• Definitions / Notation used
• Main technical argument: the survival criterion
• Chirality filtering
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

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

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