is a combination of a spatial rotation tangent and a right-unitary tangent. It has zero linear traced vector current. Its conjugate $iW_a$ generally does not. This is a distinction between two actual field disturbances, not a probability interpretation of their complex coefficients.

Define the bare normalized conjugate current by $J=2\operatorname{Re}\operatorname{tr}E_l^\dagger\alpha(iW_a)$. Direct Clifford traces give

$$\boxed{J=f(t)\,z+g(t)\,w v,} \tag{7}$$ $$f=\frac2N\left[AB-\frac ld \{t(A^2-B^2(1-t^2))-2AB(1-t^2)\}\right],$$ $$g=\frac2N\left[-BA'+AB'+\frac ld \{A^2+B^2(1+t^2)+2ABt\}\right]. \tag{8}$$

For the massive selected shell this current is multiplied by the previously derived $c_l$. If the base and variation carry physical amplitude $\sqrt{\mathcal A}$, their current is $\mathcal A c_l J$. The overall amplitude remains uncalibrated; none is chosen to fit mass.

The exact divergence formula is

$$\nabla\cdot J=w[-tf'-df+(1-t^2)g'-(d+1)tg]=0. \tag{9}$$

It follows either by substituting the Gegenbauer equation or from the same-eigenvalue Dirac current identity. Full Clifford matrices independently verify (7)–(8), and the zero orientation current, at three ordinary spatial points for every inferred assignment.

3. An Exact Finite Coexact Expansion

Introduce $\phi_L(t)=C_{L-1}^{(d+1)/2}(t)$ and

$$V_L=[-(1-t^2)\phi_L'+(d-1)t\phi_L]z +[-t\phi_L'-(d-1)\phi_L]w v. \tag{10}$$

These are divergence-free one-forms with Hodge eigenvalues $\Lambda_L=(L+1)(L+d-2)$. An explicit verification uses

$$(1-t^2)\phi_L''-(d+2)t\phi_L' +(L-1)(L+d)\phi_L=0.$$

For a general profile $f z+g w v$, the Hodge operator has coefficients

$$\widehat f=-(1-t^2)f''+dtf'+df+2tg,$$ $$\widehat g=-(1-t^2)g''+(d+4)tg'+2dg+2f'. \tag{11}$$

Substitution of the preceding ODE and its derivative gives $(\widehat f,\widehat g)=\Lambda_L(f,g)$, checked symbolically with general dimension. Distinct $L$ are orthogonal by self-adjointness of the Hodge operator on the closed sphere.

Their exact norms are

$$\boxed{\|V_L\|_2^2 =\frac{(L+1)(L+d-2)(d-1)}{2L+d-1} \binom{L+d-1}{d}.} \tag{12}$$

One derivation starts with the closed two-form $\phi_L\,dt\wedge dw$, whose codifferential is (10). Its squared norm is $(d-1)/d$ times the integral of $(1-t^2)\phi_L^2$. Its Hodge eigenvalue is the same $\Lambda_L$. Integration by parts and the normalized Gegenbauer norm give (12). The checker also compares (12) with direct spatial quadrature.

For (8), $g$ is even of degree at most $2l$, and (9) fixes $f$. There is a unique even polynomial $\phi$ such that $g=-t\phi'-(d-1)\phi$. If $g=\sum g_k t^k$, simply set $\phi=-\sum g_k t^k/(k+d-1)$. Expanding this polynomial in the Gegenbauer basis gives

$$\boxed{J=\sum_{L=1,3,\ldots,2l+1}q_L V_L.} \tag{13}$$

Every $q_L$ and every power $q_L^2\|V_L\|^2$ is rational at integer depth and index. The single checker evaluates them by exact polynomial elimination over the rational field, independently of its two numerical quadratures. Thus the harmonic cutoff and signed response do not rely on rounding a fitted decimal to a fraction.

4. Signed Response at Every Inferred Assignment

Use the charge-fixed background susceptibility. After removing its explicitly positive normalization, the leading reduced quadratic form in this bare conjugate direction is

$$\boxed{S_l=\frac{d-1}{(d+1)^2}q_1^2\|V_1\|^2 -\sum_{L=3,5,\ldots,2l+1} \frac{2q_L^2\|V_L\|^2}{(L-1)(L+d)}.} \tag{14}$$

The actual form is $\eta_l^2\mathcal A^2c_l^2S_l/\rho_B$; its value per squared canonical disturbance norm is $\eta_l^2\mathcal A c_l^2 S_l/(\rho_B G_l)$. These factors matter for any physical comparison. The following table reports only the geometry of (14), with exact rational signs:

Representative exact values are

$$S_{\rm strange}=349/1000,\qquad S_{\rm up}=54274/200655,$$ $$S_{\nu_1}=-1394536851/108143750,$$ $$S_e=-12388872366741345020897296/239441731549304284916643.$$

The full exact fractions and every individual harmonic power are in the report. The signed result varies within depth four, so it is not a depth-only classification. No independent sector-evidence claim follows.

5. What the Sign Does and Does Not Establish

Equation (14) identifies actual accessible selected-shell currents and their signed leading background response. In particular, the electron and muon have negative directions in this constrained quadratic response. A negative direction alone is not an exponentially growing mode, a decay channel or a negative complete particle inertia. The orientation tangent itself remains a symmetry direction, and the symplectic pairing of shape and conjugate disturbances must be retained in the evolution.

The two directions in this calculation have not been shown to form an invariant subspace of the full coupled dynamics. Other zero-shell polarizations and nonzero-frequency particle/background modes remain necessary for a persistence result. Higher-order excited background self-feedback also mixes harmonics through its variable density. The known mass targets and inferred indices are preserved throughout.

Reproducibility

The single script combines actual Clifford point checks, two independent positive spatial quadratures, symbolic all-dimension Hodge identities, exact norm formulas and complete rational current expansions. Its final report passes 1,192 evaluations across all fourteen assignments. Maximum Clifford current error is $1.2\times10^{-14}$, integrated norm/charge errors are below $2.4\times10^{-12}$, and the exact signed response agrees with quadrature to $2.8\times10^{-11}$ in the reported normalized error. All three successful source versions and dated reports retain their hashes.

These classical internal field disturbances do not replace the electron's literal deterministic position trajectory. The same-field block split, ordinary sphere, constitutive plateau and uncalibrated amplitudes remain the previously stated premises, with no added mediator particle.

A right-polarized horizontal derivative now supplies an explicit higher-only coexact shape current, with strictly negative leading response at every excited assignment. Its conjugate generates exact stationary shape freedom at fixed inferred clock, full matrix charge and complete stationary energy. These facts sharpen the need to retain symplectic evolution; they do not turn the signed response into a persistence or decay result.