• Definitions / Notation used
• The ambient torsionful Dirac operator
• The mixed fermion flux
• Visibility on the observer slice
• The effective operator on \(X\)
  • The direct spinor channel
  • The gamma-trace channel
  • The RS-like channel
  • Cartan-aligned coupling data
  • The electromagnetic limit
• Key takeaway

The previous chapter identified the observed fermion channels. This chapter gives them dynamics.

The ambient fermionic pair

$$ \chi=(\nu,\zeta) $$

does two things at once. It descends to effective Dirac propagation on the observer slice \(X\), and it produces a fermion flux that sources the visible adjoint geometry.

The result is a torsionful Dirac system on \(X\) for the three ground-sector algebraic channels

$$ \nu_{\vec 0}, \qquad \lambda_{\zeta,\vec 0}, \qquad \eta_{\mathrm{RS},\vec 0}, $$

together with a visible current that enters the first-order bosonic equation. In the light abelian Cartan direction, this system reduces to Maxwell-Dirac electrodynamics on the observer slice.

§Definitions / Notation used

The observer immersion is

$$ \iota:X^4\hookrightarrow Y^{14}. $$

Along \(\iota(X)\), the tangent bundle splits as

$$ TY|_X\simeq TX\oplus N_\iota, \qquad \operatorname{rank}(N_\iota)=10. $$

The ambient fermionic pair is

$$ \chi=(\nu,\zeta), \qquad \nu\in\Omega^0(Y,S_Y), \qquad \zeta\in\Omega^1(Y,S_Y). $$

The observed algebraic ground-sector channels from VI.4 are

$$ \chi_{\vec 0}^{\mathrm{obs}} = \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$

Here:

$$ \nu_{\vec 0} $$

is the direct spinor channel,

$$ \lambda_{\zeta,\vec 0} $$

is the gamma-trace channel induced from \(\zeta\), and

$$ \eta_{\mathrm{RS},\vec 0} $$

is the effective spin-\(1/2\) channel induced from the Rarita-Schwinger-like part of \(\zeta\).

§The ambient torsionful Dirac operator

The torsion-first geometry uses the rotated connection

$$ B_\omega = \varepsilon A_0\varepsilon^{-1}+T, $$

with augmented torsion

$$ T=\eta-\varepsilon^{-1}D_{A_0}\varepsilon. $$

The spinorial covariant derivative induced by \(B_\omega\) is

$$ \nabla^{S_Y}_{B_\omega}. $$

The corresponding ambient Dirac-like operator is

$$ \mathcal D_{Y,T} = c_Y\circ\nabla^{S_Y}_{B_\omega}, $$

where \(c_Y\) denotes Clifford multiplication by tangent vectors of \(Y\).

This operator acts on the full ambient pair

$$ \chi=(\nu,\zeta). $$

For \(\nu\), this is the ordinary torsionful spinorial Dirac action on \(Y\).

For \(\zeta\), the operator acts on a spinor-valued one-form,

$$ \zeta\in\Omega^1(Y,S_Y) \simeq \Gamma(T^\ast Y\otimes S_Y). $$

Along the observer slice,

$$ T^\ast Y|_X\simeq T^\ast X\oplus N_\iota^\ast. $$

The tangential part contributes directly to visible spinor kinetics. The normal part interacts with the Hermite ladder structure. This is the origin of the intermediate \(\mathcal H_{(0,1)}\) support required by the gamma-trace and RS-like channels.

The ambient fermionic equation is therefore written schematically as

$$ \mathcal D_{Y,T}\chi=0. $$

After pullback and projection, this equation becomes the effective Dirac system for the observed algebraic channels on \(X\).

§The mixed fermion flux

The same ambient pair defines an adjoint-valued one-form current:

$$ J_{\nu\zeta} = \Pi_{\mathrm{ad}} \left( \bar\nu,\zeta+\bar\zeta,\nu \right) \in \Omega^1(Y,\operatorname{ad}(P_H)). $$

The projection

$$ \Pi_{\mathrm{ad}} $$

places the bilinear in the same adjoint type as the augmented torsion. This gives the current the correct target space for the first-order bosonic equation.

With the fixed Shiab operator

$$ S_E^{(\varepsilon)}(F) = \star_Y \left( E\wedge \varepsilon^{-1}F\varepsilon \right), $$

the sourced first-order equation takes the schematic form

$$ S_E^{(\varepsilon)}(F_B)-\kappa_1T = g_{\nu\zeta}J_{\nu\zeta} +\cdots. $$

The left side is the visible torsion-curvature balance. The right side is the fermionic source. The omitted terms denote higher-order, background, or suppressed contributions that are not needed for the low-energy channel construction.

This equation is the dynamical closure. Fermions propagate through the visible geometry, and their flux sources the same projected geometry.

