This article describes the origin of the three generations of fermions sourced from the ambient spinor pair $(\nu, \zeta)$. The direct spinor $\nu$ contributes one spin-$1/2$ channel. The spinor-valued one-form $\zeta$ contributes a gamma-trace spin-$1/2$ channel and a Rarita-Schwinger-like channel whose vector index can be contracted by torsion, immersion, or transport data.
$$ \chi=(\nu,\zeta), $$
with
$$ \nu\in\Omega^0(Y,S_Y), \qquad \zeta\in\Omega^1(Y,S_Y). $$
This article identifies the fermionic channels induced on $X$ from this pair.
Definitions / Notation used
Ambient space:
$$ Y=Y^{14}, $$
with split signature $(7,7)$ and structure group
$$ \mathrm{Spin}(7,7). $$
Observed space:
$$ X=X^4. $$
Observation map:
$$ \iota:X\hookrightarrow Y. $$
Along $\iota(X)$:
$$ TY|X\simeq TX\oplus N\iota, $$
with
$$ \operatorname{rank}(N_\iota)=10. $$
The spinor bundle decomposes schematically as
$$ S_Y|X\simeq S_X\otimes S{N_\iota}. $$
The normal spinor factor $S_{N_\iota}$ supplies internal labels in the observed sector.
The ambient fermionic fields are
$$ \nu\in\Omega^0(Y,S_Y), $$
and
$$ \zeta\in\Omega^1(Y,S_Y). $$
The normal Hermite basis is denoted
$$ {\phi_k(n)}, $$
where
$$ k=(k_1,\ldots,k_{10}). $$
The strict ground span is
$$ \mathcal{H}0 := \operatorname{span}{\phi{\vec 0}}, $$
with
$$ \vec 0=(0,\ldots,0). $$
The Hermite span used by the $\zeta$ channel is
$$ \mathcal{H}_{0,1} := \operatorname{span}{\phi_k: |k|=0\text{ or }|k|=1}. $$
The adjoint selector is
$$ E. $$
The $\Theta_E$ form saturates the ten normal directions and leaves one $X$-slot. It is used to convert ambient curvature data into $X$-visible one-form data.
The Shiab operator is
$$ \mathcal{S}_e^{(\varepsilon)}(F) := \star_Y \left(e\wedge \varepsilon^{-1}F\varepsilon\right). $$
In this instantiation, $\varepsilon$ is taken to be the $E$-selected adjoint block.
The augmented torsion is
$$ T = \eta -\varepsilon^{-1}D_{A_0}\varepsilon. $$
The first-order torsion/Shiab balance has schematic form
$$ \mathcal{S}_E(F_B)-\kappa_1T=0. $$
With fermions included, the source-extended equation is written schematically as
$$ \mathcal{S}E(F_B)-\kappa_1T = g\chi J_\chi. $$
Main technical argument: the survival criterion
The survival criterion is determined by the type of the source equation.
The bosonic terms in the first-order equation take values in
$$ \Omega^1(Y,\operatorname{ad}). $$
Indeed,
$$ T\in\Omega^1(Y,\operatorname{ad}), $$
and the Shiab curvature term satisfies
$$ \mathcal{S}_E(F_B)\in\Omega^1(Y,\operatorname{ad}). $$
Thus a fermionic source coupled to the same equation must also be an adjoint-valued one-form:
$$ J_\chi\in\Omega^1(Y,\operatorname{ad}). $$
The pair $(\nu,\zeta)$ naturally produces such a source. Since
$$ \nu\in\Omega^0(Y,S_Y), $$
and
$$ \zeta\in\Omega^1(Y,S_Y), $$
their mixed bilinear has type
$$ \bar\nu,\zeta \in \Omega^1(Y)\otimes S_Y^\ast\otimes S_Y. $$
Using
$$ S_Y^\ast\otimes S_Y\simeq\operatorname{End}(S_Y), $$
and projecting the spinor endomorphism factor to the adjoint representation gives
$$ J_{\nu\zeta} := \Pi_{\operatorname{ad}}(\bar\nu,\zeta+\bar\zeta,\nu) \in \Omega^1(Y,\operatorname{ad}). $$
The fermionic source may therefore enter the first-order equation as
$$ \mathcal{S}E(F_B)-\kappa_1T = g{\nu\zeta}J_{\nu\zeta} +\cdots . $$
The visibility of this equation on $X$ is obtained by applying the appropriate $E$-selected maps.
