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: At fixed electron depth six, an explicit filled-band response selects equal nonzero skew strengths and CP3 minimum components, with positive stiffness for normal variations in matrix space but unresolved source occupation and first-order dynamics.

Fixed-depth question

The particle depths supplied by the mass formula are fixed inputs to this investigation. In particular the electron has six spatial dimensions. The question here is whether a six-dimensional interaction selects the complex-structure geometry, not whether the electron should be reassigned to a different depth.

Allow an unconstrained real antisymmetric 6x6 variable B. Its three signed canonical plane strengths are lambda_1, lambda_2, lambda_3. We do not impose B squared proportional to minus the identity. Introducing B as a collective electron order parameter is itself a modeling premise; its microscopic definition remains to be derived. An arbitrary six-dimensional particle does not automatically have this order parameter as its configuration.

A specified matrix response rather than a prescribed CP3 penalty

Introduce two six-component vector response bands with gap Delta>0 and coupling g!=0. Their real symmetric one-particle operator is

h(B)=(0gB−gBΔI6). h(B)=\begin{pmatrix}0&gB\\-gB&\Delta I_6\end{pmatrix}.

Under B -> R B R^T it transforms by conjugation with diag(R,R); no rotation plane is distinguished. Twelve is the number of source states, not a particle depth. This is a new candidate self-response model, distinct from the earlier singlet-plus-nine nuclear-contact model. Neither its source carrier nor its occupation is claimed to follow already from the V2 master kernel.

For each plane, the source eigenvalues are

e±(y)=Δ±Δ2+4g2y2,y=λi2, e_\pm(y)=\frac{\Delta\pm\sqrt{\Delta^2+4g^2y}}2, \qquad y=\lambda_i^2,

each with multiplicity two. Assume all six lower-band states are occupied, and add a positive bare cost mu times the squared skew norm, with mu>0. Equivalently use the lower-band trace as an explicitly specified effective energy. The fixed six-occupation interpretation is a possible source model, not a claim that an electron has six constituent particles.

With <B,B>=−tr(B squared)/2=sum lambda_i², the total energy is exactly

E(B)=μ⟨B,B⟩+Tr⁡−h(B)=∑i=13f(λi2),f(y)=μy+Δ−Δ2+4g2y. E(B)=\mu\langle B,B\rangle+\operatorname{Tr}_{-}h(B) =\sum_{i=1}^3 f(\lambda_i^2),\qquad f(y)=\mu y+\Delta-\sqrt{\Delta^2+4g^2y}.

The source force in each plane is bounded at large strength, while the positive quadratic cost makes the total energy coercive. Thus the energy is not an unstable truncated expansion. Near B=0 its expansion is (mu−2g²/Delta) sum lambda_i²+(2g^4/Delta³) sum lambda_i^4 plus higher orders. The equalizing quartic sign follows from this source spectrum, not a fit.

Exact global minimum: the complex-structure constraint emerges

For y>=0,

f′(y)=μ−2g2Δ2+4g2y,f′′(y)=4g4(Δ2+4g2y)3/2>0. f'(y)=\mu-\frac{2g^2}{\sqrt{\Delta^2+4g^2y}},\qquad f''(y)=\frac{4g^4}{(\Delta^2+4g^2y)^{3/2}}>0.

Strict convexity makes equal squared strengths the unique minimizer at any fixed sum of squared strengths. Allowing that sum to vary gives:

  • If mu>=2g²/Delta, the minimum is B=0, not CP3.
  • If 0<mu<2g²/Delta, each squared strength has the same positive minimum

ρ2=4g4/μ2−Δ24g2. \rho^2=\frac{4g^4/\mu^2-\Delta^2}{4g^2}.

Consequently every global minimum obeys

B2=−ρ2I6,J=B/ρ,J2=−I6. B^2=-\rho^2 I_6,\qquad J=B/\rho,\qquad J^2=-I_6.

The full minimum set is O(6)/U(3). It has two connected components, distinguished by Pfaffian sign. Each component is SO(6)/U(3), hence CP3. The response energy does not select one component or a handedness. The orthogonal-complex constraint has been obtained by minimization rather than inserted into the definition of B.

For the dimensionless diagnostic mu=Delta=g=1, rho²=.75 and E_min=−.75. The stationary states with one or two nonzero plane strengths have energies −.25 and −.5; they do not minimize the full energy.

