• Definitions / Notation used
• Main technical argument: bosons are not extra fields on \(X\)
  • What “adjoint” means operationally
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters
• Key takeaway
• Technical takeaway

In this torsion-first, transport-based instantiation, bosons are not planted onto spacetime \(X\) as independent ingredients. They live natively on the ambient manifold \(Y\) as adjoint-valued geometry.

§Definitions / Notation used

• \(Y\) is a 14D manifold with split signature \((7,7)\). \(X\) is a 4D manifold immersed by \(\iota: X \hookrightarrow Y\).
• Along \(\iota(X)\): \(TY|_X \simeq TX \oplus N_\iota\), with indices \(\mu,\nu\) on \(TX\); \(a,b\) on \(N_\iota\); and \(M,N\) on \(TY\).
• \(g_X := \iota^\ast g_Y\). We use the \(\sigma\)-split: \(g_Y \simeq g_X \oplus \sigma^2(x) \delta_{ab} \hat{n}^a \hat{n}^b\), and distinguish \(\ast_X\) from \(\ast_Y\).
• \(H\) is the gauge group, \(N := \Omega^1(Y,\mathrm{ad})\) (\(\mathrm{ad} = \mathrm{ad}(P_H)\)), and \(G := H \ltimes N\). A generic gauge-affine variable is \(\omega = (\varepsilon, \eta) \in G\).
• \(A_0\) is the chosen background connection on \(Y\). From \(\omega\) we form \(B_{\omega}\) (the transported/rotated connection built from \(A_0\) and \(\varepsilon\)), its curvature \(F_B\), and the augmented torsion \(T\) (the covariant “difference” built from \(\eta\) and \(\varepsilon\) relative to \(A_0\)).
• The Shiab operator: \(\bullet_\varepsilon\).

§Main technical argument: bosons are not extra fields on \(X\)

In this instantiation, the bosonic sector is not a list of separate spacetime fields added to \(X\). The bosonic sector is the adjoint-valued geometry on \(Y\): the connection-like transport data, its curvature, and the torsion variable that makes the first-order theory gauge-covariant.

The basic transport variable is

$$ \omega = (\varepsilon,\vartheta) \in G = H \ltimes N, $$

where

$$ N = \Omega^1(Y,\mathrm{ad}(P_H)). $$

Thus \(\vartheta\) is already an adjoint-valued 1-form on \(Y\). The rotated connection is

$$ B_\omega := A_0 \cdot \varepsilon, $$

with curvature

$$ F_B \in \Omega^2(Y,\mathrm{ad}(P_H)). $$

A connection is not a tensor. We avoid treating the raw connection as the bosonic tensorial object. Instead, we make use of the augmented torsion

$$ T := \vartheta - \varepsilon^{-1} d_{A_0}\varepsilon, $$

which is an adjoint-valued 1-form with the correct covariance properties.

So the tensorial bosonic data are

$$ T \in \Omega^1(Y,\mathrm{ad}(P_H)), \qquad F_B \in \Omega^2(Y,\mathrm{ad}(P_H)). $$

That is the operational meaning of the phrase “Bosons are adjoint geometry.”

The bosonic degrees of freedom are the components, variations, and representations of \((T,F_B)\) in \(\mathrm{ad}(P_H)\), before any low-energy decomposition into familiar particle names.

We are not assuming any symmetry at this point. The ambient structure is \(\mathrm{Spin}(7,7)\) on \(Y\), and the bosonic variables live in the adjoint geometry associated to \(H\) and its transport extension \(G=H\ltimes N\). The Standard-Model-like sector, if it appears, must appear later as a consequence of decomposition, projection, spectrum, and pullback.

The fundamental bosonic sector is generated by \((T,F_B)\) on \(Y\).

There are no fundamental bosonic fields native to \(X\). What \(X\) sees are restrictions and pullbacks of \(Y\)-native objects.

§What “adjoint” means operationally

Operationally, “adjoint” means that bosonic quantities transform in the adjoint representation of the gauge/transport structure. The curvature \(F_B\) is adjoint-valued. The torsion variable \(T\) is adjoint-valued. Linearized bosonic fluctuations are therefore variations of adjoint-valued geometric objects:

$$ \delta T \in \Omega^1(Y,\mathrm{ad}(P_H)), \qquad \delta F_B \in \Omega^2(Y,\mathrm{ad}(P_H)). $$

So a “boson” is not, at this stage, a named particle. It is an excitation direction in the adjoint-valued geometry.

After pullback to \(X\), some of these directions may look like gauge bosons. Some may look like spin-connection-like degrees of freedom. Some may become heavy. Some may be projected out of the low-energy observer sector. But none of that should be assumed here.

§Assumptions vs Consequences

§Definitional

The ambient structure is \(\mathrm{Spin}(7,7)\) on \(Y\), with split signature \((7,7)\).

The physical spacetime is an immersed four-manifold

$$ \iota:X^4\hookrightarrow Y^{14}. $$

The bosonic variables are native to \(Y\), not \(X\).

The transport group is

$$ G=H\ltimes N, \qquad N=\Omega^1(Y,\mathrm{ad}(P_H)). $$

§Ansatz

The metric split is

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

The Shiab operator \(\bullet_\varepsilon\) is fixed.

The selectors \(E\) and \(\Theta_E\) are fixed.

Axial torsion is default and non-perturbative.

§Consequence

Bosons are not independent spacetime fields placed on \(X\).

They are adjoint-valued geometric degrees of freedom on \(Y\):

$$ T,\quad F_B,\quad \delta T,\quad \delta F_B. $$

The effective bosonic fields on \(X\) arise by restriction, decomposition, and pullback.

§Why this matters

• Once bosons are identified as adjoint geometry, we need a way to choose and describe directions inside \(\mathrm{ad}(P_H)\).
• Masses should not be put in by hand. Once bosonic directions are identified, their effective masses can come from overlap, torsion, and normal-direction localization.
• Fermions couple to the pulled-back and induced bosonic geometry. Chirality selection only becomes meaningful once we know which bosonic components survive on \(X\).

§Key takeaway

Bosons are not extra fields added to spacetime. They are the adjoint-valued geometry of transport on \(Y\). Spacetime \(X\) sees them only through restriction and pullback.

§Technical takeaway

$$ T\in\Omega^1(Y,\mathrm{ad}(P_H)), \quad F_B\in\Omega^2(Y,\mathrm{ad}(P_H)). $$

$$ \Upsilon_\omega=\bullet_\varepsilon(F_B)-\kappa_1T. $$

Bosonic observables on \(X\) are derived from \(\iota^\ast T,\ \iota^\ast F_B\), and their selected components.