For curvature two-forms, define
$$ \mathcal{V}^{(2)}_{E,\Theta}(F_B) := \iota^\ast \left( \star_Y[\Theta_E\wedge E F_BE] \right). $$
This lands in
$$ \Omega^1(X,\operatorname{ad}_E). $$
For adjoint-valued one-forms, define
$$ \mathcal{V}^{(1)}_E(K) := \iota^\ast(EKE), $$
with
$$ K\in\Omega^1(Y,\operatorname{ad}). $$
The observed source equation is then
$$ \mathcal{V}^{(2)}_{E,\Theta}(F_B) - \kappa_1\mathcal{V}^{(1)}E(T) = g{\nu\zeta}\mathcal{V}^{(1)}E(J{\nu\zeta}) +\cdots . $$
A fermionic channel survives as an observed channel when it contributes to this $E$-visible equation after Hermite support, pullback, Clifford decomposition, and torsion-biased projection.
The direct spinor branch comes from $\nu$. Expanding in the Hermite basis,
$$ \nu(x,n) = \sum_k\phi_k(n)\otimes\nu_k(x). $$
The direct ground contribution is
$$ \nu_{\vec 0}(x,n) = \phi_{\vec 0}(n)\otimes\nu_{\vec 0}(x). $$
After restriction to $\iota(X)$,
$$ \nu_{\vec 0}|X(x) = \phi{\vec 0}(0)\otimes\nu_{\vec 0}(x). $$
After visibility projection, this yields
$$ \nu_{\vec 0} \in \Gamma(X,S_X\otimes S_{N_\iota}^{\mathrm{vis}}). $$
The spinor-valued one-form branch comes from $\zeta$. Locally,
$$ \zeta = \zeta_\mu dx^\mu+\zeta_a dn^a. $$
Pullback gives
$$ \iota^\ast\zeta = \zeta_\mu dx^\mu, $$
using
$$ \iota^\ast(dn^a)=0. $$
The coefficient $\zeta_\mu$ remains valued in
$$ S_X\otimes S_{N_\iota}^{\mathrm{vis}}. $$
Thus $\iota^\ast\zeta$ is a vector-spinor on $X$.
The $\zeta$ channel requires intermediate support in
$$ \mathcal{H}_{0,1}. $$
This support condition is encoded schematically by
$$ P_0,\mathcal{C}\zeta,P{0,1} : \mathcal{H}{0,1}\otimes\Omega^1(Y,S_Y) \longrightarrow \mathcal{H}0\otimes\Gamma(X,S_X\otimes S{N\iota}^{\mathrm{vis}}). $$
Here $P_{0,1}$ projects onto the Hermite span $(0,1)$, $P_0$ projects onto the final visible ground sector, and $\mathcal{C}_\zeta$ denotes the relevant combination of pullback, Clifford contraction, and torsion or immersion contraction.
The gamma-trace spinor channel is
$$ \lambda_\zeta := \gamma^\mu\zeta_\mu. $$
Including the Hermite support and final ground projection gives
$$ \lambda_{\zeta,\vec 0} := P_0 \gamma^\mu\zeta_\mu^{(0,1)}. $$
This is the second observed ground-sector spinor channel.
