In this instantiation, “boson mass” is not something you can tack on as a Proca term for a connection. The transport formalism provides a different approach, based on the covariant torsion variable

$$ T := \vartheta - \varepsilon^{-1} d_{A_0}\varepsilon \in \Omega^1(Y,\operatorname{ad}(P_H)). $$

It is (i) tensorial, (ii) gauge-covariant under the transport subgroup, and (iii) already appears quadratically in the first-order action once you expand

$$ I_1(\omega) = \int_Y \langle T,\star_Y(\bullet_\varepsilon(F_B) - \kappa_1 T)\rangle. $$

The point of this post is to put the mass-generation mechanism into a clean framework that respects the rules of the construction.

Definitions / Notation used

• Bosonic fluctuation: a perturbation of tensorial objects $(T, F_B)$ around a background solution $(\omega_0, A_0)$, not a perturbation “of $A$”.
• Observable boson field on $X$: a projected pullback of torsion,

$$ t := \iota^\ast(P_{\mathrm{obs}} T) \in \Omega^1(X,\operatorname{ad}{\mathrm{obs}}), $$ where $P{\mathrm{obs}}$ is the composition of (i) leg selection to $TX$, (ii) $\operatorname{ad}$-block selection via $E$, and (iii) the physical-corner projection used in the split-signature $(7,7)$ setting.

• Normal-mode expansion: for $n$ in the normal bundle $N_\iota$, expand components as

$$ (P_{\mathrm{obs}} T)\mu(x,n) = \sum_i t{\mu,i}(x) \varphi_i(n), $$ where $\varphi_i$ is the Hermite–Gaussian basis on the normal bundle.

Main technical argument: the torsion quadratic induces 4D mass terms on the observable corner

Start from the tensorial functional of $T$. In $I_1(\omega)$, the term

$$ -\kappa_1 \int_Y \langle T,\star_Y T\rangle $$

is already present. It is gauge-covariant (because $T$ is), it is local on $Y$, and it does not refer to $A$ anywhere. Now linearize around a background solution $\omega_0$ with torsion $T_0$ and rotated connection $B_0 := A_0\cdot\varepsilon_0$. Write

$$ T = T_0 + \delta T, F_B = F_{B_0} + \delta F_B, $$

and restrict attention to fluctuations in the observable corner $P_{\mathrm{obs}}(\delta T)$. The quadratic part of the torsion norm produces, immediately, a positive-definite (in the physical corner) quadratic functional of those fluctuations:

$$ \kappa_1 \int_Y \langle P_{\mathrm{obs}}\delta T,\star_Y P_{\mathrm{obs}}\delta T\rangle + \text{(cross terms with }T_0\text{)}. $$

The only nontrivial step is to explain why this becomes a 4D mass term rather than “just another kinetic thing”. The reason is structural: once you have imposed the observable corner, the remaining degrees of freedom are effectively 4D fields dressed by fixed normal profiles, and the $Y$-integral factorizes into an $X$ integral times overlap integrals on the normal bundle.

Concretely, along $\iota(X)$ we split legs using $TY|X \simeq TX \oplus N\iota$. Write the 1-form as

$$ P_{\mathrm{obs}}\delta T = (\delta T)_\mu(x,n) dx^\mu + (\delta T)_a(x,n) \hat n^a, $$

then apply the pullback rule (already established in the pullback/projection discussion): $\iota^\ast(\hat n^a)=0$, so only the $TX$ legs can contribute to what is seen as a 4D bosonic field. Thus on the observable corner you are effectively dealing with the $TX$ part,

$$ (P_{\mathrm{obs}}\delta T)_\mu(x,n) dx^\mu. $$

Now expand in the Hermite basis on the normal bundle:

$$ (P_{\mathrm{obs}}\delta T)\mu(x,n) = \sum_i t{\mu,i}(x)\varphi_i(n). $$

Insert this into the quadratic torsion functional and integrate over $Y$ with $\star_Y$. Using only the $\sigma$-split structure

$$ g_Y \simeq g_X \oplus \sigma(x)^2\delta_{ab} \hat n^a \otimes \hat n^b, $$

the $Y$-measure decomposes into an $X$ piece times a normal piece with $\sigma$-dependent weight. The result is a 4D quadratic form:

$$ \kappa_1 \int_X \langle t_i,\star_X (M^2)_{ij}(\sigma;T_0,E,\Theta_E) t_j\rangle + \text{(derivative terms)}, $$

where the “mass matrix” is the normal overlap integral of the selected torsion pairing:

$$ (M^2){ij}(\ldots) \sim \int{N_\iota} (\text{normal measure})\cdot\langle \varphi_i,\mathcal{O}_{\mathrm{mass}}(T_0;E,\Theta_E)\varphi_j\rangle. $$

