• Definitions / Notation used
• Main technical argument: Weyl-on-\(Y\) is a kinematic restriction; chiral-on-\(X\) is a dynamical/spectral outcome
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters
• Key takeaway
• Technical takeaway

If one wants chiral fermions in four dimensions, the familiar move is to impose a Weyl condition. In this instantiation, that move is premature. The theory’s native spinors live on \(Y\) with Spin\((7,7)\) structure, while fermions on \(X\) are already filtered objects: restrictions of ambient fields shaped by the immersion and by the torsion background. Chirality, therefore, is not stipulated at the level of \(\Gamma(Y,S_Y)\). It is read off from which pulled-back modes are light, stable, and actually visible to the observer.

§Definitions / Notation used

• \(X=X^4\), \(Y=Y^{14}\), \(\iota:X\hookrightarrow Y\), and pullback \(\iota^\ast\).
• Along \(\iota(X)\): \(TY|_X \simeq TX \oplus N_\iota,\) with indices \(\mu,\nu\) on \(TX\), indices \(a,b\) on \(N_\iota\), and indices \(M,N\) on \(Y\).
• The metric split is \(g_X := \iota^\ast g_Y, g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b.\)
• Spinors are native to \(Y\): \(\Psi \in \Gamma(Y,S_Y).\) The observed restriction is \(\Psi_\iota := \Psi \circ \iota.\)
• Chirality on \(X\) is denoted schematically by the usual 4D chirality grading \(\gamma^5\) on the pulled-back tangent Clifford sector.

§Main technical argument: Weyl-on-\(Y\) is a kinematic restriction; chiral-on-\(X\) is a dynamical/spectral outcome

A Weyl condition imposed on \(Y\) is a kinematic deletion of half of a spinor representation. It says, before solving the dynamics, that only one ambient chirality is allowed.

But the chirality relevant to the observed fermion sector is not an ambient constraint. It is a property of the effective spinor content seen on \(X\) after restriction:

$$ \Psi_\iota = \Psi \circ \iota. $$

The pulled-back sector is already shaped by three pieces of structure:

1. the immersion \(\iota:X \hookrightarrow Y\),
2. the split \(TY|_X \simeq TX \oplus N_\iota\),
3. the torsion background \(T\).

We organized ambient spinors as

$$ \Psi(x,n)=\phi_k(n)\otimes\psi_k(x). $$

This means that the observed spinor content is not the full ambient spinor bundle. It is a spectrum of normal modes with spinorial coefficients on \(X\). We sidestep Weyl and instead ask the following question:

Which \(\psi_k(x)\) remain light, stable, and pullback-stable after immersion and torsion are included?

A Weyl condition is a constraint. A chirality filter is a selection mechanism.

In the torsion-first instantiation, axial torsion is the default background. Since axial torsion couples differently to opposite 4D chiralities, it can split what would otherwise be paired left/right modes. That splitting can make one chiral sector energetically favored in the ground state while the other becomes heavy, unstable, or absent from the low-energy pulled-back sector.

So the lemma-level statement is:

Chirality is defined only for the observed sector on \(X\), and the low-energy, pullback-stable spectrum on \(X\) is effectively chiral because axial torsion selects one \(\gamma^5\) sector.

Starting from \(\Psi \in \Gamma(Y,S_Y)\) with no Weyl condition imposed by hand and with an axial torsion component treated as non-perturbative, the goal is to show that \(\Psi_\iota\) has an effectively chiral low-energy sector on \(X\).

That conclusion is not automatic. It must be earned by the spectral behavior of the effective pulled-back Dirac operator. But the point of the instantiation is precisely that axial torsion supplies the missing selection mechanism.

This is also why the construction should not be phrased as “we choose one chirality.” We do not choose it. The immersion and the torsion background choose it dynamically by changing the spectrum.

If both chiralities remain equally light and equally stable, then the model has failed to explain chirality. If one chirality is consistently selected in the pullback-stable ground sector, then chirality has emerged without a Weyl imposition.

§Assumptions vs Consequences

§Definitional

Spinors are native to \(Y\):

$$ \Psi \in \Gamma(Y,S_Y). $$

The observer sees the pulled-back or restricted field:

$$ \Psi_\iota = \Psi \circ \iota. $$

The observed decomposition is organized by

$$ TY|_X \simeq TX \oplus N_\iota. $$

§Ansatz

Axial torsion is present and non-perturbative.

The relevant observed fermions are pullback-stable and symmetry-respecting. No Weyl condition is imposed on the ambient spinor \(\Psi\).

§Consequence

Effective chirality on \(X\) is determined by the spectrum of the pulled-back operator in the torsion background.

The light observed sector can be effectively Weyl even though the ambient spinor field was not constrained to be Weyl.

Symbolically:

\(\Psi \in \Gamma(Y,S_Y)\) unconstrained by Weyl, but \(\Psi_\iota\) has a chiral low-energy sector.

§Why this matters

• It avoids confusing a 14D spinorial constraint with a 4D observed property.
• It keeps chirality geometric: it comes from immersion plus torsion, not from a representation-theoretic assumption.

§Key takeaway

Do not impose Weyl on \(Y\); instead, compute chirality on \(X\).

The chiral sector is supposed to emerge as the low-energy, pullback-stable outcome of immersion plus axial torsion.

§Technical takeaway

\(\Psi \in \Gamma(Y,S_Y)\) with no Weyl condition imposed.

$$ \Psi(x,n)=\phi_k(n)\otimes\psi_k(x), \qquad \Psi_\iota=\Psi\circ\iota. $$

Chirality on \(X\) is a spectral consequence for the surviving \(\psi_k(x).\)