The remaining tangential vector-spinor part of $\zeta$ is the gamma-traceless, Rarita-Schwinger-like component. Define
$$ \zeta_\mu^{\mathrm{RS}} := \zeta_\mu - \frac{1}{4}\gamma_\mu\gamma^\nu\zeta_\nu. $$
Then
$$ \gamma^\mu\zeta_\mu^{\mathrm{RS}}=0. $$
Equivalently,
$$ \zeta_\mu = \frac{1}{4}\gamma_\mu\lambda_\zeta + \zeta_\mu^{\mathrm{RS}}. $$
The RS-like branch contributes an effective spin-$1/2$ channel when the background supplies an $X$-visible contraction of the vector index. Write
$$ \eta_{\mathrm{RS},\vec 0} := P_0 \mathcal{C}{T,\iota}^{\mu} \zeta\mu^{\mathrm{RS},(0,1)}. $$
The contraction
$$ \mathcal{C}_{T,\iota}^{\mu} $$
is built from the relevant torsion, immersion, or transport-covariant structure. Its detailed form belongs to the later mass and mixing analysis. In this article it records the algebraic path by which the gamma-traceless vector-spinor sector of $\zeta$ contributes to the observed spin-$1/2$ ground sector.
The observed ground-sector fermionic channels are therefore
$$ \chi_{\vec 0}^{\mathrm{obs}} = \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$
Their algebraic origins are
$$ \nu_{\vec 0} \leftarrow \nu, $$
$$ \lambda_{\zeta,\vec 0} \leftarrow P_0\gamma^\mu\zeta_\mu^{(0,1)}, $$
and
$$ \eta_{\mathrm{RS},\vec 0} \leftarrow P_0\mathcal{C}{T,\iota}^{\mu}\zeta\mu^{\mathrm{RS},(0,1)}. $$
The multiplicity is thus the multiplicity of visible algebraic channels induced from $(\nu,\zeta)$.
Chirality filtering
The preceding construction identifies the observed ground-sector spinor channels. Axial torsion then selects the low-chirality sector.
On $X$, the torsion-biased Dirac operator has the schematic form
$$ \mathcal{D}^{(T)}_X = \mathcal{D}X + \gamma^\mu\gamma^5 S\mu, $$
where $S_\mu$ is the induced axial torsion one-form.
The chiral components satisfy
$$ \gamma^5\psi_L=-\psi_L, \qquad \gamma^5\psi_R=+\psi_R. $$
The axial torsion term therefore shifts the two chiralities with opposite sign. Let
$$ \Pi_{\mathrm{low}}^{(T)} $$
denote the low-sector spectral projection associated to $\mathcal{D}^{(T)}_X$. The observed low-sector fermions are
$$ \chi_{\mathrm{low}}^{\mathrm{obs}} = \Pi_{\mathrm{low}}^{(T)} \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$
This produces the chiral low-energy fermion content from the torsion-biased visible ground sector.
Assumptions vs Consequences
Assumptions
The ambient fermionic content is the pair
$$ \chi=(\nu,\zeta), $$
with
$$ \nu\in\Omega^0(Y,S_Y), \qquad \zeta\in\Omega^1(Y,S_Y). $$
The observer immersion is
$$ \iota:X\hookrightarrow Y, $$
with tangent split
$$ TY|X\simeq TX\oplus N\iota. $$
The spinor bundle restricts schematically as
$$ S_Y|X\simeq S_X\otimes S{N_\iota}. $$
The direct $\nu$ branch uses the strict Hermite ground span
$$ \mathcal{H}_0. $$
The $\zeta$ branches use the Hermite span
$$ \mathcal{H}_{0,1} $$
as intermediate support.
The final observed $\zeta$ outputs are projected to the visible ground sector by $P_0$.
