• Definitions / Notation used
• Main technical argument: factorization is the natural basis adapted to \(TY|_X \simeq TX \oplus N_\iota\)
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters
• Key takeaway
• Technical takeaway

If you let spinors live on \(Y\), then “fermions on spacetime” becomes a derived notion, not a starting axiom. The immersion \(\iota : X \hookrightarrow Y\) does two jobs at once: it induces the observed metric \(g_X = \iota^\ast g_Y\) , and it restricts every ambient field to what the observer can actually sample. For spinors, that restriction translates into a spectral filter, because the normal directions \(N_\iota\) come with their own mode structure. In this instantiation, we use the Hermite basis to turn “internal degrees of freedom” into normal-bundle oscillator modes, and it tells us what can survive on \(X\).

§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 \(\Psi_\iota := \Psi \circ \iota.\)
• Local symbol: \(n=(n^a)\) denotes normal coordinates, or a local normal section, along \(N_\iota\) near \(\iota(X)\).

§Main technical argument: factorization is the natural basis adapted to \(TY|_X \simeq TX \oplus N_\iota\)

Along \(\iota(X)\), the observer distinguishes tangent directions from normal directions. The tangent directions become spacetime directions on \(X\). The normal directions are transverse to the observer’s world-volume and are not identified with \(TX\).

In a local tubular neighborhood of \(\iota(X)\) inside \(Y\), points can be written schematically as \((x,n)\), with \(x \in X\) and \(n\) in the normal fiber \(N_\iota|_x\).

Apply this to spinors. A spinor on \(Y\) is native to \(Y\), but near \(\iota(X)\) the tangent/normal split lets us organize the ambient Clifford action into tangent and normal pieces. Globally, the spinor bundle can be twisted, so this should not be read as a global product decomposition without qualification. But locally, and spectrally, the split acts as an organizing principle.

The normal dependence is then expanded in a Hermite-Gaussian basis. These are harmonic-oscillator-like modes in the normal directions. The basic factorization is

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

Here:

• \(\phi_k(n)\) is a Hermite-Gaussian mode on the normal bundle sector \(N_\iota\).
• \(k\) is a multi-index labeling normal oscillator excitations.
• \(\psi_k(x)\) is the corresponding spinor factor on \(X\) after pullback/restriction.

An ambient spinor admits a normal-mode expansion near the immersed submanifold, and the coefficients of that expansion are spinorial fields along \(X\).

Operationally, the observer samples

$$ \Psi_\iota(x) = \Psi(\iota(x)). $$

In local normal coordinates this is the restriction of \(\Psi(x,n)\) to the normal profile selected by the immersion. That is how \(\iota\) determines which normal profile is sampled.

The metric ansatz explains why the Hermite basis is natural. The normal part of the metric is

$$ \sigma(x)^2 \delta_{ab} n^a \otimes n^b. $$

So the normal sector is locally Euclidean, scaled by \(\sigma(x)\). In the simplest spectral approximation, the normal operator behaves like a harmonic oscillator in these transverse directions. The Hermite-Gaussian modes are therefore the natural basis for transverse localization.

This also gives a precise interpretation of “internal structure.” In this construction, internal labels are not assumed to be Standard Model group representations. They arise as labels in the normal-mode sector, combined with the Clifford and transport structure native to \(Y\).

The observer’s fermion content on \(X\) is not the full ambient space \(\Gamma(Y,S_Y)\). It is the pullback-stable, symmetry-respecting part of the ambient spinor spectrum. In low-energy language, it is dominated by the lowest normal modes.

§Assumptions vs Consequences

§Definitional

Spinors are native to \(Y\):

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

The observer sees the restricted field

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

The tangent/normal split along the immersion is

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

§Ansatz

The normal bundle sector admits a harmonic-oscillator-like spectral organization.

The normal basis is taken to be Hermite–Gaussian:

$$ \{\phi_k(n)\}. $$

The low-energy observed sector is dominated by low-\(k\) modes.

§Consequence

The natural basis for observed fermions is

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

Each normal mode \(\phi_k\) carries a spinor coefficient \(\psi_k(x)\) on \(X\).

The internal labels of the observed fermion sector are not imposed as Standard Model representations. They arise from normal-mode labels, Clifford orientation, and the transport geometry.

§Why this matters

• What “survives on \(X\)” is not arbitrary 14D spinor data, but a filtered low-\(k\) spectrum.
• It makes chirality selection a spectral question rather than a kinematic projection.
• It lets torsion act differently across normal-mode sectors, which is the opening needed for mass hierarchy and selection rules.
• It sets the stage for BRST, and anomaly questions must be asked about the surviving pulled-back spectrum, not about unconstrained ambient spinors.

§Key takeaway

Spinors live on \(Y\).

The observer on \(X\) sees a restricted, mode-filtered spinor sector.

In the Hermite basis, that sector is organized as normal modes times a 4D spinor.

§Technical takeaway

$$ TY|_X \simeq TX \oplus N_\iota, \qquad g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b. $$

$$ \Psi \in \Gamma(Y,S_Y), \qquad \Psi_\iota = \Psi \circ \iota. $$

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