• Definitions / Notation used
• Selection rules
  • Selection rule 0: pullback kills normal legs
  • Selection rule 1: \((E,\Theta_E)\) block selection is not optional
  • Selection rule 2: physical-corner projection in split signature \((7,7)\)
  • Selection rule 3: normal spectral suppression
  • Selection rule 4: covariance constraints
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

If you don’t put hard selection rules on the table, the ambient story is dead on arrival: \(Y\) is 14D, \(\operatorname{ad}\) is big, and \(\Omega^1(Y,\operatorname{ad})\) is enormous. The default expectation would be a zoo of light 4D vector fields, which we do not see. So this post is about why most modes on \(Y\) are either (i) identically invisible on \(X\) or (ii) too heavy / too decoupled to function as observed bosons.

§Definitions / Notation used

• Ambient bosonic mode: any adjoint-valued 1-form field \(U \in \Omega^1(Y,\operatorname{ad})\) expressed in tensorial variables \((T, \delta T\), or other pullbackable 1-forms in the model).
• Decomposition along \(\iota(X)\): write $$U|_{\iota(X)} = U_\mu(x,n) dx^\mu + U_a(x,n) \hat n^a,$$ with \(\mu\) tangent, \(a\) normal.
• \(X\)-visible projection: the operation $$U \mapsto \iota^\ast(P_{\mathrm{obs}} U),$$ where \(P_{\mathrm{obs}}\) includes (i) killing normal legs, (ii) \(E/\Theta_E\) block selection, (iii) physical-corner projection, and (iv) normal-mode truncation to the light sector.
• “Invisible mode”: a mode with nonzero ambient existence (\(U \ne 0\) on \(Y\)) but \(\iota^\ast(P_{\mathrm{obs}} U)=0\) or dynamically decoupled in the low-energy \(X\)-effective theory.

§Selection rules

§Selection rule 0: pullback kills normal legs

This is the trivial one, but it must be stated. Along \(\iota(X)\), \(TY\) splits as \(TX \oplus N_\iota\). The pullback \(\iota^\ast\) only sees \(TX\) legs. Therefore, any ambient mode of the form $$U = U_a(x,n) \hat n^a$$ is identically invisible as a 4D 1-form: $$\iota^\ast U = 0.$$

Example A (exists on \(Y\), invisible on \(X\)): pure normal-leg torsion mode

Take an ambient fluctuation of augmented torsion with only normal legs: $$\delta T^{(A)} = \tau_a(x) \varphi_0(n) \hat n^a,$$ with \(\tau_a(x)\) \(\operatorname{ad}\)-valued coefficients and \(\varphi_0\) the normal ground state. This is a perfectly legitimate tensor on \(Y\) and can backreact on the geometry there. But as a candidate 4D boson it is dead: $$\iota^\ast(\delta T^{(A)}) = 0.$$ So it does not produce a 4D vector field, regardless of how light its normal profile is.

Now the important point: this baseline selection is not sufficient. You can still have \(U_\mu(x,n) dx^\mu\) components, and those could in principle show up. The remaining mechanisms are what prevent a flood.

§Selection rule 1: \((E,\Theta_E)\) block selection is not optional

This instantiation fixes an adjoint projector \(E\) selecting the “gravitational block”, and fixes \(\Theta_E\) as the calibrator that saturates the 10 normal directions and leaves one \(X\)-slot, with \(D\Theta_E=0\). Operationally, this means the action and field equations only “see” the components of curvature and torsion that survive the composite selection implemented by these objects.

Translated into the boson story: even if an ambient mode has \(TX\) legs, it is invisible unless its \(\operatorname{ad}\) indices lie in the selected block and it can couple through the \(\Theta_E\)-trace structure. If you denote the \(\operatorname{ad}\) decomposition as $$\operatorname{ad} = \operatorname{ad}_E \oplus \operatorname{ad}_\perp \quad (E\text{-selected plus complement}),$$ then the observable boson candidates live in \(\Omega^1(X,\operatorname{ad}_E)\) after pullback, while \(\operatorname{ad}_\perp\) modes are projected out of the dynamical corner we identify with observed physics.

Example B (exists on \(Y\), “projected away”): tangential leg, wrong \(\operatorname{ad}\) block

Consider a \(TX\)-leg ambient mode $$U^{(B)} = u_\mu(x) \varphi_0(n) dx^\mu \otimes X_\perp, \quad \text{with } X_\perp \in \operatorname{ad}_\perp.$$ This is not killed by pullback at the level of differential forms: $$\iota^\ast U^{(B)} = u_\mu(x) dx^\mu \otimes X_\perp.$$ But it is killed by the operational definition of the observable corner: $$P_{\mathrm{obs}}(U^{(B)}) = 0$$ because \(E\) annihilates \(X_\perp\) and \(\Theta_E\) enforces which contractions enter the Shiab-selected dynamics. In other words: the mode exists as geometry on \(Y\), but it is not part of what the instantiation calls “the observed bosonic sector.”

