The Spatial Mode Count Fixes Curvature and a Quantum Geometric Tensor

The spatial band used for the simplex contact also determines its changing evaluation image. On a round-sphere control, this gives an explicit connection, curvature and positive quantum geometric tensor. The same binomial weights fix all of them. The construction therefore makes the count, transport and deformation response jointly calculable rather than prescribing them independently.

This is a result for actual spinor fields on an ordinary spatial manifold. The sphere is a compatible nested control family, not an adopted particle geometry or compactification. The physical sector radius and occupied field action remain unresolved. A tensor eigenvalue below is not a measured particle property or a new mass-frequency law.

1. An Explicit Spatial Eigenspace

Let the control manifold be $S^d_R$, with spinor rank

$$r_d=2^{\lfloor d/2\rfloor}.$$

Its positive Dirac shell of degree $l$ has frequency $(l+d/2)/R$ and multiplicity $r_dg_d(l)$, where

$$g_d(l)=\binom{l+d-1}{d-1},\qquad S=\sum_{l=0}^Ng_d(l)=\binom{N+d}{d},\qquad N=n-1.$$

Killing spinors trivialize the full sphere spinor bundle. The trivialization, scalar harmonic spectrum and Dirac shell multiplicities are established in Bär, section 2, lemma 1 and theorem 1. The following polynomial construction makes the relevant evaluation and derivatives explicit.

Write $D=d+1$ for the ambient coordinate count of the sphere embedding, and initially set $R=1$. This embedding coordinate count does not change the physical dimension of $S^d$. Choose Hermitian tangent Clifford matrices with $\{\gamma_a,\gamma_b\}=2\delta_{ab}I$, and symbol maps

$$\Gamma_a=\gamma_a\quad(a=1,\ldots,d),\qquad \Gamma_D=iI.$$

They obey both adjoint Clifford symbol identities. On spinor-valued homogeneous polynomials of degree $l$, use the coefficient basis $x^\alpha/\sqrt{\alpha!}$. The monogenic condition $\sum_i\Gamma_i\partial_i\Phi=0$ is the finite linear constraint

$$C_{\beta,\alpha}=\sum_{i:\alpha=\beta+e_i} \Gamma_i\sqrt{\alpha_i},\qquad |\beta|=l-1.$$

Its kernel has dimension $r_dg_d(l)$. Monogenicity also implies that each spinor component is harmonic, by composing with the adjoint Clifford differential symbol.

For a homogeneous harmonic polynomial, the probability-normalized sphere norm is its Fischer coefficient norm divided by

$$2^l(D/2)_l=\prod_{j=0}^{l-1}(D+2j).$$

This follows by Gaussian radial integration and the harmonic polynomial pairing. Direct angular integrals of $(x_1+ix_2)^l$ independently check the factor. A physical volume measure multiplies both shell and reference norms by the same sphere volume; the normalized evaluation image is unchanged. None of these normalization conventions calibrates a physical field amplitude.

The coefficient kernel is a spatial eigenspace, not merely a matching dimension. For a unit point $x$ and tangent vector $t$, define skew Clifford multiplication

$$c_x(t)=-\Gamma(x)^\dagger\Gamma(t).$$

The Killing-frame connection is $\nabla_t=\partial_t-c_x(t)/2$. Consequently

$$\mathscr D\Phi=\sum_ac_x(e_a)\partial_a\Phi+\frac d2\Phi.$$

Monogenicity and Euler homogeneity give $\sum_ac_x(e_a)\partial_a\Phi=l\Phi$. Thus $\mathscr D\Phi=(l+d/2)\Phi$. Restoring radius divides this frequency and every first spatial derivative by $R$.

2. Actual Evaluation and Its First Derivatives

Let $E_l(x)$ evaluate the normalized shell and let

$$U_l(x)=E_l(x)^\dagger B_l(x)^{-1/2},\qquad B_l=E_lE_l^\dagger=\frac{g_d(l)}{\operatorname{Vol}(S^d_R)}I.$$

At the north pole, in the Killing trivialization and unit-radius orthonormal tangent directions, the polynomial kernel gives

$$U_l^\dagger\partial_aU_l =-\frac ld\Gamma_D^\dagger\Gamma_a =i\frac ld\gamma_a,$$

$$(\partial_aU_l)^\dagger\partial_bU_l =\begin{cases} \displaystyle\frac{l(l+d-1)}dI,&a=b,\\[4pt] \displaystyle-\frac ld\gamma_a\gamma_b,&a\ne b. \end{cases}$$

Homogeneity transports these local identities. The diagonal derivative moment also agrees with the degree-$l$ scalar harmonic Laplacian in the Killing trivialization. The ground image $U_0$ is constant in this frame; its band-image curvature and quantum tensor vanish.

