One Spatial Band for Counts, Contact and Transport

A finite band of spatial fields supplies more than a number of modes. Its evaluation map fixes a contact operator, its moving image fixes a connection, and overlaps of those images fix transition strengths. These constructions can therefore be tested together without independently prescribing a count, a connection and a mixing matrix.

The geometric hypotheses below concern ordinary spatial manifolds. A spinor fibre or a finite band describes fields on that space. The round-sphere comparison is a compatible nested control family.

1. Evaluation Gives a Contact Operator

Let $\mathcal H_N$ be a finite spectral band of smooth sections of a Hermitian rank-$r$ bundle on a spatial manifold. Use its specified physical $L^2$ norm. Evaluation at $x$ is

$$E_{N,x}:\mathcal H_N\longrightarrow F_x,\qquad B_N(x)=E_{N,x}E_{N,x}^\dagger.$$

Here $B_N(x)$ is the diagonal of the reproducing kernel. Suppose a fixed reference band has $B_0(x)>0$ on the chosen fibre. Define

$$T_{N,x}=E_{N,x}^\dagger B_0(x)^{-1}E_{N,x}.$$

Its nonzero eigenvalues are those of $B_0^{-1/2}B_NB_0^{-1/2}$. Consequently,

$$B_N(x)=S(n,d)B_0(x) \quad\Longrightarrow\quad T_{N,x}^2=S(n,d)T_{N,x},\qquad N=n-1.$$

The nonzero contact eigenvalue is exactly $S(n,d)$, repeated $r$ times. This is an evaluation theorem, not the assertion that every band vector has that eigenvalue. The contact's kernel is retained.

A frequency contribution $\mu_dT_{N,x}$ then has the maintained nonzero frequency $\mu_dS(n,d)$ under one band-independent coefficient. Deriving that contact contribution, its reference normalization and its occupied state from the same-field action remains the physical problem. A finite-band point form is mathematically well-defined; it does not make a smooth self-source into a delta distribution.

2. Actual Spatial Examples

In units $c=\hbar=1$, on a round $S^d$ of radius $R$, a positive Dirac shell has frequency and multiplicity

$$\omega_l=\frac{l+d/2}{R},\qquad M_l=r_dg_d(l),\qquad r_d=2^{\lfloor d/2\rfloor},\qquad g_d(l)=\binom{l+d-1}{d-1}.$$

The full positive band through $N$ has dimension $r_dS(N+1,d)$. These are the multiplicities in Bär's sphere theorem.

Homogeneity and isotropy give

$$B_N(x)=\frac{S(N+1,d)}{\operatorname{Vol}(S^d)}I_{r_d}.$$

At odd $d$ the isotropy spinor is irreducible. At even $d$, its two irreducible chiral pieces have equal diagonal coefficients: a nonzero Dirac eigenstate has zero integrated chirality, while homogeneity makes the diagonal chirality trace constant. The two equal-rank pieces therefore contribute equally. Thus the same formula holds at every $d\geq2$.

With $B_0=I/\operatorname{Vol}(S^d)$, the normalized contact has eigenvalue $S(n,d)$ on $r_d$ channels. The normalization cancels both the spinor multiplicity and physical volume through a specified reference operator, rather than silently discarding them.

3. Retain the Kinetic Operator and Calculate Its Dressing

At one sphere point, each evaluation polarization has the same coupled radial matrix. In shell coordinates,

$$H_N=\operatorname{diag}(\omega_0,\ldots,\omega_N)+\mu vv^\dagger, \qquad v_l=\sqrt{g_d(l)},\qquad v^\dagger v=S.$$

The other directions in each shell remain explicit uncoupled states. Put $b=v/\sqrt S$ and $\Lambda=\mu S$. The exact coupled frequencies away from the free poles satisfy

$$1=\mu\sum_{l=0}^N\frac{g_d(l)}{z-\omega_l}.$$

This follows either from the matrix determinant lemma or the evaluation resolvent. It supplies the correction that an isolated contact eigenvalue omits.

The uniform cumulative state count gives exact kinetic moments:

$$\bar\omega=\frac1R\left(\frac d2+\frac{dN}{d+1}\right),\qquad \sigma_\omega^2= \frac{dN(N+d+1)}{R^2(d+1)^2(d+2)}.$$

For example, $\sum_l l(l-1)g_d(l)/S=dN(N-1)/(d+2)$, which with the first moment gives the displayed variance. The same variance is the squared bright-to-complement kinetic coupling:

$$\|(I-bb^\dagger)H_0b\|^2=\sigma_\omega^2.$$

If $\mu>0$ and $\Lambda>W=N/R$, the top frequency has the controlled form

$$z_{\max}=\Lambda+\bar\omega+\delta,\qquad 0\leq\delta\leq\frac{\sigma_\omega^2}{\Lambda-W}.$$

To prove it, use the Rayleigh value $\Lambda+\bar\omega$ as a lower bound and the Schur complement on $b^\perp$. The complementary kinetic operator is at most $\omega_N$, so its resolvent denominator is at least $\Lambda-W$. The source norm in that Schur complement is exactly $\sigma_\omega$. The complementary component of a normalized top eigenvector is at most $\sigma_\omega/(\Lambda-W)$.

