Spatial Source Saturation and the Complete Spinor Self-Response

The Same Spatial Carrier and Its Source

The spatial projector construction provides actual positive Dirac bands on a round ordinary spatial sphere S^d_R. Its classical spinor field describes field structure; its squared norm is not an electron-position probability law. Literal deterministic electron trajectories remain a separate physical requirement.

The theorem concerns a static source operator. A point source following a deterministic path retains the instantaneous point-contact strength at each position; its time-ordered field response need not equal the response to a static source distributed along that path. The moving spatial contact calculation makes that distinction explicit, retaining bare spatial frequencies and reciprocal source energy. Neither the static theorem nor an orbit average alone establishes a moving electron's mass or coupling.

Let H_N contain the positive Dirac shells 0 through N, let V be the physical sphere volume, and put

$$g_l=\binom{l+d-1}{d-1},\qquad S_N=\sum_{l=0}^Ng_l=\binom{N+d}{d},\qquad r_d=2^{\lfloor d/2\rfloor}.$$

The shell frequencies are (l+d/2)/R, and their multiplicities are r_d g_l. The spectral setup is given by Camporesi and Higuchi, Sections 3–5. The positive shell kernel and complete-source comparisons below are calculated explicitly on that setup.

Physical evaluation E_N,x obeys E_N,x E_N,x†=(S_N/V)I. For a nonnegative Borel measure ρ with total mass one, define the reference-normalized source operator by

$$\langle\psi,T_{\rho,N}\psi\rangle =V\int |\psi(x)|^2\,d\rho(x).$$

The existing evaluation bound gives 0≤T_ρ,N≤S_N I. A single point source attains S_N. The question here is precisely when a spread source can attain that same bound, followed by explicit tests of finite spread and self-density.

An Explicit Off-Diagonal Spinor Kernel

Use the global trivialization supplied by the positive ground Killing spinors, and temporarily use probability-normalized sphere volume. Fix a north pole x. Write a second point y as

$$y=(\sin\theta\,n,\cos\theta),\qquad |n|=1,\qquad t=\cos\theta.$$

For Hermitian tangent Clifford matrices γ_a, write γ(n)=Σ_a n_a γ_a. The positive shell kernel in this frame is

$$K_l(y,x)=a_l(t)I-i\sin\theta\,b_l(t)\gamma(n),$$

$$a_l(t)=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad b_l(t)=C_{l-1}^{(d+1)/2}(t),\qquad b_0=0.$$

Here C is a Gegenbauer polynomial. This formula is fixed by the regular rotationally equivariant monogenic polynomial and its evaluation normalization: a_l(1)=g_l. One direct check writes its homogeneous extension as A(z,r)I−iγ(x_perp)B(z,r). The ambient Dirac equation reduces to

$$A_z-dB-rB_r=0,\qquad A_r/r+B_z=0.$$

The Gegenbauer derivative and recurrence identities verify these equations. Its scalar components are harmonic polynomials of degree l. The existing Killing-frame construction therefore gives the positive spatial Dirac eigenvalue l+d/2. Gegenbauer orthogonality, with weight (1−t²)^((d−2)/2), supplies the physical shell norm g_l. Independent projection onto actual polynomial Dirac kernels checks the matrix formula at degrees zero, one and two for every occupied depth.

Put A_N=Σ_{l=0}^N a_l and B_N=Σ_{l=0}^N b_l. Since γ(n)²=I,

$$K_N(y,x)^\dagger K_N(y,x) =\bigl[A_N(t)^2+(1-t^2)B_N(t)^2\bigr]I.$$

Thus every polarization has the same overlap singular value

$$\boxed{s_N(\theta) =\frac{\sqrt{A_N(\cos\theta)^2+ \sin^2\theta\,B_N(\cos\theta)^2}}{S_N}.}$$

This is the overlap of normalized evaluation images, not a new interaction field. Its role in a physical response still depends on the common action.

Exact Saturation Theorem

For N≥1, the source operator attains its maximal eigenvalue S_N if and only if ρ is a single point measure. At N=0 every probability source attains the bound.

The existing equality condition says that some unit field must saturate the evaluation bound at ρ-almost every point. At a saturation point x, that field has the form

$$\psi=\sqrt{V/S_N}\,E_{N,x}^\dagger u, \qquad |u|=1.$$

Suppose the same field saturates at a second point y, with polarization v. Comparison of the ground-shell components, which are constant in the chosen Killing frame, forces u=v. The first positive shell has kernel

$$K_1(y,x)=d\cos\theta\,I-i\sin\theta\,\gamma(n).$$

Its two evaluation vectors with this common unit polarization obey, in probability-normalized volume,

$$\|E_{1,x}^\dagger u-E_{1,y}^\dagger u\|^2 =2d-2\operatorname{Re}(u^\dagger K_1(y,x)u) =2d(1-\cos\theta).$$

The imaginary Clifford term has zero real expectation. The difference can vanish only when y=x. Thus a saturating excited-band field has exactly one saturation point. The support condition for ρ forces a point measure; the converse follows from the evaluation identity.

At N=0 the ground evaluation map is square and unitary after multiplication by √V. Every unit ground field has constant density 1/V, so T_ρ,0=I for every probability source. This boundary case is essential.