The normalized cumulative image is the shell stack

$$U_N=\bigoplus_{l=0}^N\sqrt{\frac{g_d(l)}S}\,U_l, \qquad U_N^\dagger U_N=I.$$

It supplies the same contact $T=S U_NU_N^\dagger$ as the reference-normalized spatial evaluation. Its nonzero contact eigenvalue is $S$, repeated on the evaluation fibre. The bare spatial kinetic operator must still accompany that contact in a physical frequency calculation.

3. Binomial Moments Determine the Geometric Tensor

For the cumulative shell probability $g_d(l)/S$, exact count identities give

$$\langle l\rangle=\frac{dN}{d+1},\qquad \operatorname{Var}(l)=\frac{dN(N+d+1)}{(d+1)^2(d+2)}.$$

Define the inverse-area coefficient

$$c_N=\frac{N(N+d+1)}{R^2(d+1)^2}.$$

The actual matrix-valued quantum geometric tensor is

$$\mathcal Q_{ab}=(\partial_aU_N)^\dagger (I-U_NU_N^\dagger)(\partial_bU_N).$$

Insert the shell derivative moments and subtract the projected connection product. This yields

$$\boxed{\mathcal Q_{ab}=G_N\delta_{ab}I -\frac{c_N}{2}[\gamma_a,\gamma_b],\qquad G_N=\left(d-1+\frac1{d+2}\right)c_N.}$$

Its curvature is

$$F_{ab}=\mathcal Q_{ab}-\mathcal Q_{ba} =-c_N[\gamma_a,\gamma_b].$$

The full spinor trace of this curvature is zero. A specified polarized line can have nonzero curvature, but the trace-free matrix connection is not itself the independent color action or a derived electric-charge line.

For any fixed unit polarization $u$, the real spatial metric is isotropic:

$$\operatorname{Re}(u^\dagger\mathcal Q_{ab}u)=G_N\delta_{ab}.$$

The normalized curvature scale $c_N/G_N=(d+2)/(d^2+d-1)$ depends on depth and is independent of cutoff for $N>0$. Its use as a physical response coefficient still requires an action and gap-weighted reduction.

4. Two Clifford Components of the Positive Complex Tensor

As a Hermitian block operator on $T^*S^d\otimes\Sigma_d$,

$$\mathcal Q=(G_N+c_N)I-c_N(\gamma_a\gamma_b)_{ab}.$$

The block $(\gamma_a\gamma_b)_{ab}$ is the adjoint product of Clifford contraction $(v_a)\mapsto\sum_a\gamma_av_a$. It has eigenvalue $d$ on one rank-$r_d$ component and zero on its orthogonal rank-$(d-1)r_d$ component. Therefore

$$\boxed{\lambda_{\rm Clifford}=\frac{c_N}{d+2},\qquad \lambda_{\rm twistor}=\frac{(d+1)^2c_N}{d+2}.}$$

The multiplicities are $r_d$ and $(d-1)r_d$. For $N>0$,

$$\boxed{\lambda_{\rm twistor}/\lambda_{\rm Clifford}=(d+1)^2.}$$

These are eigenvalues of a positive complex matrix-valued quantum tensor. They are not two eigenvalues of the real scalar translation metric, two particle masses or two derived oscillation frequencies. The Clifford component equals the bare kinetic spectral variance divided by $d$:

$$\lambda_{\rm Clifford}=\frac{\operatorname{Var}(\omega_l)}d, \qquad \omega_l=(l+d/2)/R.$$

A single shell has zero frequency variance and a vanishing Clifford component. The cumulative band has a strictly positive one whenever $N>0$. The mode distribution therefore contributes geometric information beyond its total dimension.

5. The Same Rule at Every Maintained Depth

Representative exact values are given in units $R^{-2}$. Physical radii are not assumed equal by this table.

The neighboring depths obey the same formulas, with no depth-five coefficient. Depth five has nonzero geometry; the small neutrino response in the finite-gap sawtooth comparison instead used its slow source frequency and specified interaction. This tensor does not establish neutrino electric neutrality or weak rates.

The electron and muon share their dimension-only normalized tensor spectrum, while the tau has the different depth-ten ratio. Thus the construction does not by itself explain exact charged-lepton charge universality. Its absolute curvature also retains the cutoff.

The down branch has $N=0$ and zero evaluation-image geometry. A claim that this curvature alone generates all quark color would fail that control. The depth-three positive sphere ground has spinor rank two and no independent triplet multiplicity. A geometric color completion must supply the actual equivalent-mode carrier and invariant occupied states, as discussed in the paired-color calculation.

6. Physical Scope and Reproducibility

One actual spatial band now supplies its count, contact, kinetic moments, induced connection, curvature and positive quantum tensor. These are linked consequences under a common spatial operator. They are not independent experimental evidence for the sector assignments. Applying them to particle properties requires the physical spatial geometry, the occupied field action, source maps and the spectral gaps that weight an eliminated-mode response.

