Real Spatial Counts and Canonical Contact

The simplex count survives a construction using real fields and real shape coordinates throughout. This supplies a conditional geometric realization of the count in ordinary space. A temporal frequency additionally requires a specified action; a dimension count alone does not give that action.

Real Parity Bands

Let $d\in\{2,3,4,5,6,10\}$, $N=n-1\geq0$, and let $\sigma$ be uniform probability measure on the real amplitude sphere $S^d$. A shape has one scalar and $d$ directional amplitude coordinates. These coordinates do not add physical spatial directions. The restricted real homogeneous polynomials decompose as

$$V_N=\bigoplus_{k=N,N-2,\ldots}\mathcal H_k(S^d),\qquad h_k=\binom{k+d}{d}-\binom{k+d-2}{d}.$$

The second term is zero for $k<2$. The sum telescopes:

$$D=\dim V_N=\binom{N+d}{d}=S(n,d).$$

This is a parity band, not the full spherical band through $N$. Its reproducing kernel, in the stated normalization, has the closed form

$$K_N(x,w)=C_N^{(d+1)/2}(x\cdot w),\qquad K_N(w,w)=D.$$

Indeed, with $\lambda=(d-1)/2$, the degree-$k$ addition kernel is $Z_k=(k+\lambda)C_k^\lambda/\lambda =C_k^{\lambda+1}-C_{k-2}^{\lambda+1}$; summing the parity sequence gives the displayed expression. The identity holds for arbitrary $N$, including the constant band $N=0$.

What Covariance Fixes

Suppose a positive rank-one filter $\Lambda(w)$ is $SO(d+1)$ covariant and resolves the identity, $\int\Lambda\,d\sigma=I_{V_N}$. Taking its trace fixes its sole nonzero eigenvalue to $D$. The full classification is

$$\Lambda_s(w)=D v_w^s(v_w^s)^T,\qquad v_w^s=D^{-1/2}\sum_{k=N,N-2,\ldots}s_k Z_k(\cdot,w),\quad s_k=\pm1.$$

To derive it, the stabilizer of $w$ has one fixed line in every harmonic summand. Write its unit vector as $\sum c_k Z_k/\sqrt{h_k}$. Averaging the rank-one operator gives harmonic block $Dc_k^2/h_k$, so identity resolution requires $c_k^2=h_k/D$. Relative signs remain free: there are $2^{\lfloor N/2\rfloor}$ projector families after removing a common sign. Point evaluation selects the all-positive choice, for which $\langle f,\Lambda_+(w)f\rangle=f(w)^2$.

The enlarged $SO(d+1)$ shape symmetry mixes the scalar amplitude with the directional amplitudes. Ordinary spatial $SO(d)$ isotropy alone does not imply it. The sphere-integral polynomial norm is another explicit premise; the unmodified symmetric-tensor norm need not give this tight frame.

A real canonical pair with Hamiltonian $H=\mu(q^T\Lambda q+p^T\Lambda p)/2$ has bright-mode frequency $\mu D$. A second-order oscillator with stiffness proportional to $\Lambda$ and fixed kinetic coefficient instead has frequency proportional to $\sqrt D$. The difference follows by varying the action. Thus the real geometry permits the desired count frequency, conditional on the canonical contact law, rather than forcing it for every kinetic law.

Actual Profiles in Flat Space

Choose a length $L$ and ordinary coordinates $x\in\mathbb R^d$. Define

$$w(x)=\left(\frac{2Lx}{L^2+|x|^2}, \frac{|x|^2-L^2}{|x|^2+L^2}\right),\quad \rho_L(x)=\frac{[2/(1+|x|^2/L^2)]^d}{|S^d|L^d},\quad (Uf)(x)=\sqrt{\rho_L(x)}f(w(x)).$$

Since $d\sigma=\rho_L\,dx$, $U$ preserves real squared norm. This is a formula for smooth spatial profiles with finite norm and gradient energy in flat space; it does not turn physical space into a sphere. The spatial projector satisfies

$$P_N(x,z)=\sqrt{\rho_L(x)\rho_L(z)}K_N(w(x),w(z)),\qquad P_N(x,x)=D\rho_L(x),\qquad\int P_N(x,x)\,dx=D.$$

A point contact per ordinary spatial volume therefore has bright value $D\rho_L(x)$. Uniformity in the normalized shape measure does not remove this factor. In $\mathbb R^{10}$ a normalized transverse envelope preserves the count but contributes its own gradient and interaction energy. A constant envelope instead requires a per-transverse-volume interpretation.

Finite contact width also changes the bright value. Near a selected shape, the leading fractional deficit is $N(N+d+1)\operatorname{tr}\Sigma/(d+2)$, where $\Sigma$ is the small tangent displacement covariance. This follows from differentiating the Gegenbauer kernel and expanding the normalized projector overlap. Exact equality for a positive averaged contact requires support on the same antipodal shape pair when $N\geq1$. No such restriction occurs at $N=0$.

Deterministic Occupation and Energy

The identity $\operatorname{Tr}(\epsilon I\Lambda)=\epsilon D$ requires equal energy per occupied mode. Equal total energy per band, represented by $\epsilon I/D$, gives $\epsilon$ instead. Neither preparation is selected by the frame identity.

Deterministic multiplets can realize the first preparation. On the comparison sphere, put one real component in each harmonic basis function $Y_{kj}$ with common block phase and $\omega_k^2=c^2 k(k+d-1)+m_0^2$. The addition theorem gives $\sum_jY_{kj}^2=h_k$ and $\sum_j|\nabla Y_{kj}|^2=k(k+d-1)h_k$. Consequently the standing multiplet $A_kY_{kj}\cos(\omega_k t+\theta_k)$ has exactly constant block energy $A_k^2\omega_k^2h_k/2$. Taking $A_k^2\omega_k^2/2=\epsilon$ gives total $\epsilon D$, with no probability ensemble or time average. It assumes the required real components and their occupation. A scalar sum introduces interference terms and does not inherit this conclusion. For $m_0=0$ the constant harmonic is not a bounded positive-energy oscillator.

Under the flat profile map the energy density becomes $\epsilon D\rho_L(x)$, while total energy remains $\epsilon D$. Converting that total into one particle oscillation still requires reciprocal dynamics. The real quartic branches test such dynamics directly, rather than assigning the kernel profile a frequency from its fourth moment alone.

Evidence and Scope

The same real construction accommodates every maintained depth and all fourteen assigned bands. It establishes compatibility of the count and canonical contact with real spatial profiles. It does not select the occupied bands, their scales, a preferred depth-five response, or physical self-support. Electron center motion remains a separate deterministic trajectory problem.

The independent real-filter diagnostic performs 881 exact checks of kernels, signed frames, polynomial moments, all fourteen counts and deterministic multiplet identities. The finite-width calculations compare all fourteen assigned bands. Their exact source versions and dated reports are retained in the research catalog under the real-energy-count run of 14 October 2026. The harmonic decomposition and addition formula use the standard conventions explained by Dai and Xu, Spherical Harmonics; the physical interpretation and spatial map above are conditional IDWT constructions.

Reproduce the exact identities with the real-filter checker, and the contact-width comparison with the width checker.