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: Positive-energy projection of a massive six-spatial-dimensional Dirac scalar current supplies the four-component polarization, Berry kinetic term, and a calculable momentum-transfer bivector source.

Source and fixed-depth reduction

Electron depth six remains fixed. This note calculates which of its ingredients come from the Dirac kinetic term and scalar current already written in the self-contact module and binding module.

Work in a local orthonormal spatial frame, with one Hamiltonian time and six spatial momenta. In units c=hbar=1, take the massive Dirac symbol H0(p)=alpha dot p+m beta, with m>0 fixed by the electron mass assignment, Hermitian alpha_i and beta, {alpha_i,alpha_j}=2 delta_ij, and {alpha_i,beta}=0. This is the explicit conventional Hamiltonian realization of the written Dirac-type kinetic term.

Positive energy gives the needed four-component polarization

The minimal matrices are 8x8. The beta=+1 rest eigenspace has dimension four; write V for an isometric basis of it. Spatial rotation generators commute with beta, so this rest space carries one chiral Spin(6) representation. This is spatial chirality, not an assumed Lorentz chirality in seven space-time dimensions. At nonzero momentum the energy band is not a pure beta eigenspace.

For E_p=sqrt(m²+|p|²), the exact positive-energy frame is

U(p)=Ep+m2Ep(I8+α⋅pEp+m)V. U(p)=\sqrt{\frac{E_p+m}{2E_p}} \left(I_8+\frac{\alpha\cdot p}{E_p+m}\right)V.

Direct checks give U dagger U=I4, H0 U=E_p U, and

U(p)†βU(p)=mEpI4. U(p)^\dagger\beta U(p)=\frac{m}{E_p}I_4.

Thus, for a fixed-momentum projected amplitude Psi=U(p)w, the written scalar current bar(Psi)Psi equals (m/E_p) w dagger w. It equals positive density at rest. The current normalization is calculated, not replaced by notation. This identifies the four-component polarization used in the effective construction as an available positive-energy Dirac degree of freedom. It does not identify its projective ray with spatial position.

For a fixed normalized mode and fixed frame U, the kinetic action reduces exactly to i hbar w dagger dot w. At fixed norm it supplies the projective Berry term. If U depends on time, momentum, or position, its connection i hbar w dagger U dagger dot U w must also be retained. This is not a proof that the field's spatial envelope or polarization remains separable.

Nonforward scalar current contains a bivector

In an interaction, incoming and outgoing momenta need not agree. The full current matrix element is

C(p,p′)=U(p)†βU(p′)=NpNp′[I4−V†(α⋅p)(α⋅p′)V(Ep+m)(Ep′+m)],Np=Ep+m2Ep. C(p,p')=U(p)^\dagger\beta U(p') =N_pN_{p'}\left[I_4- \frac{V^\dagger(\alpha\cdot p)(\alpha\cdot p')V} {(E_p+m)(E_{p'}+m)}\right], \quad N_p=\sqrt{\frac{E_p+m}{2E_p}}.

The Clifford product separates exactly:

V†(α⋅p)(α⋅p′)V=(p⋅p′)I4+∑i<j(pipj′−pjpi′)V†αiαjV. V^\dagger(\alpha\cdot p)(\alpha\cdot p')V =(p\cdot p')I_4+ \sum_{i<j}(p_ip'_j-p_jp'_i)V^\dagger\alpha_i\alpha_jV.

The second term is a genuine spin-bivector operator with a fixed coefficient. It vanishes in the forward element p=p', explaining why the earlier fixed-momentum/zeroth-gradient scalar audit did not see it. Reciprocity C(p,p') dagger=C(p',p) makes the interaction Hermitian when the scalar source has the corresponding reciprocal kernel.

For a real scalar mean field S(x), the positive-band interaction kernel is S-tilde(p−p') C(p,p'). In the self-consistent scalar equation S is the coefficient multiplying the Dirac mass channel, such as kappa J; for a nuclear contribution it is the projected cross-depth scalar source. This is a projection of the scalar vertex, not an inserted magnetic force.

Leading orientation operator

At |p|,|p'| much smaller than m,

C(p,p′)=I4−∣p∣2+∣p′∣28m2I4−V†(α⋅p)(α⋅p′)V4m2+O(p4/m4). C(p,p')=I_4-\frac{|p|^2+|p'|^2}{8m^2}I_4 -\frac{V^\dagger(\alpha\cdot p)(\alpha\cdot p')V}{4m^2} +O(p^4/m^4).

For Hermitian Sigma_ij=−i V dagger alpha_i alpha_j V, its bivector part becomes the position-space operator

Hbiv=−14m2∑i<jΣij[(∂iS)pj−(∂jS)pi],pj=−i∂j. H_{\rm biv}=-\frac1{4m^2}\sum_{i<j}\Sigma_{ij} \left[(\partial_iS)p_j-(\partial_jS)p_i\right], \qquad p_j=-i\partial_j.

The sign here is for a scalar mass-channel potential, not a time-component vector potential. Antisymmetry cancels the derivative-ordering correction from the symmetric second derivatives of S. The accompanying scalar momentum corrections remain present; the bivector term is not the entire effective Hamiltonian. Physical units must be restored before a rate fit.

At the level of a local position/momentum symbol the source is the simple bivector grad(S) wedge p. Its real antisymmetric rank is at most two, not three independently adjustable plane strengths. If S depends only on the nuclear O three-plane, its mixed O/H block is proportional to grad_O(S) p_H^T, of rank at most one. This single contribution does not already supply the full equal-strength self-response, a locked frame, or the demo's prescribed frequency pattern. Contributions at different positions/momenta can combine; their self-consistent source still needs evaluation.

What is now connected

The Dirac reduction supplies an explicit route to the four-component polarization and its kinetic phase term. The nonforward scalar vertex supplies a calculable linear bivector source. These are concrete pieces of the master-to-effective connection. The positive quadratic B response cost, its amplitude kinetics, the full nuclear solution, and the mapping from polarization/collective coordinates to spatial configuration have not been obtained by this local projection.

The new calculation determines which of its inputs are present in the specified Dirac projection and which remain to be supplied.

The subsequent magnetic bivector bridge supplies a Pauli coupling and quadratic cost from the minimally gauged Dirac–Maxwell candidate, retaining the magnetic field constraint.

Verification

master_chiral_bridge_checks.py checks explicit Clifford matrices, rest rotation invariance, positive-band eigenvectors, current normalization, kinetic normalization, the exact nonforward current, reciprocity, scalar/bivector decomposition, its rank, and fourth-order convergence of the low-momentum expansion. The shared script also tests the separate exchange reduction and its counterexample. Normal and optimized runs completed 610 evaluations across 29 runtime conditions. Rescaling the positive-band frame and reversing the exchange energy in memory were each rejected by independent checks in both modes. The complete 46-script suite passed after the bridge additions. These are consistency tests of the stated reductions, not validation of the remaining physical identification.