• Definitions / Notation used
• Main technical argument: the torsion quadratic induces 4D mass terms on the observable corner
  • What sets relative scales (framework, not fit)
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

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 \(t_i = \iota^\ast(P_{\mathrm{obs}} T)_i\).