This line is deliberately schematic. But it illustrates the principle: a mass matrix is an overlap of normal modes against an operator built out of background fields and selectors. Here $\mathcal{O}_{\mathrm{mass}}$ is constructed from the only admissible ingredients: the $\operatorname{ad}$ pairing restricted by $E$, and the calibrator $\Theta_E$ that enforces which components actually contribute to the “Einstein-like” contraction and, correspondingly, which torsion components can couple coherently to $X$-visible bosons. The key physical picture is: the normal bundle is not just “extra directions”; it is the selection apparatus that turns a covariant $Y$-quadratic into a hierarchy of 4D masses.

What sets relative scales (framework, not fit)

There are three “knobs” that appear without cheating:

1. Commutator selection in $\operatorname{ad}$ (unbroken vs broken directions). If the background torsion $T_0$ has support in certain $\operatorname{ad}$ directions, then fluctuations whose $\operatorname{ad}$ generators commute with that background direction remain effectively massless in the observable corner, while non-commuting directions pick up a mass through the adjoint action. This is the Higgs logic, but implemented purely via a background tensor $T_0$ rather than a scalar field.

2. Normal-mode overlap (geometric Yukawa analogue for bosons). Even if a direction is “allowed” in $\operatorname{ad}$, it only couples if its normal profile overlaps the profiles that $\Theta_E$ and the background solution actually pick out. In practice: most $\varphi_i$ are orthogonal to what the calibrator sees, or integrate to zero by parity, or are suppressed by being energetically high in the normal oscillator.

3. $\sigma$-scaling from the split metric. Because the normal metric is scaled by $\sigma(x)^2$, the effective 4D coefficient you read as $m^2$ is $\sigma$-dependent. Once $\sigma$ is dynamical, boson masses become environment-dependent in principle (constrained in practice by stabilization).

Assumptions vs Consequences

Assumptions

• There exists a background solution $\omega_0$ with nontrivial torsion $T_0$ in the observable corner (torsion-first ansatz).
• Observable fields are defined by $P_{\mathrm{obs}} = (TX\text{-leg selection}) \circ (E \text{ block selection}) \circ (\text{physical-corner projection})$, consistent with $\mathrm{Spin}(7,7)$ split signature.
• Normal-mode basis ${\varphi_i}$ (Hermite–Gaussian) is a faithful spectral decomposition for the normal dependence.

Consequences

• A gauge-covariant quadratic term $\kappa_1\langle T,\star_Y T\rangle$ becomes, after mode reduction, a 4D quadratic form in the observable bosons $t_i(x)$, i.e. a mass matrix.
• “Massless vs massive” is not an imposed symmetry breaking; it is a statement about commutants in $\operatorname{ad}$ and overlaps in the normal bundle.
• $\sigma$ necessarily enters the mass sector through the $Y$-measure and the normal metric scaling, tying boson masses to cosmology in a controlled way.

Why this matters

• The same normal-overlap technology that sets boson masses is what sets Yukawa textures once fermions are expanded in the Hermite basis and coupled torsion-first.
• Because $\sigma$ rescales normal geometry, any stabilization mechanism will also stabilize mass ratios; conversely, $\sigma$-dynamics would imprint correlated drifts unless locked down.

Key takeaway

Boson masses in this instantiation are not “added”; they are the unavoidable 4D shadow of a covariant torsion quadratic once you (i) select the observable corner with $(E,\Theta_E)$ and (ii) integrate out the normal bundle via overlap integrals.

Technical takeaway

A schematic mass-matrix statement is:

$$ (m^2){ij} \sim \kappa_1 \int{N_\iota} \langle \varphi_i,\mathcal{O}_{\mathrm{mass}}(T_0;E,\Theta_E,\sigma)\varphi_j\rangle, $$

with the 4D mass term appearing as $\int_X \langle t_i,\star_X (m^2){ij} t_j\rangle$ for $t_i = \iota^\ast(P{\mathrm{obs}} T)_i$.