Minimal Spatial Carriers and Their Count-Compatible Rotation Forms

The spatial fields used in the forward count derivation do not require a full square matrix of balanced columns at every depth. A weaker, precisely specified seed condition suffices. Its minimum column ranks at occupied depths two, three, four, five, six and ten are respectively

$$\boxed{2,\;2,\;4,\;1,\;1,\;1.} \tag{1}$$

The reduced fields retain the full spatial equation, complete energy and count clock. The same fields have dimension-dependent local spin-density forms. In the pure balanced class, depth five has two equal planes and a kernel axis, whereas depth six has three equal planes. These are exact conditional geometric results, without a coefficient added to favor five or six. A form representing local rotation generators is not by itself a literal moving trajectory or the particle's observed total spin.

1. The Geometric and Field Assumptions

Use the conditional ordinary round spatial sphere of dimension d and radius R, the canonical Dirac kinetic term, and the specified effective count interaction in the forward derivation. Its field describes internal classical structure; the electron's deterministic six-dimensional position trajectory remains a distinct variable.

Let alpha_i and beta be Hermitian Clifford matrices, with beta parallel, beta squared equal to identity and beta anticommuting with every alpha_i. Introduce the convenient Clifford generators

$$\Gamma_0=\beta,\qquad \Gamma_i=-i\alpha_i\beta, \qquad \{\Gamma_A,\Gamma_B\}=2\delta_{AB}I. \tag{2}$$

There are d+1 algebraic generators because the grading beta is included. This adds no physical spatial direction or occupied sector. A seed C is an r by k classical matrix, with tr(C dagger C)=1. Its column Gram matrix is a field-component construction, not a probability distribution. Require

$$\operatorname{tr}(C^\dagger\Gamma_A C)=0,\qquad \operatorname{tr}(C^\dagger i\Gamma_A\Gamma_B C)=0 \quad(A<B). \tag{3}$$

This condition is sufficient and imposed here. A proof that every possible particle field needs it is not claimed. The count k is the number of columns, not physical depth, number of particles or number of colors.

2. Why the Actual Spatial Fields Preserve This Balance

For shell level l, put h_l=l+d/2 and g_l=binomial(l+d-1,d-1). The actual positive Dirac kernel has the local form

$$E_l=\frac{M_lKC}{\sqrt{g_l}},\qquad M_l=A_l(t)I-iB_l(t)\alpha(v),\quad |v|^2=1-t^2,$$ $$A_l=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad B_l=C_{l-1}^{(d+1)/2}(t), \tag{4}$$

with B_0=0. K is the Killing-spinor frame, satisfying covariant derivative_i K=i alpha_i K/2 in unit-radius conventions. Its values belong to the Spin(d+1) representation generated by (2).

The essential simplification is

$$M_l^\dagger M_l=p_lI,\quad p_l=A_l^2+B_l^2(1-t^2)>0,\qquad \frac{M_l}{\sqrt{p_l}} =\frac{A_l+B_l\Gamma(v)\Gamma_0}{\sqrt{p_l}} \in\operatorname{Spin}(d+1). \tag{5}$$

Spin conjugation preserves vector and bivector degree. Therefore (3) survives both K and M_l. In particular,

$$E_l^\dagger\beta E_l\text{ has zero trace},\qquad \operatorname{tr}E_l^\dagger\alpha_iE_l=0,\qquad \rho_l=\operatorname{tr}E_l^\dagger E_l=p_l/g_l. \tag{6}$$

The scalar polynomials obey A_l'=(l+d-1)B_l and dt B_l-(1-t squared)B_l'=l A_l. Using these identities and the actual covariant jets in (4) gives D_R E_l=(h_l/R)E_l. Multiplying on the right by fewer constant columns does not change that equation.

Now use the original massive-shell map

$$\Phi=(a+b\beta)E_l e^{-i\omega_l t},\qquad a^2=(1+c_l)/2,\quad b^2=(1-c_l)/2, \quad x_l=\sqrt{1-c_l^2}.$$

The identities (a+b beta) squared=I+x_l beta and (a+b beta)alpha_i(a+b beta)=c_l alpha_i imply

$$\rho_\Phi=\rho_l,\quad s_\Phi=x_l\rho_l,\quad j_\Phi^i=0. \tag{7}$$

These are exactly the invariants entering the full count source. The forward stationary equations remain

$$\chi=h_l,\qquad c_l=\epsilon_dP_d'(h_l),\qquad \boxed{\omega_l=\mu_d\binom{l+d}{d},\quad E=\omega_l Q.} \tag{8}$$

No particle-dependent amplitude is fitted. Equation (8) is derived from the specified constitutive count law, whose independent master-field origin remains open.