Full fluctuation calculation

Here "normal variations in matrix space" means changes away from the CP3 minimum family in the auxiliary matrix B, not extra spatial directions. A skew 6x6 matrix has 6(6−1)/2=15 independent rotation-plane components: six variations are tangent to the CP3 family and nine are normal to it in this parameter space. The electron's assigned spatial depth remains six.

The gradient in the skew pairing is

∇E=2μB−4g2B(Δ2I6−4g2B2)−1/2. \nabla E=2\mu B-4g^2B(\Delta^2 I_6-4g^2B^2)^{-1/2}.

At B=rho J, skew perturbations H split into six matrix-space variations satisfying HJ+JH=0 and nine satisfying HJ−JH=0. The first are tangent to the CP3 component. The Hessian is zero on them and is a positive scalar on the nine normal variations in matrix space:

k⊥=16g4ρ2(2g2/μ)3>0. k_\perp=\frac{16g^4\rho^2}{(2g^2/\mu)^3}>0.

This tests all fifteen skew coordinates, not just the three canonical strengths. The diagnostic has six zero eigenvalues and nine eigenvalues equal to 1.5. Flat orientation directions are expected along a family of minima; they do not imply attraction to a unique orientation.

What dynamics this energy can and cannot supply

A commutator-only equation dot B=[F(B,t),B] preserves all spectral invariants tr(B^n). It cannot turn unequal strengths into equal ones. For this invariant energy, [gradient E,B]=0 identically. Thus simply substituting E into the earlier coadjoint equation would give no amplitude-selection dynamics.

As an explicit additional kinetic assumption, introduce canonical momentum Pi for B and Hamiltonian <Pi,Pi>/(2M)+E(B). This allows strength fluctuations. Near the selected family the nine oscillation modes for normal variations in matrix space have frequency sqrt(k_perp/M). Direct evolution verifies their linearized oscillations and total-energy conservation. There is no damping and no claim of relaxation from arbitrary initial states. Low-energy confinement near the minimum manifold is distinct from dissipative capture into it.

Restricting the flat kinetic term to B=rho J gives an induced CP3 metric. For a normalized geometric ray Z and horizontal dZ, the existing Pauli map obeys <dJ,dJ>=8||dZ||². With the convention ds_FS²=dZ dagger dZ−|Z dagger dZ|², the kinetic term is therefore 4M rho² ds_FS²/dt². The metric is proportional to Fubini–Study under this specific kinetic assumption, rather than separately named by fiat.

This is still a collective-coordinate construction. If B is a field over an already chosen particle base, CP3 is its target/order-parameter space, not automatically that base.

The source does not generate the needed first-order term

The filled-band projector of h(B) is real symmetric. Its derivative is real symmetric, so the determinant-band Berry curvature i tr(P[dP,dP]) vanishes: a symmetric matrix has zero trace against a real antisymmetric commutator. The calculation checks this using independent projector differences in all skew directions.

Therefore this filled real source cannot by itself supply the nonzero first-order symplectic term used in the previous CP3 precession equation. The canonical kinetic example is an inertial model, not a derivation of that first-order equation. An existing electron action term or a suitable complex/chiral response would have to provide it. Geometry selection and the origin of the precession action must not be conflated.

Occupation is a substantive premise: counterexample

Keep the same h(B) but occupy only its lowest state. The source energy depends only on the largest squared plane strength. All other strengths cost positive mu energy without lowering that occupied level. When the coupling exceeds its corresponding threshold, this version prefers rank two B, not three equal nonzero strengths.

For mu=.5, Delta=g=1, the single occupied state gives one squared strength .75 and minimum energy −.125. Both the global bound and the cost of adding other planes are checked. This is not an alternative particle-depth assignment. It shows precisely why the filled-band premise must be derived: rotational symmetry and the displayed source operator alone do not select CP3.

Verification and next derivation

cp3_geometry_selection_checks.py checks the complete twelve-state spectrum, exact energy, global bounds, finite-difference gradient, full Hessian, pure-ray metric, zero filled-band Berry curvature, partial-rank stationary states, threshold behavior, single-occupation counterexample, and canonical dynamics of normal variations in matrix space. All tests use explicit runtime failures and remain active under Python -O. Normal and optimized runs each completed 631 evaluations across 24 runtime conditions. Reversing the gradient in memory was rejected by the independent energy-gradient check in both modes; no mutated source was saved.

The next source-level task is to derive or reject this response operator, its filled-band occupation, and an amplitude kinetic term from the V2 master interaction. Then determine the first-order electron term. The result already obtained is a conditional forward mechanism selecting CP3 components at fixed depth six, not a completed physical identification of the electron geometry.