At large contact strength, $\delta=\sigma_\omega^2/\Lambda+O(W^3/\Lambda^2)$. The mean kinetic term and this dressing must be included in a physical frequency prediction. They cannot be removed by relabeling the contact eigenvalue. The construction now makes their size and mode dependence calculable.

The same measure also constructs a complete finite Jacobi chain. Starting Lanczos orthogonalization of $H_0$ from $b$ gives first diagonal $\bar\omega$ and first off-diagonal $\sigma_\omega$. Subsequent coefficients come from orthogonalizing polynomials against $g_d(l)/S$. The contact shifts only the first Jacobi diagonal by $\Lambda$. Thus recurrence coefficients, endpoint condition and normalization can be derived together from the spatial band.

4. The Same Evaluation Image Has a Connection

Normalize the evaluation image by

$$U_N(x)=E_{N,x}^\dagger B_N(x)^{-1/2},\qquad U_N^\dagger U_N=I,\qquad P_N=U_NU_N^\dagger.$$

In a local fibre frame its induced connection and curvature are

$$A=U_N^\dagger dU_N,\qquad F=dU_N^\dagger(I-P_N)\wedge dU_N.$$

Changing the fibre frame changes $A$ by the usual connection law. These formulas define transport of actual band images; they require no extra physical coordinate.

The tensor

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

simultaneously supplies $F_{ab}=\mathcal Q_{ab}-\mathcal Q_{ba}$ and the positive shape metric $\operatorname{Re}\operatorname{tr}\mathcal Q_{ab}$. Consequently,

$$|\operatorname{tr}F_{ab}| \leq2\sqrt{\operatorname{tr}\mathcal Q_{aa}\, \operatorname{tr}\mathcal Q_{bb}}.$$

The same changing spatial modes determine both quantities. A real normalized line can have a nonzero shape metric and zero Berry curvature, as in the Gaussian calculation.

5. Overlap Gives an Exact Mixing Defect

Suppose two canonically normalized frames $U_a,U_b$ have been embedded by explicit maps into the same field Hilbert space. For the overlap vertex $K=U_a^\dagger U_b$,

$$I-K^\dagger K=U_b^\dagger(I-P_a)U_b\geq0,\qquad \|K\|_{\rm op}\leq1.$$

For equal finite ranks, $K$ is unitary precisely when the two frame images coincide. Their bases may still differ. The defect measures amplitude in directions omitted by the receiving band. It cannot be removed by replacing $K$ with its polar unitary factor.

This is a concrete geometric test for the overlap contribution to generation mixing. A general physical vertex need not be a pure overlap; its actual kinetic metrics, interaction weights and eliminated channels must be retained. The formula then applies to the explicitly identified overlap part.

6. The Same Spectral Measure Controls Response and Truncation

Eliminating complementary modes gives

$$\Sigma(z)=PHQ(z-QHQ)^{-1}QHP.$$

Its discrete poles give dressing, while a continuous spectral component gives the imaginary part associated with outgoing response. This reuses the same spectral measure rather than introducing a separate decay kernel. An instantaneous effective interaction needs its own low-frequency and propagation justification.

For a frequency tower $\mu S(n,d)$, the multiplicity must be stated. The unweighted reciprocal sum is finite for $d>1$. With complete oscillator-shell weights, however,

$$\frac{g_d(n-1)}{S(n,d)}=\frac{d}{n+d-1}.$$

Thus a uniformly coupled, shell-weighted static response has a logarithmic baseline contribution. Its frequency-dependent subtraction is better controlled:

$$\sum_{n=N}^{\infty}g_d(n-1) \left[\frac1{\mu S(n,d)-z}-\frac1{\mu S(n,d)}\right].$$

For $|z|\leq\mu S(N,d)/2$, its absolute value is at most

$$\frac{2|z|}{\mu^2}\, \frac{d}{(d-1)S(N,d)}.$$

Indeed $g/S^2=d/[(n+d-1)S]$, and the exact reciprocal tail bounds its sum by $d/[(d-1)S(N,d)]$. A uniformly bounded per-state squared vertex multiplies this bound by that bound on the vertex. The static subtraction still needs a physical reference prescription. This gives a usable error estimate for response calculations under declared weights.