§Visibility on the observer slice

The visible one-form current is obtained by applying the \(E\)-projected visibility map:

$$ \mathcal V_E^{(1)}(J_{\nu\zeta}) = \iota^\ast (EJ_{\nu\zeta}E) \in \Omega^1(X,\operatorname{ad}_E). $$

In the Cartan-aligned frame,

$$ \operatorname{ad}_E \simeq \mathfrak t_E \oplus \bigoplus_{\alpha\in\Delta_E}\mathfrak g_\alpha, $$

this current decomposes as

$$ J_{\mathrm{vis}} = J_{\mathfrak t} + \sum_{\alpha\in\Delta_E}J_\alpha. $$

The Cartan part \(J_{\mathfrak t}\) carries diagonal charge currents. The root components \(J_\alpha\) carry off-diagonal interaction currents.

For the observed channels

$$ c \in {\nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} }, $$

the current has the schematic form

$$ J_{\mathrm{vis}}^\mu = \sum_c \bar\psi_c\gamma^\mu\psi_c\otimes\rho_c + \sum_{c\neq c'} \bar\psi_c\gamma^\mu\psi_{c'}\otimes\rho_{cc'}. $$

Here \(\rho_c\) denotes the Cartan weight of the channel \(c\), while \(\rho_{cc'}\) denotes the root-space transition data coupling distinct channels or internal states.

The conservation law is inherited from the ambient fermionic equation and the sourced bosonic balance. In Cartan components, one may write

$$ D_\mu^{(\mathfrak t_E)}J_{\mathfrak t}^{\mu} = \mathcal A_T+\mathcal B_{\mathrm{mix}}, $$

where \(\mathcal A_T\) denotes torsion-induced axial contributions and \(\mathcal B_{\mathrm{mix}}\) denotes exchange with root or mixing sectors. In the low-energy adiabatic regime, these terms are suppressed, absent, or absorbed into effective constitutive data.

§The effective operator on \(X\)

The effective Dirac operator on \(X\) is obtained by restricting the ambient operator to \(\iota(X)\), pulling back by \(\iota^*\), projecting to the low Hermite sector, and applying the channel maps.

The resulting schematic operator is

$$ \mathcal D^{(T)}_{X,\mathrm{eff}} = \gamma^\mu \left( \nabla_\mu^{S_X} + A_\mu^{\mathrm{vis}} + \Phi_{E,\mu} \right) + \alpha\gamma^\mu\gamma^5S_\mu + \mathcal M_\perp. $$

The visible connection decomposes in the Cartan-aligned frame as

$$ A_\mu^{\mathrm{vis}} = A_\mu^{\mathfrak t} + \sum_{\alpha\in\Delta_E}A_\mu^\alpha. $$

The Cartan component gives diagonal minimal couplings. The root components generate off-diagonal transitions.

The term

$$ \Phi_{E,\mu} $$

collects background torsion and curvature contributions visible through the \(E/\Theta_E\) projection. These terms later contribute to mass shifts, charge splittings, and channel-dependent spectral data.

The axial torsion term is

$$ \alpha\gamma^\mu\gamma^5S_\mu. $$

Here \(S_\mu\) is the axial torsion one-form on \(X\). This term implements the chirality bias in the low sector.

The normal-overlap term

$$ \mathcal M_\perp $$

contains Hermite matrix elements and channel-specific overlap integrals from the normal bundle.

For each observed channel \(c\), the effective equation is

$$ \mathcal D^{(T)}_{X,c}\psi_c=0. $$

The three operators share the same structural form and differ by Cartan weights, root couplings, torsion overlaps, and normal matrix elements.

§The direct spinor channel

The direct channel comes from the ambient spinor \(\nu\). Its observed field is

$$ \nu_{\vec 0} \in \mathcal H_{\vec 0}\otimes \Gamma(X,S_X\otimes S_{N_\iota}^{\mathrm{vis}}). $$

Its effective operator is

$$ \mathcal D^{(T)}_{X,\nu} = \gamma^\mu \left( \nabla_\mu^{S_X} + A_\mu^{\mathrm{vis}} + \Phi_{E,\mu}^{(\nu)} \right) + \alpha_\nu\gamma^\mu\gamma^5S_\mu + \mathcal M_\perp^{(\nu)}. $$

The Cartan weights of this channel are read directly from its placement in the visible normal spinor representation. Axial torsion selects the surviving low-energy chirality.

§The gamma-trace channel

The one-form spinor \(\zeta\) contains a gamma-trace component. Along \(X\), the channel map is

$$ \lambda_\zeta = \gamma^\mu\zeta_\mu + \text{normal-overlap terms}. $$

After Hermite support and low-sector projection,

$$ \lambda_{\zeta,\vec 0} \in \mathcal H_{\vec 0}\otimes \Gamma(X,S_X\otimes S_{N_\iota}^{\mathrm{vis}}). $$

Its effective operator is

$$ \mathcal D^{(T)}_{X,\lambda} = \gamma^\mu \left( \nabla_\mu^{S_X} + A_\mu^{\mathrm{vis}} + \Phi_{E,\mu}^{(\lambda)} \right) + \alpha_\lambda\gamma^\mu\gamma^5S_\mu + \mathcal M_\perp^{(\lambda)}. $$

The one-form origin modifies its coupling strengths through the tangential-normal split of \(T^*Y|_X\). The resulting Cartan weights, root evaluations, and normal overlaps differ from the direct \(\nu\) channel.

