Signed Scalar Current and Polarization in an Actual Spatial Band

Use the Written Scalar Interaction

The positive-source calculation explicitly uses Ψ†Ψ. The local scalar Dirac interaction instead uses J=Ψ†βΨ. These are different functions and require separate variations. On the same ordinary round-sphere spatial control, take

$$E[\Psi]=\langle\Psi,\mathscr D\Psi\rangle +\frac g2\int J^2\,dv,\qquad i\partial_t\Psi=\mathscr D\Psi+gJ\beta\Psi.$$

Here β is a parallel Hermitian grading, β²=I and {β,D}=0. At even spatial depth the usual intrinsic chirality supplies it. At odd depth the full graded double of the intrinsic spinor carrier is used; that is an explicit carrier choice. Its complex ranks at d=2,3,4,5,6,10 are 2,4,4,8,8,32. Spinor rank is not physical spatial dimensionality. This calculation uses both spatial Dirac signs and does not identify either sign with antiparticles or electric charge.

The sphere and its undetermined radius remain a conditional actual spatial control. No electron position probability interpretation or mediator is introduced.

The existing constant-current scalar-gap solution uses a mixture of both spatial Killing signs. The present calculation instead tests cumulative positive-band bright fields and their complete nonlinear sources. It does not inherit that exact solution or its stability.

The Grading Is Not Constant in the Killing Frame

Let t=cos θ, s=sin θ, γ(n)=n_aγ_a, with Hermitian spatial Clifford matrices. In the positive Killing frame, the shell kernel is

$$K_l=a_l(t)I-is b_l(t)\gamma(n),\qquad a_l=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad b_l=C_{l-1}^{(d+1)/2}(t).$$

Put b₀=0, A=Σ₀ᴺa_l, B=Σ₀ᴺb_l and S=binomial(N+d,d). The normalized bright field is ψ=(A−isBγ(n))u/√S for a unit pole polarization u, using normalized volume dν=dv/V.

The grading is parallel in the physical spin connection, but the Killing frame is a different connection. Along the radial geodesic its explicit frame is U=exp(−iθγ(n)/2), hence

$$\boxed{\beta_K(y)=U^\dagger\beta_0U =(tI+is\gamma(n))\beta_0.}$$

Here β₀ anticommutes with every γ_a. Keeping β₀ constant throughout this frame would calculate the wrong scalar source. The formula is Hermitian and squares to I. It also gives the exact shell map

$$\beta_K K_l=\widetilde K_l\beta_0,\qquad \widetilde a_l=t a_l-s^2b_l,\quad \widetilde b_l=-(a_l+t b_l).$$

These are precisely the negative spatial Dirac shells already retained in the source calculation. The map verifies the physical anticommutation with D, rather than replacing the scalar grading by an arbitrary matrix.

All Polarizations in the Bright Family

Define real polarization moments

$$q=u^\dagger\beta_0u,\qquad v_a=u^\dagger i\gamma_a\beta_0u.$$

They are coefficients of a scalar field, not electric charge or position. Direct matrix multiplication gives

$$\boxed{J(t,n)=F(t)q+s H(t)n\cdot v,}$$

$$F=\frac{t(A^2-s^2B^2)-2s^2AB}{S},\qquad H=\frac{A^2+2tAB-s^2B^2}{S}.$$

For complex A,B, replace their squares by absolute squares and AB by Re(A*B). The formula therefore retains distinct phases between spatial shells. Since the matrices (β₀,iγ₁β₀,…,iγ_dβ₀) form a Hermitian Clifford family, q²+|v|²≤1 for every unit pure polarization. To prove the bound, the expectation of a linear combination with a unit real coefficient vector is at most its operator norm, which is one.

Angular averaging now gives the entire polarization dependence of the quartic on this bright family:

$$\int J^2d\nu=L_0q^2+L_1|v|^2,\qquad L_0=\int F^2d\nu,\quad L_1=\frac1d\int s^2H^2d\nu.$$

Both coefficients are strictly positive in the tested bands. Exact symbolic expansion for arbitrary complex u proves q²+|v|²=1 at depths two and four. Consequently the minimum and maximum are min(L₀,L₁) and max(L₀,L₁). Polarizations that are eigenvectors of β₀ or of one iγ_aβ₀ attain those endpoints.

The rank-four identity is the same complete Clifford identity already used in spatial spinor compatibility. Its application here is to the parallel scalar grading and the actual cumulative Dirac bright family, rather than a new general identity.

At odd depths three and five, the even Clifford family has a chirality C that anticommutes with all d+1 moment matrices. A C-eigenvector has q=v=0. At even depths six and ten, an explicit pure vector also gives q=v=0: in the standard tensor-Pauli frame, take a balanced superposition of two computational basis states differing in three bits. β₀ is the product of the Z matrices, while iγ_aβ₀ connects only states differing in one bit. The β expectations cancel and the directional expectations vanish. Thus the minimum at these four depths is zero and the maximum is max(L₀,L₁).