The single checker passes 4,456 conditions across 51 cases, preserved in its report. It explicitly constructs normalized polynomial eigenspaces at depths two through ten, including neighboring controls, and checks cumulative moments through depth twelve. All maintained assigned massive branches are evaluated analytically. The explicit matrix cutoffs range from $N=6$ at depth two to $N=2$ at depths seven through ten; the much larger assigned bands are covered by the derived moment identities, rather than represented as enormous matrices. No particle simulation or amplitude fit is used.

7. Kinetic Dressing and the Actual Complement Gaps

The contact alone has a bright eigenvalue $\mu S$. Retaining the positive sphere kinetic operator gives the radial matrix

$$H_{\rm rad}=\operatorname{diag}(\omega_l)+\mu vv^\dagger, \qquad v_l=\sqrt{g_d(l)},\qquad 1=\mu\sum_{l=0}^N\frac{g_d(l)}{z-\omega_l}.$$

The top eigenvalue $z$ is above every bare frequency for $\mu>0$. Its normalized spatial frame has shell coefficients

$$a_l=\frac{v_l/(z-\omega_l)} {\sqrt{\sum_jg_d(j)/(z-\omega_j)^2}},\qquad p_l=a_l^2.$$

These weights are fixed by the same operator, rather than chosen to preserve the earlier geometric ratio. The top multiplet retains rank $r_d$. For general normalized shell probabilities, in unit-radius units put

$$\beta=\langle l\rangle_p/d,\qquad c=\beta(\beta+1),\qquad G=(d-1)c+\operatorname{Var}_p(l)/d.$$

The same derivative calculation gives $\mathcal Q_{ab}=G\delta_{ab}I-c[\gamma_a,\gamma_b]/2$. Its two positive tensor eigenvalues are

$$\lambda_{\rm Clifford}=\operatorname{Var}_p(l)/d, \qquad \lambda_{\rm twistor}=dc+\operatorname{Var}_p(l)/d.$$

Their ratio generally differs from $(d+1)^2$. That exact value belongs to the cumulative count weights. Contact-dominated dressed states approach those weights; retaining the kinetic complement calculates the departure.

An Exact Gap-Weighted Tensor

For a response that eliminates the other spatial eigenstates, their actual gaps also matter. Define the positive top-complement tensor

$$\mathcal K_{ab}=(\partial_aU_z)^\dagger Q_z(z-Q_zHQ_z)^{-1}Q_z(\partial_bU_z), \qquad Q_z=I-U_zU_z^\dagger.$$

The inverse is only on the complement; the top multiplet is excluded. Split a shell derivative into its evaluated fibre and orthogonal spatial part. With $R=1$, let

$$c_l=l(l+d)/d^2,\qquad h_l=a_l(l/d-\beta),$$

$$t_0=\sum_l\frac{p_lc_l}{z-\omega_l},\qquad t_1=h^\dagger(z-H_{\rm rad})^+h =\sum_l\frac{h_l^2}{z-\omega_l}.$$

The last equality is exact. Since $h\perp a$ and $a\propto\operatorname{diag}(z-\omega_l)^{-1}v$, the singular rank-one resolvent's Fredholm condition removes its additional quadratic term. An independent radial eigendecomposition verifies the result.

The complete tensor is

$$\mathcal K_{ab}=[(d-1)t_0+t_1]\delta_{ab}I +\frac{t_1-t_0}{2}[\gamma_a,\gamma_b].$$

Its Clifford and twistor eigenvalues are $dt_1$ and $dt_0$, with the same multiplicities as before. This uses the full complement gaps instead of replacing them with one constant.

For a quantum adiabatic reduction with this top branch occupied, the inertia correction uses denominators $E_{\rm other}-z$. Its real symmetric part is therefore $-2I\operatorname{Re}\mathcal K$, with $I$ the unspecified occupation/action scale. The negative correction is a partial collective response of an excited branch. It is not the complete particle inertia, a negative particle mass or a demonstrated instability. A passive ground-branch reduction has the opposite denominator sign. A physical particle calculation must include the complete same-field kinetic and stress contribution rather than identifying this correction alone with inertia.

One Common Strength Across All Assigned Branches

The comparison uses shared $\mu R=1/10,1,10$. It does not fit each particle. At $\mu R=1$, representative tensor ratios are:

The gap-weighted ratio's relative departures from the count reference are approximately $0.0477\%$ for electron, $0.000558\%$ for muon and $0.0000179\%$ for tau. Smaller-count branches have larger corrections. These remain geometric response-tensor comparisons, not physical charge ratios. The same source note supplies all other assigned branches and neighboring-depth controls. The cutoff-zero down control remains flat.

The dressing checker passes 1,974 conditions across 87 cases, recorded in its report. It compares the analytic weighted formulas with actual spatial eigenspaces, full active Hamiltonians, independent radial resolvents and 60-digit secular roots. Explicit spatial matrices use $N\le2$ at depths two through six; the derived moment and resolvent formulas evaluate every maintained massive branch. Spatial dark modes orthogonal to every evaluation and first-derivative source have exactly zero contribution to this response and retain their bare frequencies. No particle simulation is used.