The preferred-time convex envelope also retains its nonlinear spatial persistence proof on these reduced fields. That proof needs positive background density, an open winner chart and a self-adjoint operator acting on each column; it does not need square-matrix balance. The same p_l/g_l has a positive minimum on the sphere. Its graph-norm and Sobolev argument therefore applies unchanged to a fixed reduced number of columns. This does not transfer the different responding-background Hessian theorem or establish localization in an unbounded containing sector.

3. The Minimum Rank Is an Exact Algebraic Result

Write H=CC dagger, positive semidefinite with unit trace. At depth two the three vector generators span all traceless Hermitian two-by-two matrices, so (3) forces H=I_2/2 and rank two. At depth four the five vector and ten bivector generators span all traceless Hermitian four-by-four matrices, forcing H=I_4/4 and rank four.

At depth three the ten constrained matrices leave five complementary Hermitian matrices Sigma_a. They mutually anticommute and square to identity. Consequently every allowed H has the form

$$H=\frac14(I+v_a\Sigma_a),\qquad |v|\le1. \tag{9}$$

Its eigenvalues are (1+|v|)/4 and (1-|v|)/4, each repeated twice. The least possible nonzero rank is two, attained at |v|=1. A convenient example is H=(I+Gamma_)/4 with Gamma_=Gamma_0 Gamma_1 Gamma_2 Gamma_3.

At depths five, six and ten, the standard Jordan-Wigner Clifford basis admits the explicit one-column seed

$$C=(e_0+e_{r-1})/\sqrt2, \qquad r=8,8,32\text{ respectively}. \tag{10}$$

Every vector and bivector expectation in (3) vanishes exactly. One is the smallest nonzero rank. Equations (9) and (10), together with the complete low-degree matrix bases, prove (1). This is minimum rank under (3), rather than a classification of all mass-supporting field ansatzes.

4. Higher Moments Become Local Rotation Forms

The vector and bivector seed moments vanish, but the three-form need not:

$$\varphi_{ABC}=\operatorname{tr}C^\dagger i\Gamma_A\Gamma_B\Gamma_C C. \tag{11}$$

Let T=M_lK/sqrt(p_l) and define the orthonormal Clifford vectors X=T dagger Gamma_0 T and U_i=T dagger Gamma_i T. Here X is a unit vector in the algebraic Clifford vector space, with the U_i perpendicular to it. The massive map in (7) gives the exact local identity

$$\boxed{\mathcal J_{ij} =\frac{\operatorname{tr}\Phi^\dagger i\alpha_i\alpha_j\Phi} {x_l\rho_\Phi}=\varphi(X,U_i,U_j).} \tag{12}$$

The normalized skew form is the contraction A_X=iota_X varphi restricted to X perpendicular. Its coefficients are local spin-generator densities. Their plane strengths follow from the eigenvalues of -J squared. They are independent of the selected count and overall field amplitude.

Depth Five: Two Equal Planes and a Local Axis

For (10), the six-generator three-form is the real complex-volume form of an orthogonal complex structure J. In an explicit basis it has four unit coefficients. Exact tensor multiplication gives, for every vector X,

$$\boxed{A_X^2=XX^T+(JX)(JX)^T-|X|^2I,\quad J^2=-I.} \tag{13}$$

For unit X, both X and JX are in its kernel. Restricting to the physical five-dimensional tangent space X perpendicular leaves one kernel axis JX and two orthogonal equal-strength rotation planes. Thus the squared singular values are (0,1,1,1,1).

Depth Six: Three Equal Planes

The seven-generator seed three-form has seven unit coefficients and is the standard positive G2 form. Its exact contraction identity is

$$\boxed{A_X^2=XX^T-|X|^2I.} \tag{14}$$

For unit X, the restriction to X perpendicular squares to minus identity. All six squared singular values are one: three equal orthogonal planes, without an inherent split of the six physical dimensions into two groups.

These identifications use standard spinor geometry; see Agricola, Chiossi, Friedrich and Höll, Section 2. Equations (13) and (14) are also checked here directly in the actual Clifford convention. They do not assume a new force associated with SU3 or G2, or identify a stabilizer with observed color.