The cancellation is not limited to the common-phase trial. With arbitrary complex A,B the same q=v=0 moments make J vanish. In particular, ordinary free evolution with phases exp[−i(l+d/2)t/R] preserves this zero-current zonal family. Finite explicit matrix checks verify distinct shell phases; the displayed current formula establishes the general statement. This extends the earlier six-dimensional zero-current Killing example to its zonal spectral superpositions. It supplies no self-interaction count enhancement on that family and no localized particle solution.

These are exact differences within a declared carrier and trial family. They do not single out depth five. Zero scalar current is not electric neutrality or absence of all same-field interaction. Nothing here selects which polarization an observed particle occupies. General spatial fields outside this bright family remain outside this classification.

The Complete Radial Scalar Source

Choose β₀u=u, so q=1 and v=0. The bright field has scalar current F and positive intensity h=(A²+s²B²)/S. Because β maps the positive and negative spatial spaces into one another,

$$\int F\,d\nu=0,\qquad L_0=\int F^2d\nu>0.$$

Thus a vanishing integrated signed current does not eliminate local self-interaction. The source in the field equation is Fβ_Kψ, not Fψ. Its radial polynomial components before division by √S are

$$A_s=F(tA-s^2B),\qquad B_s=-F(A+tB).$$

For g_l=binomial(l+d−1,d−1), the complete coefficients are

$$c_l=\frac{\int[a_lA_s+s^2b_lB_s]d\nu}{\sqrt{g_lS}}, \qquad \widetilde c_l= \frac{\int[\widetilde a_lA_s+s^2\widetilde b_lB_s]d\nu} {\sqrt{g_lS}}.$$

Positive shells through 3N+2 and negative shells through 3N+1 suffice; the highest allowed positive coefficient vanishes. Leading polynomial coefficients of A_s and B_s have opposite signs, putting that top degree in the negative family. Exact norm checks retain

$$\sum_l|c_l|^2+\sum_l|\widetilde c_l|^2 =\int F^2h\,d\nu,\qquad \langle\psi,F\beta_K\psi\rangle=L_0.$$

At N=0, F=t and h=1. The complete expansion is especially simple:

$$L_0=\frac1{d+1},\qquad c_0=\frac1{d+1},\qquad \widetilde c_1=\frac{\sqrt d}{d+1}.$$

All other coefficients vanish. The omitted squared source fraction is exactly d/(d+1). The ground positive-density comparison was closed, but this signed scalar-current bright trial is not. The separate mixed-sign constant-current scalar-gap branch remains a valid different solution.

Maintained Assignments Under One Scalar Law

The interaction coefficient, physical norm Q and sphere volume V remain shared within a sector. The trial's scalar nonlinear frequency component is gQ L₀/V, with a residual given by its complete source. It is not an exact nonlinear stationary frequency. Representative results are:

For the extreme assigned q=1 branches in each shared sector, the quartic ratio's signed relative drift from the simplex-count ratio is −0.00516% at depth two, −53.15% at three, −17.0734% at four, −2.55879% at five and −2.63249% at six. No within-sector comparison follows from the one maintained depth-ten assignment. These preserve the same normalization; individual amplitudes were not adjusted to repair the ratios.

The pointwise bound |J|≤h gives L₀≤∫h², but the nonnegative-source saturation theorem cannot be transferred directly to J, whose integral is zero here. The full scalar calculation, rather than an intensity substitution, establishes the displayed residuals. It does not favor the assigned depths or supply a mass operator or coupling rule by itself.

Reproduction and Scope

scalar-current-band-checks.py and its results retain all fourteen maintained massive assignments and all six occupied depths. The final run passes 796 evaluations across twenty-four conditions. It verifies the physical frame transformation, Clifford grading, opposite-sign shell map, complete pure-polarization identities at depths two/four, explicit zero-current states, distinct shell phases, quartic variation and complete source Parseval.

Gauss–Jacobi quadrature integrates the finite source polynomials. Twenty-three additional nodes verify refinement, with maximum relative discrepancy below 2.5×10⁻¹⁰. Complete Parseval differs by less than 1.6×10⁻¹⁰ relative. The initial absolute check for a vanishing highest coefficient failed at the top-quark cutoff: 2.91×10⁻⁷ was only 1.15×10⁻¹² of its complete source norm. The corrected criterion uses that norm; the source equations and coefficients did not change. The failed code and successful pilot are preserved with their exact execution hashes.

The signed scalar calculation does not establish positive complete Dirac energy, a stationary nonlinear particle, localization, lifetime or relativistic inertia. Its role is to test the actual written local source and identify precisely which bands and polarizations it does and does not preserve before attempting a self-consistent spatial solution.

The later local current-ratio construction changes the action and uses balanced opposite-sign shell multiplets. Its complete local source closes exactly despite an inhomogeneous density, but a general nonnegative-potential bound excludes the maintained count ratios on that connected shell family. It does not repair the current quartic action or transfer a full spatial stability result to the new law.