The polarization geometry at neighboring depths can be connected explicitly. At depth five, a spinor seed determines a unit axis and two equal-strength rotation planes perpendicular to it. With a chosen embedding into six ordinary spatial directions, that axis pairs with the additional direction to complete a three-plane rotation structure. The same construction joins depths three and four, with one perpendicular plane becoming two. Thus an axis in an odd number of directions and a full rotation structure in the neighboring even number need not be unrelated ingredients.

Here is the common calculation. Set D=d+1D=d+1, with (d,D)=(3,4)(d,D)=(3,4) or (5,6)(5,6). Take the even-dimensional Hermitian Clifford matrices and a chiral seed χ\chi of squared norm s>0s>0. On that chirality define

Γi=−iγiγD,vi=χ†Γiχ,Bij=−iχ†ΓiΓjχ(i≠j),Bii=0.(1) \Gamma_i=-i\gamma_i\gamma_D, \qquad v_i=\chi^\dagger\Gamma_i\chi, \qquad B_{ij}=-i\chi^\dagger\Gamma_i\Gamma_j\chi \quad(i\ne j),\qquad B_{ii}=0. \tag{1}

The Γi\Gamma_i obey the Clifford relation in dd dimensions. The same seed's even-dimensional polarization matrix is

Beven=(Bv−vT0),Beven2=−s2ID.(2) B_{\rm even}=\begin{pmatrix}B&v\\-v^T&0\end{pmatrix}, \qquad B_{\rm even}^2=-s^2I_D. \tag{2}

The square identity holds for every chiral seed at four and six. Multiplying the blocks gives

∣v∣2=s2,Bv=0,B2=vvT−s2Id.(3) |v|^2=s^2,\qquad Bv=0,\qquad B^2=vv^T-s^2I_d. \tag{3}

In plain terms, v/sv/s is the axis. The matrix B/sB/s vanishes along it and acts as a right-angle rotation on its perpendicular directions. The extra direction completes the missing plane. Overall field amplitude cancels from these normalized geometric quantities. The five/six comparison uses the same four complex spinor coefficients; the three/four comparison uses two. These coefficients describe the field on ordinary space, not another physical space.

This relates the polarization structures used by the spatial-current and normal-geometry constructions. It does not require different particles to share one seed or derive their relative embedding. It is shared by depths three and five, so it does not uniquely explain neutrino neutrality. At ten, the corresponding full rotation structure requires the additional pure-seed restriction; chirality alone does not suffice. The result adds a concrete compatibility relation, not a new independent charge or mass prediction.

Derivation and Review Provenance

This maintained account presents Astra's adjacent-depth block identity from connection-1-spinor-axis-and-plane.md, SHA 2ef65aac0a0cbb2ce9e50946ca199c70828eed3ab3e1996fa2a10e4bb293a278, preserved with its verification addendum and full evidence at database snapshot 4088 with its original exact source paths and hashes. The exact source pins identify the maintained spatial-current, normal-structure and normal-charge inputs at snapshot 4086. The root-free charge construction and the new curvature source keep their separate premises.

The block multiplication follows directly from the Clifford definitions. Every chiral seed at four and six is pure, giving the square identity; no generic ten-dimensional purity claim is used. The author checked the identities symbolically for arbitrary spinor coefficients at the two odd depths, within a 27-condition packet. A separate 35-condition checker concerns the optional spatial-derivative vector and is not evidence for a solved field equation.

Claude's scoped review agents/claude/astra-axis-plane-review-20261104/review-1.md, SHA d2affdd92c78c58cab9f449035b77f298838f15b8d96fcaf923c444e1c5b036a, independently checks the block identity and derivative-vector algebra with 427 numerical controls and hand analysis. Its sampled tests do not replace the general block proof. The original review did not cover the later full-profile cross-product expression. That extension now has a separate independent analytical assessment, presented below. Claude originated the literature lead and an erroneous identical-spinor vector claim; his agreement with that correction is clarification, whereas his check of the new block identity is a separate mathematical pass.

The full packet also explains why an alternating vector map vanishes on two identical commuting chiral spinors at depth six, and how pairing a field with its spatial derivative supplies a conditional plane description. Those results remain retrievable in the research database. They are not needed for the adjacent-depth identity here and do not establish physical response.

For the existing spatial-current and normal geometry, see Spinor Polarization and Spatial Current and Canonical Normal Structure. This is an explicit synthesis of their geometry, not independent observational evidence for the assigned depths.

A Derivative Vector on the Actual Radial Profile

The subsequent calculation uses the negative-chirality component ψ−=iFγ(x)χ/a\psi_-=iF\gamma(x)\chi/a, a supplied real direction XX, and the antisymmetric matrix CγiC\gamma_i. Its derivative bilinear is

ψ−TCγi∂Xψ−=−F2a2(γ(x)χ)TCγiγ(X)χ. \psi_-^T C\gamma_i\partial_X\psi_- =-\frac{F^2}{a^2}(\gamma(x)\chi)^T C\gamma_i\gamma(X)\chi.

The envelope derivative cancels because an antisymmetric bilinear vanishes on two copies of the same commuting spinor. For the resulting vector QQ with that scalar prefactor removed, a unit pure seed gives

Q†Q=2[∣x∣2∣X∣2−(x⋅X)2−(Jx⋅X)2],Q⋅Q=0. Q^\dagger Q=2\bigl[|x|^2|X|^2-(x\cdot X)^2-(Jx\cdot X)^2\bigr], \qquad Q\cdot Q=0.

For nonzero xx, its exact zero directions are the real plane spanned by x,Jxx,Jx. It is orthogonal to x,Jx,X,JXx,Jx,X,JX; reversing the chirality convention reverses its complex rotation sign. The unnormalized bilinear is smooth at the origin, where it vanishes. There is no need to divide by a component that becomes zero. A common field phase doubles in this bilinear; that does not introduce another physical oscillator.

This supplies a concrete direction-sensitive diagnostic on the existing radial profile. Its probe direction and absolute amplitude are not selected, and a reciprocal interaction energy remains to be derived. The result is independent of a physical weak or color-force interpretation.

The later independent analytical assessment, SHA-256 b88f326f0c03055838f739793bfb783081b67929552edb6c65b8dfbd6f619475, checks the supplied actual-profile extension separately from the original adjacent-depth block identity. Its derivation uses the pure-seed rotation to a Fock vacuum and a two-coordinate transverse check, followed by covariance; the author's matrix outputs were inspected without numerical replay.