Projective rays here describe field coefficients. Particle positions remain ordinary spatial coordinates in the infinite-dimensional containing space.

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.

Summary: A normalized six-dimensional chiral spinor coupled to an unconstrained bivector selects a CP3 minimum family with a Berry term and nonlinear energy confinement, conditional on its master-field derivation.

Fixed-depth route using chiral polarization

Electron depth six is fixed by the mass-formula assignment in this model. Instead of requiring the filled twelve-state response band of the preceding candidate model, consider one normalized chiral Spin(6) polarization z in C4 and an unconstrained real skew 6x6 bivector B. A six-dimensional Dirac spinor has chiral halves of this size; selecting one half and representing the relevant polarization by a pure normalized vector are additional premises, not consequences of dimension alone.

The Pauli bilinear map already checked in the geometric-response note gives J(z)=J(zz dagger), with J squared=−I and <J,J>=3. It is linear in the density matrix. We do not impose B squared=−rho² I. The purpose of the coupling is to determine whether that condition follows for B.

The chiral spinor here is a polarization variable. It must not silently be identified with a global spatial Dirac field on an already assumed CP3 base. The earlier spinor-bundle obstruction remains in force.

Linear bivector coupling and exact energy minimum

Let mu>0, g>0, and <A,B>=−tr(AB)/2. Define the Hermitian chiral operator h(B) by the expectation identity

z†h(B)z=−g⟨B,J(z)⟩. z^\dagger h(B)z=-g\langle B,J(z)\rangle.

It is the bivector representation on the chosen chiral spinor. In the explicit traceless two-Pauli basis P_A, its checked matrix is h(B)=−(g/4) sum_A <B,J(P_A)> P_A. The total joint energy is

E(B,z)=μ⟨B,B⟩−g⟨B,J(z)⟩=μ∥B−ρJ(z)∥2−3g24μ,ρ=g2μ. E(B,z)=\mu\langle B,B\rangle-g\langle B,J(z)\rangle =\mu\|B-\rho J(z)\|^2-\frac{3g^2}{4\mu},\qquad \rho=\frac{g}{2\mu}.

This is an exact completed square for every B and normalized z. Therefore the global minima have B=rho J(z), and B squared=−rho² I follows. The family of minima modulo the common spinor phase is CP3. The positive quadratic cost and the bilinear chiral coupling are the effective interaction assumptions; their presence and normalization remain to be derived from the master theory.

For a chosen chirality and the stated signs, the map selects one component of the orthogonal-complex structures. If both chiralities are admitted symmetrically, the opposite component returns with the same energy. Thus the construction does not derive a physical chirality selection.

At a minimum the source operator has the especially simple form

h(ρJ(z))=−gρ(4zz†−I4). h(\rho J(z))=-g\rho(4zz^\dagger-I_4).

Its occupied ray has eigenvalue −3g rho and the other three eigenvalues are +g rho. The gap is 4g rho=2g²/mu. Unlike the preceding real-band model, there is no separately assumed twelve-state filling or externally supplied gap Delta; the gap follows from the self-consistent bivector amplitude.

Normal variations in matrix space and conservative confinement

The auxiliary skew 6x6 matrix B has 15 independent rotation-plane components. Six parameter variations stay tangent to the CP3 minimum family; the other nine are "normal variations in matrix space," meaning deviations away from that family. These are not nine additional spatial directions. The electron's assigned spatial depth remains six.

Eliminating the spinor into the lowest eigenstate of h(B) gives E_eff(B)=mu||B||²+lambda_min(h(B)). Near a selected minimum the source ground ray is nondegenerate. The full fifteen-parameter matrix-space Hessian has six zero tangent eigenvalues and nine positive eigenvalues 2mu. These are checked through finite differences of the Hellmann–Feynman gradient 2mu B−gJ(z_ground(B)).

An explicit joint action is

L=iℏz†z˙+M2⟨B˙,B˙⟩−E(B,z),z†z=1, L=i\hbar z^\dagger\dot z+\frac{M}{2}\langle\dot B,\dot B\rangle-E(B,z), \qquad z^\dagger z=1,

with the phase multiplier understood. It gives the reciprocal equations

MB¨=−2μB+gJ(z),iℏz˙=h(B)z. M\ddot B=-2\mu B+gJ(z),\qquad i\hbar\dot z=h(B)z.

