Rotations of nested spatial sectors

Summary: A rotation preserving every fixed member of the nested spatial chain has a restricted block structure, while rotating the whole chain requires the moving embeddings in the projected dynamics.

The same spatial chain must support the interaction

Local geometric coupling operators have to fit together on the one nested manifold. Consider a common point with a nested chain of tangent spaces

$$V_{d_1}\subset V_{d_2}\subset\cdots\subset V_{d_N}, \qquad d_1<\cdots<d_N=D. \tag{1}$$

All directions are ordinary macroscopic spatial directions. The orthogonal differences between these spaces record their relation to other sectors; they give no particle an inherent division into different kinds of space. The following theorem concerns rotations of this finite tangent flag. It does not classify all interactions, differential operators, curved global embeddings, or dynamical sector projectors of the master field.

Rotations preserving every fixed member

Let $P_r$ be the orthogonal projector onto $V_{d_r}$ and $K^T=-K$ a real spatial rotation generator. Preservation of $V_{d_r}$ is equivalent to

$$(I-P_r)KP_r=0.$$

Skew symmetry also annihilates its transpose block, so equivalently

$$[K,P_r]=0. \tag{2}$$

Set $P_0=0$ and $E_r=P_r-P_{r-1}$. Nesting makes these mutually orthogonal projectors. If (2) holds for every $r$, $K$ commutes with every $E_r$ and hence has precisely the form

$$\boxed{K=\bigoplus_{r=1}^N K_r,\qquad K_r\in\mathfrak{so}(d_r-d_{r-1}).} \tag{3}$$

Conversely every such block diagonal generator preserves the flag. Therefore the Lie algebra dimension is

$$\sum_{r=1}^N\frac{(d_r-d_{r-1})(d_r-d_{r-1}-1)}2. \tag{4}$$

For the working depths $(2,3,4,5,6,10)$, the increments are $(2,1,1,1,1,4)$. The algebra is

$$\mathfrak{so}(2)\oplus\mathfrak{so}(4), \qquad \dim=1+6=7. \tag{5}$$

At depths three through six, its only possible nonzero tangential rotation is the shared two-plane. The four-dimensional final block acts in directions outside $V_6$. This result depends on requiring all listed sectors to stay fixed under the same infinitesimal rotation.

Consequence for a rotation-neutral construction

Rotation-neutral spatial modes uses two equal rotation planes at depth five and three at depth six. In an adapted six-dimensional frame these are generated by

$$K_5=J_{12}+J_{34},\qquad K_6=J_{12}+J_{34}+J_{56}, \tag{6}$$

where $J_{ab}=e_ae_b^T-e_be_a^T$. The first generator moves the fixed three-dimensional member; the second also moves the five-dimensional one. For the complete equal-plane generator in an even containing depth,

$$\|[K,P_r]\|_F^2= \begin{cases}0,&r\text{ even},\\2,&r\text{ odd},\end{cases} \tag{7}$$

for the coordinate flag with the last incomplete plane crossing its boundary. The crossing term cannot be discarded while calling (6) the restriction of a rotation that fixes every member of the chain.

Indeed, under the fixed-flag assumption a nonzero five-dimensional spin generator has only one plane: $Q_5=\kappa H_{12}/2$. Since $H_{12}^2=I$, it has no neutral spinor subspace. Including the ordinary single-valued orbital functions does not produce a neutral mode of this one-plane rotation: the orbital weights are integers and the spin weights are half integers. Twists or further field structure change that premise.

This locates a definite requirement for the two-plane cancellation: it must come from dynamics that move the nested embeddings, a derived effective operator, or other specified geometric structure beyond (3).

Moving the entire chain preserves nesting

Let $R(t)$ be an orthogonal spatial motion with $\dot R=K(t)R$, and transport every sector together:

$$P_r(t)=R(t)P_r(0)R(t)^T, \qquad \dot P_r=[K,P_r]. \tag{8}$$

Then $P_r(t)P_s(t)=P_r(t)$ for $r\leq s$, and every rank is unchanged. Thus the full chain can rotate while remaining nested. Preserving the nesting does not require preserving each member's original position.

For $u(t)\in\operatorname{ran}P_r(t)$, differentiate $P_ru=u$:

$$(I-P_r)\dot u=\dot P_ru=(I-P_r)Ku. \tag{9}$$

This normal component describes motion of the embedding in the same physical space. It is not an additional spatial dimension or a separate interaction particle. Omitting it freezes part of the proposed geometry.

The induced mode equation retains the moving basis

Spatial tangent projectors and projectors on field amplitudes are different objects. Once a physical field-mode inclusion has been derived, write it as an isometry $I(t)$ with $I^\dagger I=I_{\rm amp}$, where $I_{\rm amp}$ is the amplitude-space identity. Decompose the one field into

$$\Psi=I(t)a+h,\qquad I^\dagger h=0, \qquad i\dot\Psi=H(t)\Psi.$$

Differentiating $a=I^\dagger\Psi$ gives the exact projected row

$$\boxed{i\dot a= \left(I^\dagger HI-iI^\dagger\dot I\right)a +I^\dagger Hh+i\dot I^\dagger h.} \tag{10}$$

The connection term $-iI^\dagger\dot I$ is Hermitian. The last term records motion relative to the omitted modes. Neither can be suppressed solely because $h$ was initially zero. The complement equation must preserve that condition, or its response must be retained or eliminated consistently.

For a complete unitary basis $U(t)$, the corresponding generator is $U^\dagger HU-iU^\dagger\dot U$. A change of description transforms probes as well; it does not create a new physical motion or coupling by itself.

Use in generalized coupling

The fixed-chain result provides a precise check on proposed common rotation generators. The moving-chain result supplies a constructive alternative when the higher spatial pattern rotates the nested geometry. Equation (10) then identifies the additional terms that a downward source calculation must include.

Neither result derives a preferred embedding motion, electromagnetic charge, or the occupied five-dimensional neutral states. They keep the proposed cancellation compatible with the single nested chain of spatial submanifolds while making its next dynamical calculation explicit.

Verification

spatial_coupling_mechanism_checks.py solves all fixed-flag commutator constraints as a linear system, obtains the seven-dimensional stabilizer, checks its restrictions at every depth, and tests moving-projector ranks and derivatives. The projected field equation is also checked against direct evolution and basis differentiation.

Related

• Rotation-neutral spatial modes — supplies the equal-plane cancellation whose compatibility is tested here.
• Descending photon source — gives the fixed-mode projected row extended by (10).
• Nested sector dynamics — states the spatial lift and intertwining conditions needed for a closed reduced evolution.