§Selection rule 2: physical-corner projection in split signature \((7,7)\)

\(\mathrm{Spin}(7,7)\) is not Euclidean comfort; it is defining and it forces discipline. Not every direction in the ambient gauge algebra corresponds to a physical propagating mode with positive energy in the 4D effective theory. The project therefore uses a “physical corner” projection (the same one already motivated earlier when discussing stability/ghost avoidance in the split-signature setting): you restrict propagating bosonic modes to the appropriate compact or Cartan-involution-selected subspace, so that the effective kinetic and mass forms have the correct sign.

Operationally: even within \(\operatorname{ad}_E\), you do not keep every generator. You keep the ones whose induced pairing (after the Cartan-involution-adjusted inner product) yields a positive kinetic term on \(X\). The rest are either non-propagating, constrained, or reinterpreted as nonphysical directions in the effective description.

This is not aesthetic; it is forced if you want the low-energy \(X\) theory to be unitary in the usual sense.

§Selection rule 3: normal spectral suppression

Even after you keep \(TX\) legs and the correct \(\operatorname{ad}\) sector, you are still not done: most modes are heavy because they live in excited normal shells. Expand a tangential-leg ambient mode in Hermite–Gaussian normal profiles: $$U_\mu(x,n) = \sum_k u_{\mu,k}(x) \varphi_k(n).$$ Typically, the effective 4D mass squared includes a contribution that grows with the normal excitation number \(|k|\) (the “oscillator energy” of the normal bundle). In plain terms: the tower is there, but it is heavy, and at low energies you only see the ground state and perhaps a very small set of near-ground excitations.

You also get exact zeros from overlap/parity. Many couplings and mixings require integrals of products of Hermite functions; those vanish unless the parity/occupation selection rules are satisfied. So “doesn’t show up” can mean either “too heavy” or “cannot couple”.

Example C (exists on \(Y\), decouples on \(X\)): tangential leg, high normal excitation

Take $$U^{(C)} = u_\mu(x) \varphi_k(n) dx^\mu \otimes X_E, \quad \text{with } |k| \gg 1 \text{ and } X_E \in \operatorname{ad}_E.$$ Pullback alone doesn’t kill it: $$\iota^\ast U^{(C)} = u_\mu(x) dx^\mu \otimes X_E.$$ But in the effective 4D action, its mass receives an additive contribution from the normal excitation, so it sits far above the observable scale and decouples. Moreover, if the couplings require overlap with the normal ground state, the relevant integrals can vanish outright for odd-parity \(k\).

§Selection rule 4: covariance constraints

In this transport-based GU instantiation, the only mixing terms you can write must be gauge-covariant under the inhomogeneous symmetry \(G=H\ltimes N\) and must be built from tensorial objects \((T, F_B\), selectors). This forbids a large class of would-be mass/mixing terms that would otherwise activate extra 4D bosons.

Concretely: if two candidate modes live in different \(E\)-block sectors, or in different normal-mode parity sectors, or transform incompatibly under the residual covariance, then there is no admissible bilinear that mixes them in the effective action. This is a selection rule “by symmetry of the allowed tensors,” not a postulated phenomenological constraint.

§Assumptions vs Consequences

§Assumptions

• Observable fields are defined by a specific projection \(P_{\mathrm{obs}}\) that is compatible with the fixed \((E,\Theta_E)\) choice and the split-signature physical corner.
• Normal modes are organized in a Hermite–Gaussian basis on \(N_\iota\), with a meaningful low-energy truncation.
• The effective low-energy sector is identified with modes that both survive projection and remain light under normal excitation.

§Consequences

• Pure-normal-leg ambient modes are exactly invisible on \(X\) (\(\iota^\ast\) kills them).
• Many tangential-leg modes are operationally removed because they lie outside the \(E/\Theta_E\)-selected block or outside the physical corner.
• The remaining tangential-leg, \(E\)-selected modes still mostly decouple because the Hermite tower is heavy and/or overlap selection rules force their couplings to vanish.
• The net effect is that “4D bosons” are not generic; they are a tightly filtered residue.

§Why this matters

• The fermion sector will reuse the same selection logic, but with an additional torsion-driven chirality filter. The point is to show the fermion spectrum is also a residue, not an assumption.
• \(\sigma\) controls the normal geometry, hence the spacing of the Hermite tower and the strength of overlap integrals. Stabilizing \(\sigma\) is therefore stabilizing the selection rules themselves.

§Key takeaway

“Normal legs vanish” is only the first gate. The observed 4D boson sector is the residue after at least four filters: pullback leg-killing, \((E,\Theta_E)\) block selection, split-signature physical-corner projection, and normal spectral suppression (plus covariance-forbidden mixing).

§Technical takeaway

A compact way to say it is: a \(Y\)-mode \(U\) contributes a 4D boson iff $$\iota^\ast(P_{\mathrm{obs}} U) \ne 0$$ and its normal expansion has support in the low Hermite shells with nonzero overlap against the \(\Theta_E\)-selected contractions; otherwise it is either identically invisible (normal legs), projected away (wrong block/ghosty corner), or decoupled (heavy/spectrally orthogonal).