7. Temporal Occupation and the Contact Readout

A spatial band also supplies the operator that reads an occupied temporal spectrum. For harmonic coefficient vectors $a_r$ and a Hermitian source $B$, its bilinear beat is $J_p=\sum_r a_r^\dagger B a_{r+p}$. Orthogonality under an integrated identity source need not imply orthogonality under point evaluation.

In the explicit finite control $T=ee^\dagger$ with $e=(1,\ldots,1)^T$, both $T^2=ST$ and $\sum_{r=1}^{S-1}T_{r,r+1}/[r(r+1)]=1-1/S$ hold. This connects the contact form to the fundamental beat of a specified harmonic preparation. The sawtooth coupling calculation retains the canonical metric and distinguishes its temporal cutoff from the spatial shell index. It does not assign evenly spaced temporal frequencies to the sphere's individual degenerate states without a dynamics establishing that map.

Verification and Use

The moving spatial-contact calculation now tests actual transport around a great circle of this sphere band. Its exact positive Fourier weights retain shell and rotation labels separately. Their rotation variance is the same directional geometric-tensor coefficient that controls finite-width source loss. Time-ordered moving contact differs from a static orbit average. A reciprocal collective rotor retains its complete source energy and Noether charge, while an additional convective action shows the physical connection required for exact bright following. That connection does not remove bare spatial kinetic dressing or derive the source, occupied cutoff, sector unit or a preferred depth.

The second-order continuation uses those same spatial weights in the inverse restoring square λ²(4I−3P). Both oscillator frequency sectors give a quadratic pencil and slow rotating shift −7Ω²G_N/(6λ). Complete reciprocal energy supplies a conditional nonlinear orbital-stability bound for the slow relative equilibria, including the electron assignment. Faster prescribed sources can supply work to growing modes. This isolated inverse law omits bare dispersion and does not explain observed lifetimes or select the sector depths.

One calculation verifies cumulative counts, exact first and second moments, projective reference ratios, normalized contact projectors, independent dressed eigenvalues and overlap defects. It compares depths 2 through 12 with the same rules and coefficients.

This construction joins the spatial count, its contact operator, kinetic corrections, geometric transport and transition accessibility. Physical completion supplies the occupied band and common interaction; the formulas specify what those choices jointly entail.

Explicit Connection and Quantum Tensor on the Spatial Control

The spherical-band continuation constructs the actual spinor polynomial eigenspaces and calculates their evaluation derivatives. The same cumulative shell moments give matrix-valued curvature and a positive quantum tensor whose Clifford components have ratio $(d+1)^2$ for nonzero cutoffs. This is a conditional spatial operator identity, not an empirical particle property; the real translation metric is isotropic, depth-five geometry is nonzero, and the down branch at cutoff zero has flat evaluation-image geometry. The complete action and spectral gaps remain necessary for an induced physical response.

A Sharp Bound for Normalized Spatial Sources

Retain the sphere band and its physical norm from Section 2. Write $V=\operatorname{Vol}(S^d_R)$ and $S_N=\binom{N+d}{d}=S(N+1,d)$. Let $\rho$ be any nonnegative Borel probability measure on that sphere. Define a finite-band operator by its quadratic form,

$$\langle\psi,A_{\rho,N}\psi\rangle =\int_{S^d_R}|\psi(x)|^2\,d\rho(x).$$

If $d\rho=\rho(x)\,dv(x)$, this is $P_NM_\rho P_N$. A point measure is covered by the form definition without pretending that a delta distribution is an ordinary multiplication density. Since $E_{N,x}E_{N,x}^\dagger=(S_N/V)I$, the evaluation norm gives

$$|\psi(x)|^2\leq\frac{S_N}{V}\|\psi\|^2, \qquad 0\leq A_{\rho,N}\leq\frac{S_N}{V}I.$$

Integrating the pointwise inequality proves the operator bound for every probability source. For a point source at $x$, every unit fibre vector $u$ gives the unit bright field

$$\psi_{N,x,u}=\sqrt{\frac{V}{S_N}}\,E_{N,x}^\dagger u, \qquad |\psi_{N,x,u}(x)|^2=\frac{S_N}{V}.$$

Thus the bound is sharp. The reference-normalized point contact satisfies

$$T_{N,x}=V E_{N,x}^\dagger E_{N,x},\qquad T_{N,x}^2=S_NT_{N,x},\qquad \operatorname{rank}T_{N,x}=r_d,\qquad \operatorname{tr}T_{N,x}=r_dS_N.$$

