Magnetic bivector master bridge

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.

Source action and scope

The projected-current note already considers a minimally gauged Dirac diagnostic action and a Maxwell kinetic coefficient. Those remain candidate terms of the projected theory, not independently derived electromagnetic charge or gauge identification. This note asks what they supply once present. Electron spatial depth remains six.

Use c=hbar=1, a positive magnetic normalization Z_B, and the convention pi_i=p_i−e A_i. Here e is the signed coupling defined by that convention. For the earlier D=partial+i q a convention with the same spatial covector potential, e=−q. The result is independent of this relabeling when expressed consistently; the induced-energy coefficient depends on e².

For F_ij=partial_i A_j−partial_j A_i, [pi_i,pi_j]=i e F_ij. Squaring the six-dimensional Dirac spatial term and projecting to the massive positive band gives, at leading inverse mass order,

$$H_+=m+\frac{\pi^2}{2m} -\frac{e}{2m}\sum_{i<j}\Sigma_{ij}F_{ij}+\cdots, \qquad \Sigma_{ij}=-iV^\dagger\alpha_i\alpha_jV.$$

Thus g=e/(2m) is the Pauli bivector coefficient in this convention. The magnetic part of −Z_B F_AB F^AB/4 supplies positive energy Z_B sum_(i<j) F_ij²/2. With <F,F>=sum_(i<j) F_ij², it corresponds to the auxiliary cost coefficient mu=Z_B/2. This identifies a possible origin for both coefficients, not just their rotational tensor structure. The nonrelativistic expansion, gauge-field normalization, and charged action are explicit premises.

The remainder of the calculation isolates the magnetostatic spin term. Orbital magnetic coupling in pi², electric energy, radiation, and the spatial kinetic/self-binding energy are not thereby removed from the complete problem. This is not a completed nuclear induced-current model.

The field constraint changes the response

A physical magnetic two-form must satisfy dF=0. In a flat Fourier patch with nonzero wavevector k, admissible F has k wedge F=0. Define

$$P_k=\frac{kk^T}{|k|^2},\qquad \mathcal P_k M=P_kM+MP_k.$$

This is the orthogonal projector in skew-matrix space onto the closed two-forms at that k. It has rank five and nullity ten. These are counts of field amplitudes at a wavevector, not a change to spatial dimension. Each projected matrix is a simple bivector of rank at most two. The zero mode and harmonic/flux sectors depend on boundary conditions and are not covered by this nonzero-k formula.

For a prescribed magnetization M(k), minimizing the spin-field energy gives

$$F_*(k)=\frac{g}{Z_B}\mathcal P_k M(k),\qquad E_{\min}(k)=-\frac{g^2}{2Z_B}\|\mathcal P_k M(k)\|^2.$$

For any admissible F, the difference from this minimum is exactly Z_B||F−F_*||²/2. For complex Fourier amplitudes use the Hermitian norm and real part of the pairing, or apply the same identity to independent real sine/cosine components. This replaces the unconstrained auxiliary rule F=(g/Z_B)M with a derived, generally nonlocal response kernel.

The isotropic angular average of the projector in six dimensions is I/3 on skew matrices. This follows directly from <kk^T/|k|²>=I6/6 and is checked with an exact orthogonal-axis average. The average induced energy is one third of the unrestricted matrix result, not the same local energy.

Fixed-envelope CP3 joint minimum family survives

For one normalized positive-band polarization z, define its physical Pauli magnetization by J_ij=z dagger Sigma_ij z. The explicit Dirac matrices give J squared=−I6 and ||J||²=3. This checks the complex-structure identity in the same physical Clifford basis as the Pauli coupling.

For every nonzero k,

$$\|\mathcal P_kJ\|^2=\frac{|Jk|^2}{|k|^2}=1.$$

Now fix a spatial density profile n(x) with finite squared integral, and restrict to a single polarization z that is uniform across that envelope. Then M(k)=n-tilde(k)J(z). Completing the constrained field square and using Parseval gives

$$E[F,z]=\frac{Z_B}{2}\int\|F(k)-F_*(k;z)\|^2\,dk -\frac{g^2}{2Z_B}\int|\widetilde n(k)|^2\,dk,$$

with a consistent Fourier normalization. The last term is independent of z. Every pure ray z therefore has its associated minimizing field, and the joint minima in this fixed-envelope, uniform-polarization sector are parameterized by CP3. They have positive restoring cost for departures of the admissible field from F_*(z). No assertion that the physical field itself satisfies F squared proportional to minus the identity is needed or valid mode by mode.

The electron kinetic phase term survives from the positive Dirac band. The result neither selects one ray nor drives precession by itself; nuclear or other orientation-dependent interactions remain relevant.

Boundaries needed for the next spatial calculation

The uniform-polarization restriction is substantive. Allowing z(x) to vary changes the nonlocal functional and may permit lower-energy textures. The subsequent frozen magnetic texture test now provides explicit lower-energy textures in predeclared stronger-coupling cases, while the inherited weak benchmark survives the limited family tested. The original uniform-sector square remains valid; it must not be extended to unrestricted spatial-polarization stability. The square above does not prove a global minimum over all spinor fields. Likewise n(x) was fixed: the magnetic attraction alone does not prove a stable localized profile, and the orbital/electric/kinetic terms must be included before testing localization or collapse.

This model concerns magnetization and its field on the assumed local six-dimensional space. It does not by itself identify polarization rays with the six spatial coordinates.

The new next step is well specified: evaluate the constrained kernel with the actual electron/nuclear profile and the remaining kinetic and current terms. Replacing its projector by the identity would discard precisely the field equation that this bridge has now supplied.

Verification

Explicit runtime conditions remain active under Python optimization. Normal and optimized runs completed 1,100 evaluations across 15 runtime conditions. Replacing the closed-field projector by the identity in memory was rejected at the Fourier Bianchi check in both modes. No mutated source was saved. The complete 46-script suite also passed after these additions.

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