Both spinor norm and the total energy are conserved. The exact square gives

μ∥B−ρJ(z)∥2+∥Π∥22M=Etotal−Emin⁡. \mu\|B-\rho J(z)\|^2+\frac{\|\Pi\|^2}{2M} =E_{\rm total}-E_{\min}.

This controls distance to the entire minimum family and the momentum for small excess energy, not merely an eigenvalue test at one point. Direct coupled evolution verifies conservation and the bound. It is conservative stability of the family, not attraction to one orientation, and not a proof of stability for an electron orbit after adding the nuclear interaction.

Phase dynamics now survive

For the rank-one projector P=zz dagger and horizontal variations u,v,

itr⁡(P[δuP,δvP])=−2Im⁡(u†v). i\operatorname{tr}\bigl(P[\delta_uP,\delta_vP]\bigr) =-2\operatorname{Im}(u^\dagger v).

This is nonzero in general and is checked against an explicit canonical pair. It supplies the projective Berry/symplectic term missing from the filled real-band candidate. Direct differentiation of J under the spinor equation gives

J˙=−2gℏ[B,J], \dot J=-\frac{2g}{\hbar}[B,J],

matching the existing geometric commutator convention with I_e=hbar/2. At an isolated minimum B=rho J this commutator vanishes, as it should: the self-coupling creates the geometry and gap, not a preferred moving orientation. Nuclear orientation energy is still needed to drive precession.

Reduction to the earlier CP3 dynamics

In the algebraic M=0 limit, B=rho J(z) exactly. The self-energy becomes a constant, leaving the Berry term and any added nuclear energy V_N(J). The latter then produces the previously checked first-order CP3 precession. Thus a consistent effective route to the earlier action exists within this model; a real-band source with zero Berry curvature could not do this.

For small nonzero M, substituting the leading B=rho J into the kinetic term gives 4M rho² ds_FS²/dt², using <dJ,dJ>=8 ds_FS². This is the leading derivative expansion, not an exact constraint at finite frequency. Frequency scales must be small relative to the eliminated amplitude dynamics before discarding its corrections. Source and kinetic parameters are not fitted to an observed electron frequency here.

What has and has not been selected

The construction supplies a CP3 collective minimum family, positive stiffness for normal variations in matrix space, a self-generated polarization gap, and the projective phase term, without assuming the defining constraint on B or a filled real source band. It does assume a pure chiral polarization, a quadratic bivector cost, and their coupling. The pure chiral ray already carries CP3 polarization geometry; the nontrivial selection result is that an initially unrestricted B locks to it and acquires controlled normal variations in matrix space. This is not a derivation of CP3 without any spinorial premises.

Most importantly, a CP3 polarization/collective space is not automatically the electron's spatial base.

The normalization used here is for a finite collective spinor. If z is instead a local field with variable density, global charge conservation does not fix its pointwise norm. Eliminating B then produces an attractive density-quartic contribution, and density stabilization must be supplied and checked separately. The finite normalized calculation is not a proof of spatial localization or absence of field collapse.

The remaining physical identification concerns z and B, the chiral sector and its coupling, and the configuration reduction—not another particle-depth comparison. No geometry is ruled out merely because it differs from the V1 assignments.

There is a useful algebraic route to examine at that next step. For an unnormalized pure chiral amplitude w, let n=w dagger w. The bilinear identity becomes ||J(ww dagger)||²=3n². Algebraically eliminating B therefore gives the local term −3g² n²/(4mu). An attractive density quartic can thus admit this auxiliary-bivector representation at the classical commuting-amplitude level. This observation does not supply an independent B kinetic term or establish an operator identity for anticommuting quantum fields.

The written master-current construction uses bar(Psi)Psi, not automatically w dagger w for a single chiral amplitude. Their projection and normalization must be evaluated before claiming this auxiliary channel is already derived from the master scalar interaction.

Verification

chiral_geometry_selection_checks.py checks the chiral operator, bilinear constraint, completed square, minimum spectrum, gap, full effective Hessian, induced metric, Berry curvature, commutator normalization, opposite-chirality degeneracy, and reciprocal spinor/bivector evolution with the nonlinear confinement bound. Tests use explicit runtime failures, including with Python optimization enabled. Normal and optimized runs each completed 788 evaluations across 20 runtime conditions after adding the unnormalized-density checks. Reversing h(B) in memory was rejected by the independent bivector-pairing condition in both modes. The full 44-script diagnostic suite also completed successfully; the final density additions and cached constant matrices were separately rechecked in normal and optimized execution. These checks establish the stated effective-model calculations, not the missing physical identification.