Counting, Positive Degree, Reciprocal Uniformization, and Mellin-Fourier Reflection of a Spherical Casimir Spectrum
Year: 2026
Type: preprint
Abstract: For $d\ge2$, let $\lambda_n^{(d)}=\frac{n(n+d-2)}2,$ $\Pi_d(u)=\prod_{n=1}^{\infty}\left(1-\frac{u}{\lambda_n^{(d)}}\right).$ These are the simple zonal eigenvalues of $-\tfrac12\Delta_{S^{d-1}}$ after removal of the constant mode. Let $\Lambda_d$ be the Dirichlet-to-Neumann operator of the unit ball. Harmonic extension of a degree-$n$ spherical harmonic is $r^nY_n$, hence $\Lambda_dY_n=nY_n$, and therefore $\boxed{-\frac12\Delta_{S^{d-1}}=\frac12\Lambda_d\bigl(\Lambda_d+(d-2)I\bigr).}$ Equivalently, $\Lambda_d=\sqrt{-\Delta_{S^{d-1}}+\frac{(d-2)^2}{4}I}-\frac{d-2}{2}I.$ This positive square root makes the primitive counting spectrum and the branch selection operator-theoretic, not merely coordinate choices. Extending degree to the signed line gives the involution $n\mapsto-D-n$: the two exterior rays carry the same nonnegative Casimir values, the finite chamber $-D<n<0$ carries negative values, and $\Lambda_d\ge0$ selects the physical exterior ray canonically. The reflection-completed centered lattice also has an exact Poisson-summed theta heat law, so Gaussian duality already appears at the counting/Casimir level before Gamma closure. With $D=d-2$, $\lambda_n^{(d)}=T_n+\frac{d-3}{2}n,$ $\lambda_n^{(d)}+\frac{D^2}{8}=\frac12\left(n+\frac D2\right)^2.$ Thus dimension changes only the linear counting term while preserving the unit second finite difference of the triangular law. The same centered square is the Bessel order produced by radial separation of the Euclidean Helmholtz equation. We realize $\Pi_d$ as a Fredholm determinant and close it under the reflection coordinate $u_d(z)=\frac{(z-1)(z-d+1)}2,$ $\Pi_d(u_d(z))=\frac{\Gamma(d-1)}{\Gamma(z)\Gamma(d-z)}.$ For $d>2$, the projective coordinate $r=\frac{z-1}{d-1-z}$ conjugates $z\mapsto d-z$ to reciprocal inversion $r\mapsto1/r$. In this coordinate $u_d(r)=-\frac{(d-2)^2}{2}\frac{r}{(1+r)^2},$ $ u_d(r)+\frac{(d-2)^2}{8}=\frac{(d-2)^2}{8}\left(\frac{r-1}{r+1}\right)^2.$ Hence the fixed line $\Re z=d/2$ is exactly the unit circle $|r|=1$, the two half-planes become the inside and outside of that circle, and the finite fixed point becomes the positive unit radius $r=1$. Writing $r=e^{2y}$ straightens reciprocal inversion to $y\mapsto-y$ and gives the hyperbolic law $u_d=-(d-2)^2\operatorname{sech}^2y/8$. After the dimensionless normalization $\tau_B:=-\frac{2u_d}{(d-2)^2},$ one has the exact chain $\boxed{\tau_B=p(1-p)=\frac{r}{(1+r)^2}=\frac14\operatorname{sech}^2y.}$ This is the same hyperbolic coordinate used in the spherical-to-Euclidean limit of the supplied OpenAI source. On the positive half-orbit $y\ge0$, the two OpenAI probability measures become $\nu=2\,d p,$ $\rho=\frac{2}{\pi\sqrt{p(1-p)}}\,d p,$ $\frac{d\nu}{d\rho}=\pi\sqrt{p(1-p)}.$ Thus the coordinate and measure transport agree exactly, while no claim is made that the Fredholm determinant itself generates the external variational measures. For a Schwartz function in a single spherical harmonic degree, $f(r\omega)=f_n(r)Y_n(\omega)$, the same positive degree operator also resolves the Mellin--Fourier reflection: $M_{\widehat f_n}(z)=(-i)^n\pi^{d/2-z}\frac{\Gamma((n+z)/2)}{\Gamma((n+d-z)/2)}M_{f_n}(d-z).$ Equivalently, $n$ is replaced by $\Lambda_d$ through spectral functional calculus. The fixed-line multiplier is unitary in every degree, and the degree-zero specialization is exactly the radial formula used in the supplied OpenAI proof. Fourier projection followed by the standard Gaussian origin correction therefore enforces the Fourier eigenvalue and $g(0)=0$ constraints on the radial rail; the last-sign observable is not controlled by these algebraic steps. At $r=1$ every determinant factor becomes a perfect square and $P_d:=\Pi_d\!\left(-\frac{(d-2)^2}{8}\right)=\frac{\Gamma(d-1)}{\Gamma(d/2)^2}=\binom{d-2}{(d-2)/2}_{\!\Gamma}.$ When $d-2$ is even this is an ordinary central binomial coefficient and counts the equatorial layer of a hypercube. Consequently $\sqrt{d-2}\,2^{-(d-2)}P_d\longrightarrow\sqrt{\frac2\pi}.$ The factorially normalized even central layers generate $I_0$ and $J_0$, while the twin central layers of odd cubes generate $I_1(z)/z$ and $J_1(z)/z$. More generally, shifted simplex counts have $e^{-u}L_k^{(\alpha)}(u)$ as their alternating factorial generating function, and their hard-edge scaling forces $J_\alpha$. In particular the companion Paper~A profile $2J_1(t)/t$ is both a counting limit and the coordinate characteristic function of the unit sphere $S^3$, hence it defines a positive semidefinite Gram kernel. At finite $n$, the Paper~A one-sided transform is a Pascal polynomial of the same reciprocal unit-circle coordinate that uniformizes the Casimir fixed line. At the generator level, $L_{Q_m}/2$ and $\Lambda_d$ intertwine; exponentiating shows that the Boolean noise operators $q^{L_{Q_m}/2}$ and the spherical Poisson dilations $q^{\Lambda_d}$ intertwine for $|q|<1$, and their unit-circle continuations give the finite Paper~A unitary trace. A heat-regularized spherical Fourier map then carries every harmonic degree $n$ to the Euclidean Bessel order $n+(d-2)/2$, giving an exact sphere-to-Fourier Casimir adapter. The note also retains the exact cubical-to-spherical harmonic intertwiner, Gaussian norm defect, Legendre--conical--elliptic specialization, and Mellin reflection comparison of the previous revision. Repeated summation is retained beyond the triangular case. For $S_p(n)=\binom{n+p-1}{p}$, the canonical product $\mathcal P_p(u)=\prod_{n\ge1}(1-u/S_p(n))$ has an algebraic Gamma closure obtained from the $p$ roots of $S_p(x)=S_p(a)$, and has order $1/p$ with explicit type. The vertex-exact cube extension has the exact polar decomposition $E_d=U_d\mathsf D_d^{-1}$, with all norm loss carried by one positive degree multiplier. In Euclidean space, the harmonic Gaussian--Laguerre modes diagonalize the $2\pi$-Fourier transform with eigenvalue $(-i)^{k+2m}$. On the reflection fixed line, the determinant amplitude factors through adjacent bare Gaussian Mellin envelopes; its reciprocal is the characteristic function of a log-Gamma ratio and of an $L^2$-convergent sum of independent centered Laplace variables. These extensions are included to preserve the full theorem genealogy even where a shorter publication version may later move them to companion notes.
Author: Huckstead, Jeffery
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202128185