Equality Does Not Require a Unique Point Source

The exact equality condition is

$$\lambda_{\max}(A_{\rho,N})=\frac{S_N}{V} \quad\Longleftrightarrow\quad \exists\,\psi\in\mathcal H_N,\ \|\psi\|=1: |\psi(x)|^2=\frac{S_N}{V}\quad\rho\text{-almost everywhere}.$$

Indeed, the nonnegative deficit $S_N/V-|\psi(x)|^2$ has zero integral precisely when it vanishes almost everywhere for the source measure. Finite dimensionality guarantees a maximizing unit vector. Since the fields are continuous, the condition can also be stated as support of $\rho$ on that field's saturation set. The explicit saturation theorem now proves that a single point is necessary and sufficient at every excited cutoff $N\ge1$. The ground cutoff has the different equality class below.

There is an exact smooth equality case at $N=0$. Both the ground band and the spinor fibre have dimension $r_d$. Consequently $\sqrt V E_{0,x}$ is square and unitary, and

$$E_{0,x}^\dagger E_{0,x}=\frac1V I_{\mathcal H_0},\qquad |\psi(x)|^2=\frac1V\quad(\|\psi\|=1),\qquad A_{\rho,0}=\frac1V I_{\mathcal H_0}.$$

Every probability source attains the bound at this cutoff, including the smooth uniform density $1/V$ and the normalized self-density of any ground field. The proof applies at all six occupied comparison depths, $d=2,3,4,5,6,10$, with the same normalization. It neither proves excited-band smooth saturation nor selects any of those depths preferentially.

Excited-Band Saturation and the Actual Smooth Source

In the positive ground Killing frame, equality at two points would require their normalized bright fields to coincide. Their constant ground components force equal polarizations. The first positive shell then separates the points: at angle $\theta$, the squared difference of its evaluation vectors with that common polarization is $2d(1-\cos\theta)$ in probability-normalized volume. It vanishes only for the same point. This proves the excited-band point-source theorem without assuming that an arbitrary nonconstant field has a unique maximum.

The same explicit spinor kernels give the two-equal-point contact eigenvalues $S_N(1\pm s_N(\theta))/2$, with one copy per spinor polarization. Their top fraction is $1-G_N\theta^2/4+O(\theta^4)$, where $G_N$ is the existing spatial geometric tensor coefficient. Contact width and mode geometry therefore constrain the same operator response.

For a smooth bright-field density source, the projected quartic coefficient is instead $I_4=\int |\psi_N|^4d\nu<S_N$ at every excited cutoff. The actual sphere-spinor calculation gives a $-4.52\%$ electron/muon ratio drift from the simplex ratio at shared norm, coupling and volume. Its cubic source has about $61.7\%$ and $62.9\%$ of its squared norm outside the selected positive bands, including negative Dirac modes. This is a specified classical density contact, not the Lorentz-scalar master interaction or an exact stationary particle. The full source and kinetic dressing must remain in the physical equation. All occupied depths are compared, with no special depth-five zero and no new favorable sector preference.

Cumulative Consistency and the Continuum Domain

For the actual fixed spatial-band inclusion $I_N:\mathcal H_N\hookrightarrow\mathcal H_{N+1}$, evaluation obeys $E_{N+1,x}I_N=E_{N,x}$. Therefore

$$I_N^\dagger T_{N+1,x}I_N=T_{N,x}.$$

The reference factor $V$ is independent of cutoff. This is the bare-form consistency of consistent band contacts; a changing normalized coherent-ray identification requires its own inclusion law.

Consistency alone does not provide a closed continuum point interaction. The union of the positive bands is dense in the positive spectral Hilbert subspace $\mathcal H_+=\overline{\bigcup_N\mathcal H_N}$, rather than the full spinor $L^2$ space. On that union, consider $q_x(\psi)=V|\psi(x)|^2$. For a fixed unit fibre vector $u$, set

$$w_N=\frac{V}{S_N}E_{N,x}^\dagger u.$$

Then

$$w_N(x)=u,\qquad \|w_N\|^2=\frac{V}{S_N}\longrightarrow0,\qquad q_x(w_N)=V,\qquad q_x(w_N-w_M)=0.$$

This violates the closability criterion for a positive quadratic form on $\mathcal H_+$ with its physical $L^2$ norm. A stronger kinetic form domain, a finite-width source, or a renormalized point-interaction construction must supply the missing domain control. The argument excludes only this bare continuum form in this norm; it does not exclude nonlinear same-field parents.

These results separate contact strength, source saturation, cumulative consistency and continuum domains. They organize the existing spatial evaluation identity and clarify its scope. A physical same-field source, its sustaining dynamics and the kinetic dressing of Section 3 remain necessary to derive a particle frequency. No new sector preference or empirical particle property follows from this bound alone.