The curvature term is made $X$-visible using the $E,\Theta_E$ Shiab-type map
$$ \mathcal{V}^{(2)}_{E,\Theta}. $$
Torsion and fermionic one-form sources are made visible using
$$ \mathcal{V}^{(1)}_E. $$
Axial torsion defines the low-chirality spectral filter
$$ \Pi_{\mathrm{low}}^{(T)}. $$
Consequences
The observed ground-sector channels are
$$ \chi_{\vec 0}^{\mathrm{obs}} = \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$
The direct spinor branch is
$$ \nu_{\vec 0}. $$
The gamma-trace branch is
$$ \lambda_{\zeta,\vec 0} = P_0\gamma^\mu\zeta_\mu^{(0,1)}. $$
The RS-derived branch is
$$ \eta_{\mathrm{RS},\vec 0} = P_0\mathcal{C}{T,\iota}^{\mu} \zeta\mu^{\mathrm{RS},(0,1)}. $$
The observed multiplicity is the multiplicity of algebraic channels induced from $(\nu,\zeta)$.
The Hermite basis supplies support and overlap data. The final observed channels are counted after the visible ground-sector projection.
The chiral low-energy sector is
$$ \chi_{\mathrm{low}}^{\mathrm{obs}} = \Pi_{\mathrm{low}}^{(T)} \chi_{\vec 0}^{\mathrm{obs}}. $$
Why this matters
The mass and mixing story acts on
$$ \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$
The relevant sources of splitting are algebraic origin, overlap integrals, $E$-visible torsion couplings, charge/projector structure, and coupling to the axial torsion background.
The Hermite basis supplies localization and overlap structure. The channel count is determined by the algebraic decomposition of the ambient pair $(\nu,\zeta)$.
The $\zeta$ branches require $\mathcal{H}_{0,1}$ as construction support, while the corresponding observed fermions are counted in the final ground sector. This keeps the fermion content aligned with the torsion-first source equation rather than with a normal oscillator tower.
Key takeaway
The ambient fermionic content is
$$ \chi=(\nu,\zeta). $$
The visible ground-sector channels are
$$ \nu_{\vec 0}, \qquad \lambda_{\zeta,\vec 0}, \qquad \eta_{\mathrm{RS},\vec 0}. $$
Their multiplicity comes from algebraic origin inside $(\nu,\zeta)$.
The $\zeta$ channels require intermediate Hermite support in
$$ \mathcal{H}_{0,1}, $$
and their observed outputs lie in the final ground sector after projection.
The low-energy chirality is selected by axial torsion through
$$ \Pi_{\mathrm{low}}^{(T)}. $$
Technical takeaway
The fermionic source current generated by the ambient pair has the schematic form
$$ J_{\nu\zeta} = \Pi_{\operatorname{ad}}(\bar\nu,\zeta+\bar\zeta,\nu) \in \Omega^1(Y,\operatorname{ad}). $$
This has the same type as augmented torsion:
$$ T\in\Omega^1(Y,\operatorname{ad}). $$
The source-extended first-order equation is
$$ \mathcal{S}E(F_B)-\kappa_1T = g{\nu\zeta}J_{\nu\zeta} +\cdots . $$
The observed equation is
$$ \mathcal{V}^{(2)}_{E,\Theta}(F_B) - \kappa_1\mathcal{V}^{(1)}E(T) = g{\nu\zeta}\mathcal{V}^{(1)}E(J{\nu\zeta}) +\cdots . $$
The three ground-sector channels are
$$ \nu_{\vec 0}, $$
$$ \lambda_{\zeta,\vec 0} = P_0\gamma^\mu\zeta_\mu^{(0,1)}, $$
and
$$ \eta_{\mathrm{RS},\vec 0} = P_0\mathcal{C}{T,\iota}^{\mu}\zeta\mu^{\mathrm{RS},(0,1)}. $$
The final observed low-sector content is
$$ \chi_{\mathrm{low}}^{\mathrm{obs}} = \Pi_{\mathrm{low}}^{(T)} \left( \nu_{\vec 0}, \lambda_{\zeta,\vec 0}, \eta_{\mathrm{RS},\vec 0} \right). $$