An independent overlap version uses the first cumulative band:

$$s_1^2=\frac{(1+d t)^2+1-t^2}{(d+1)^2},$$

$$(d+1)^2-[(1+d t)^2+1-t^2] =(1-t)[(d^2-1)t+d^2+2d-1]>0\quad(-1\le t<1).$$

Splitting a larger cumulative evaluation image into this band and its remaining shells gives

$$s_N(\theta)\le 1-\frac{d+1}{S_N}[1-s_1(\theta)]<1\quad(\theta\ne0).$$

The theorem concerns this source class and sphere carrier. Signed scalar currents, matrix-valued sources, other spatial geometries and different nonlinear actions require separate arguments. It is not a general theorem that IDWT particles must contain a physical delta source.

A Finite-Spread Contact and Its Geometric Correction

For two equal point sources, T=(T_x+T_y)/2. The overlap formula gives its nonzero eigenvalues

$$\boxed{\lambda_\pm=\frac{S_N}{2}(1\pm s_N(\theta)),}$$

each repeated r_d times when the two evaluation images are distinct. The ground or coincident case is degenerate and the minus values vanish. A small 2r_d-dimensional Gram matrix verifies these eigenvalues independently from the polynomial spatial modes.

The existing spatial geometric tensor supplies

$$G_N=\left(d-1+\frac1{d+2}\right) \frac{N(N+d+1)}{(d+1)^2}.$$

Its same-direction component gives

$$s_N(\theta)=1-\frac12G_N\theta^2+O(\theta^4),\qquad \boxed{\lambda_+/S_N=1-\frac14G_N\theta^2+O(\theta^4).}$$

Thus the contact's finite-spread correction is fixed by the same mode geometry already governing transport and deformation. Physical separation is Rθ. With the common angular separation θ=0.01, the exact contact fractions are 0.99940412 for electron's assigned band and 0.99636616 for muon's. These are contact contributions under a prescribed source, not complete dressed particle frequencies. The kinetic term must still be kept.

Smooth Self-Density and the Full Cubic Response

The normalized bright field at x has dimensionless density

$$h_N(t)=\frac{A_N(t)^2+(1-t^2)B_N(t)^2}{S_N}, \qquad \int h_N\,d\nu=1,$$

where dν=dv/V. This is a classical field density. For the explicit density interaction g∫|Ψ|⁴/2 and canonical norm Q, its projected nonlinear frequency contribution on this trial field is

$$\frac{gQ}{V}I_4(N,d),\qquad I_4=\int h_N^2\,d\nu.$$

This action uses Ψ†Ψ, rather than the Lorentz scalar bar(Ψ)Ψ. It is a specified classical comparison and does not inherit a Lorentz boost law. The separate signed scalar-current calculation retains the spatially varying parallel grading in the Killing frame. It gives different source coefficients, opposite-sign leakage even at the ground cutoff, and a complete polarization dependence within the bright family. Neither calculation substitutes for the other's source variation. For N≥1, the saturation theorem immediately gives I₄<S_N. More importantly, the complete source is h_N ψ_N, not just its component along ψ_N.

The opposite Dirac signs can be retained explicitly in the same frame. Their radial shell functions are

$$\widetilde a_l=t a_l-(1-t^2)b_l,\qquad \widetilde b_l=-(a_l+t b_l).$$

They describe (tI+i sin(θ)γ(n)) times the opposite ambient-monogenic polynomial and have spatial eigenvalue −(l+d/2). Their norms are g_l and they are orthogonal to every positive shell. This follows from the ambient Clifford anticommutator and homogeneous degree, and is independently checked by angular quadrature. No physical antiparticle identification is assigned to a negative spectral sign.

The cubic source coefficients on the normalized positive shell are

$$c_l=\frac1{\sqrt{g_lS_N}} \int h_N[a_l A_N+(1-t^2)b_l B_N],d\nu,$$

with the analogous expression using the tilded functions for coefficients \widetilde c_l on the negative shells. Polynomial degree makes the expansion finite: positive degrees through 3N and negative degrees through 3N−1 suffice. Their complete Parseval identity is

$$\sum_l|c_l|^2+\sum_l|\widetilde c_l|^2 =\int h_N^3\,d\nu.$$

The fraction outside the selected positive band is therefore

$$\eta_{\rm out}=1- \frac{\sum_{l=0}^N|c_l|^2}{\int h_N^3,d\nu}.$$

Representative results for the maintained assignments are:

At shared g, Q and V within each sector, the extreme assigned-band quartic ratios differ from the simplex ratios by −0.00734% at depth two, −65.24% at three, −26.08% at four, −4.45% at five and −4.52% at six. The single depth-ten assignment has no within-sector ratio comparison. These projected coefficients are not exact nonlinear stationary frequencies: the displayed omitted source is a direct residual of that restriction. Bare spatial kinetics and the complete reciprocal source remain part of the field equation.

Depth five has neither a zero source nor a uniquely small residual. The same law and normalizations were used at every occupied depth. The result does not favor the assigned sectors and supplies no new favorable sector-case evidence. It narrows the required source mechanism: this local density contact does not replace the assumed point contact while retaining its exact clock and selected spatial band.

Reproduction

spatial-source-saturation-checks.py and its recorded results retain all fourteen massive assignments, six occupied-depth controls, eighteen checked conditions and 314 evaluations. The script verifies exact low-degree monogenic identities, kernels from independent polynomial spatial modes, both-sign angular norms, two-point spectra and the existing geometric width coefficient. Gauss–Jacobi quadrature integrates the required finite polynomials; nineteen additional nodes provide an independent refinement. The largest refinement discrepancy is below 3.4×10⁻¹⁰ relative, and complete source Parseval errors are below 3.5×10⁻¹⁰ relative.

No particle simulation, amplitude fit, new adviser call or assumed source gain is used. The source sustaining dynamics, complete supported particle and selected occupied bands remain physical tasks for the common action.