Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Source and scope

Transcribed from Appendix B §20–20a, with representation-theory overclaims omitted. The associated historical code includes overlap_J near line 2282. These are scalar flat-space oscillator modes, not automatically V2 spinor, polarization, or orbital-section states.

Explicit normalized basis

For H=-Delta+lambda r^2, w=sqrt(lambda)>0, and angular harmonics normalized to unit integral on S^(d-1),

χnrℓm(r,Ω)=NnrℓrℓLnr(ℓ+d/2−1)(wr2)e−wr2/2Yℓm(Ω),Nnrℓ2=2wℓ+d/2nr!Γ(nr+ℓ+d/2). \chi_{n_r\ell m}(r,\Omega)= N_{n_r\ell}r^\ell L_{n_r}^{(\ell+d/2-1)}(wr^2) e^{-wr^2/2}Y_{\ell m}(\Omega), \qquad N_{n_r\ell}^2=\frac{2w^{\ell+d/2}n_r!}{\Gamma(n_r+\ell+d/2)}.

N=2nr+ℓ,EN=w(2N+d),⟨r2⟩N=2N+d2w. N=2n_r+\ell,\qquad E_N=w(2N+d),\qquad \langle r^2\rangle_N=\frac{2N+d}{2w}.

The level degeneracy is binomial(N+d-1,d-1); the cumulative count through N=n-1 is S(n,d)=binomial(n+d-1,d). This count is not E_N. Equality of dimensions with another representation does not establish an operator intertwiner or an equivalence of physical state spaces.

Density overlap versus transition matrix element

For isotropic normalized densities in the same R^d,

Oij=∬(x⋅y)2∣χi(x)∣2∣χj(y)∣2 dx dy=⟨r2⟩i⟨r2⟩jd. O_{ij}=\iint(x\cdot y)^2|\chi_i(x)|^2|\chi_j(y)|^2\,dx\,dy =\frac{\langle r^2\rangle_i\langle r^2\rangle_j}{d}.

Isotropy must hold for the actual density, for example an s-wave or a uniform shell mixture; an arbitrary individual oscillator eigenstate need not have it. With one mode fixed, this overlap grows linearly with the other shell index, unlike the degree-d cumulative count. It is a density-density expectation, not a normalized charged-current amplitude.

For actual coherent transitions use off-diagonal moments such as integral chi_i^*(x) x_a x_b chi_j(x) dx, retaining phases, angular indices, selection rules, and any spinor vertex.

A reusable radial integral

For s>0 and integer n_r>=0,

∫0∞u1/2e−suLnr(1/2)(u) du=Γ(nr+3/2)nr!(s−1)nrsnr+3/2. \int_0^\infty u^{1/2}e^{-su}L_{n_r}^{(1/2)}(u)\,du =\frac{\Gamma(n_r+3/2)}{n_r!} \frac{(s-1)^{n_r}}{s^{n_r+3/2}}.

At s=1 it vanishes for n_r>=1; at s=3/2 no finite radial level is eliminated. The weight must be obtained from the actual vertex before using this as a selection rule. It does not select a particle tower by itself.

Product shell counts

Shell degeneracies obey a useful convolution. For gd(N)=(N+d−1d−1)g_d(N)=\binom{N+d-1}{d-1},

∑k=0Nga(k)gb(N−k)=ga+b(N). \sum_{k=0}^{N}g_a(k)g_b(N-k)=g_{a+b}(N).

The generating functions multiply as (1−z)−a(1−z)−b=(1−z)−(a+b)(1-z)^{-a}(1-z)^{-b}=(1-z)^{-(a+b)}. This counts product states of the specified separated oscillators with fixed total excitation. Using cumulative counts as though they were shell degeneracies would give a different convolution.

The partition into aa and bb directions is a calculation choice among equivalent ordinary spatial directions. The a=b=3a=b=3 example does not give the electron an inherent 3+3 split or prove colour neutrality. The simplex band moments instead average shell observables across a whole cumulative band.

Constant-background overlaps

A uniform continuation and an oscillator ground state have different overlap weights. For the physicists' Hermite polynomials in a dimensionless coordinate,

∫−∞∞H2m(y)e−y2/2 dy=2π(2m)!m!,∫−∞∞H2m+1(y)e−y2/2 dy=0. \int_{-\infty}^{\infty}H_{2m}(y)e^{-y^2/2}\,dy =\sqrt{2\pi}\frac{(2m)!}{m!},\qquad \int_{-\infty}^{\infty}H_{2m+1}(y)e^{-y^2/2}\,dy=0.

To derive this, integrate the generating function e2ty−t2=∑jHj(y)tj/j!e^{2ty-t^2}=\sum_jH_j(y)t^j/j! against e−y2/2e^{-y^2/2}. The result is 2πet2\sqrt{2\pi}e^{t^2}.

Thus overlap against a constant background removes odd modes by parity, but every even mode has a nonzero overlap. Normalized oscillator modes add nonzero prefactors and do not change these zeros. The constant is a generalized background on the line, not a normalizable oscillator state. A ground-state bra would supply another Gaussian and a different orthogonality relation.

This distinction matters for the cylindrical continuation premise: a low-degree insertion and parity alone do not exclude excitation in several added coordinates when the relevant modes are even. The actual vertex and integration weight must be supplied.

Finite spatial operators

The complete source marginal uses these normalized constant-background integrals in a finite matrix; its singular strengths follow a weighted Hockey-Stick identity. Physical bilinear currents must be formed before that linear map is applied. The spatial evaluation kernel uses the same paired-occupation weights at the oscillator center and fixes the distinction between spatial point contact and projective ray contact.

Review checks

Independent quadrature in the preceding review checked unit normalization and the radial second moment for (d,n_r,ell,w)=(3,2,0,2.195), (4,1,2,1.314), and (6,6,0,0.5), with errors below 1e-13. This was a representative check, not a rerun of the entire historical script or validation of its conclusions.

The mode counting checks verify shell convolutions and independently integrate the Hermite overlaps by Gaussian quadrature.