Why These Five/Six Results Do Not Depend on One Lucky Pure Seed

The eight-component Clifford modules at five and six admit an antiunitary involution B conjugation that anticommutes with all Gamma_A and commutes with real spin rotations. Its fixed vectors form a real eight-dimensional space. After one phase choice, seed (10) belongs to that space.

The exact real infinitesimal stabilizer dimensions are eight in Spin(6) and fourteen in Spin(7), respectively. In each case the orbit dimension is seven. The compact orbit on the unit real seven-sphere is both open and closed, so it is the whole sphere. Its tangent rotation vectors span the orthogonal complement of any nonzero real spinor u.

Write any complex pure seed as u+i v in that real structure. Vanishing bivector expectations requires u transposed A v=0 for every real skew spin generator A. Since A u spans u perpendicular, v must be proportional to u. The seed is therefore real up to an overall phase. Spin covariance now proves that every pure seed obeying (3) has the same identities (13) or (14), respectively. The pure balanced five/six results are not a choice among otherwise inequivalent seeds. This argument does not classify the depth-ten pure balanced class.

Depth Ten: One Full-Strength Plane and Four Variable Planes

For the explicit seed (10), the eleven-generator form is

$$\varphi=\eta\wedge J_5,\qquad J_5^2=-I+\eta\eta^T,\quad |\eta|=1. \tag{15}$$

Its own metric contraction is delta_AB+4 eta_A eta_B, so it determines the distinguished axis eta without an independently inserted direction. For unit X put t=eta dot X. The two-plane in span(eta,X,J_5 X) perpendicular to X has unit strength. On its eight-dimensional orthogonal complement the contraction is t J_5. Hence its ten-dimensional squared singular values are

$$\boxed{1,1,\;t^2,t^2,t^2,t^2,t^2,t^2,t^2,t^2.} \tag{16}$$

At |t|=1 all five strengths agree; at t=0 only one plane remains. This distinguishes the explicit ten-dimensional field from the universal equal-plane pure balanced six-dimensional class. The algebraic possibility of rank loss is not a physical decay channel or a demonstrated instability.

Lower-Depth Controls

The required full rank depth-two seed has a nonzero central top form and one unit plane in (12). The canonical depth-three chirality seed in (9) has zero three-form, as does the forced full rank depth-four seed. Other allowed depth-three minimum-rank seeds can have a different three-form; zero is a statement about that displayed seed, not the entire class. None of these local forms establishes observed boson or fermion spin.

5. Verification and Scope

A direct proper-current spatial comparison now keeps the affine chart current Hessian on both carriers. The stated full-matrix ground controls grow at every occupied depth in an exact reducing family; the minimum-carrier finite test retains growth at two, four, five and six, while its depth-ten space fails closure. These are ground controls, not the maintained excited assignments or a change to the preferred-time nonlinear theorem.

The full observer-rotation calculation now retains both orbital transport and the spin connection. The actual count fields at matter depths three, four, five, six and ten have observer Casimir 3/4. Their balanced preparations have zero mean canonical rotation charge; the transformation result does not identify that charge with a measured particle spin.

minimal-count-carrier.py and its report pass 742 evaluations. They check exact seed moments, minimum-rank matrix bases, actual spatial covariant jets, complete nonlinear count sources, and local form spectra at all fourteen maintained assignments. Spatial Dirac residuals are below 3.6e-15 relatively; full field residuals scaled by the large source coefficient are below 3.5e-12. The independent high-precision clock proof remains the one in the forward action note.

count-carrier-rotation-identities.py and its report verify the three-form coefficients, real structures, stabilizer dimensions and the complete tensor polynomial identities with exact arithmetic. The identities hold for every normal, including directions not sampled in the field checker. An intermediate implementation compared unexpanded complex phase products structurally; the corrected comparison simplifies their exact difference. Its source and failure record remain preserved.

This supplies a common geometric consequence of the count-supporting fields: a sufficient balance condition both closes their mass equation and constrains their local spin-density planes. It strengthens conditional compatibility of the proposed five/six spatial patterns with an explicit effective action. It does not independently infer the occupied depths. Observed total angular momentum includes orbital rotation; spin-one-half, literal orbit motion, fixed encounter-axis dynamics, physical interaction rates and localized support require further derivation. The count constitutive law and sector scales remain stated inputs to this downstream realization of the mass-inferred assignments.