§The RS-like channel

The RS-like component is the gamma-traceless vector-spinor part of \(\zeta\) before observation. The effective spin-\(1/2\) channel is produced by a geometric contraction with torsion and immersion data:

$$ \eta_{\mathrm{RS}} = \mathcal C_{T,\iota}^{\mu}\zeta_{\mu}^{\mathrm{RS}}. $$

After Hermite support and low-sector projection,

$$ \eta_{\mathrm{RS},\vec 0} \in \mathcal H_{\vec 0}\otimes \Gamma(X,S_X\otimes S_{N_\iota}^{\mathrm{vis}}). $$

Its effective operator is

$$ \mathcal D^{(T)}_{X,\eta} = \gamma^\mu \left( \nabla_\mu^{S_X} + A_\mu^{\mathrm{vis}} + \Phi_{E,\mu}^{(\eta)} \right) + \alpha_\eta\gamma^\mu\gamma^5S_\mu + \mathcal M_\perp^{(\eta)}. $$

This channel naturally carries distinct overlap suppression and distinct root-evaluation data. These differences become hierarchy seeds in the mass and mixing calculation.

The three observed channels are all ground-sector outputs. Their multiplicity is algebraic, not oscillator-level multiplicity.

§Cartan-aligned coupling data

The Cartan-aligned frame makes the effective operator computational. For a visible channel \(c\), the covariant derivative can be written schematically as

$$ D_\mu^{(c)} = \partial_\mu + \omega_\mu^{S_X} + i\sum_I q_c^I A_\mu^I + \sum_{\alpha\in\Delta_E} A_\mu^\alpha R_c(E_\alpha). $$

Here:

$$ q_c^I=\rho_c(H_I) $$

are Cartan eigenvalues,

$$ E_\alpha\in\mathfrak g_\alpha $$

are root generators, and

$$ R_c(E_\alpha) $$

is the action of the root generator on channel \(c\).

The diagonal Cartan data determine charge assignments. The root data determine interaction selection rules. The torsion and normal-overlap data determine channel-dependent spectral shifts.

This is the useful form of the operator for later computation. The interaction basis is defined by the Cartan-root decomposition. The mass basis is defined by the full effective operator.

§The electromagnetic limit

The low-energy electromagnetic limit is obtained by selecting the light abelian Cartan direction. Let

$$ Q_{\mathrm{em}}\in\mathfrak t_E $$

be the Cartan generator whose gauge fluctuation remains massless after including \(\Phi_E\), torsion shifts, and normal overlaps.

The electromagnetic gauge field is the corresponding Cartan projection:

$$ A_\mu^{\mathrm{em}} = \langle A_\mu^{\mathfrak t},Q_{\mathrm{em}}\rangle. $$

The abelian curvature is

$$ F_{\mathrm{em}}=dA_{\mathrm{em}}. $$

The electromagnetic current is the same visible fermion flux projected onto \(Q_{\mathrm{em}}\):

$$ J_{\mathrm{em}}^\mu = \left\langle J_{\mathrm{vis}}^\mu, Q_{\mathrm{em}} \right\rangle = \sum_c q_c\bar\psi_c\gamma^\mu\psi_c + \text{suppressed cross terms}. $$

The charge of channel \(c\) is

$$ q_c=\rho_c(Q_{\mathrm{em}}). $$

Projecting the sourced first-order equation onto the electromagnetic direction gives

$$ \mathcal V_{E,\Theta}^{(2)}(F_B^{\mathrm{em}}) \simeq \kappa_1\mathcal V_E^{(1)}(T^{\mathrm{em}}) + g,\mathcal V_E^{(1)}(J_{\nu\zeta}^{\mathrm{em}}). $$

In the weak, adiabatic, low-energy observer-slice limit, augmented torsion either decouples or enters as effective constitutive data. The sourced equation reduces to

$$ d\star_XF_{\mathrm{em}}=J_{\mathrm{em}}. $$

The homogeneous equation follows from the Bianchi identity:

$$ dF_{\mathrm{em}}=0. $$

The channel derivative becomes the standard minimally coupled form

$$ D_\mu^{(c)} = \partial_\mu + \omega_\mu^{S_X} + ieq_cA_\mu^{\mathrm{em}}. $$

Thus each visible channel satisfies, at leading order,

$$ \left( i\gamma^\mu D_\mu^{(c)} + \alpha_c\gamma^\mu\gamma^5S_\mu + \mathcal M_c \right)\psi_c = 0. $$

The leading light abelian sector is Maxwell-Dirac theory on \(X\). Residual torsion terms supply chiral corrections, possible axion-like contributions, or small effective photon-mass constraints once a concrete background for \(S_\mu(x)\) and \(\sigma(x)\) is chosen.

§Key takeaway

The observed fermions remain ground-sector algebraic channels. Their dynamics and currents are not added as independent four-dimensional structures. They descend from the ambient pair \((\nu,\zeta)\) through Hermite support, pullback, visibility projection, and Cartan-aligned spectral reduction.