Nested sector dynamics

What nesting supplies

All spatial sectors in V2 belong to one nested chain in the same infinite physical manifold. Their dimensions are ordinary macroscopic spatial directions. Describing their relationship through functions needs two steps: defining the lift of a function and checking the lift of its dynamics.

For an inclusion $i:Q_a\hookrightarrow Q_b$, the canonical function map is restriction, in the opposite direction:

$$i^*:C^\infty(Q_b)\longrightarrow C^\infty(Q_a),\qquad i^*F=F\circ i.$$

To lift functions from $Q_a$ to $Q_b$ cylindrically, specify a smooth projection $\pi:Q_b\to Q_a$ with $\pi\circ i=\mathrm{id}$. Then

$$J=\pi^*,\qquad Jf=f\circ\pi,\qquad i^*Jf=f$$

is an injective algebra map. In a product chart $(x,y)$, this is simply $Jf(x,y)=f(x)$. Nested inclusions alone do not supply a global $\pi$. The sector geometry audit already identifies this geometric limitation.

When a lower-sector equation is preserved

For a time-independent lift $J$, an exact evolution lift requires

$$H_bJf=JH_af$$

on a domain preserved by the relevant evolutions. For unbounded operators, domain compatibility is part of the assertion; a symbolic differential identity alone is not a complete operator theorem.

A direct differential example is

$$H_a=-\Delta_x+V_a(x),\qquad H_b=-\Delta_x-\Delta_y+V_a(x).$$

Since $\Delta_yJf=0$, the displayed intertwining identity holds on smooth cylindrical fields. If instead $H_b$ contains $\kappa xy$ in a two-coordinate example, then

$$H_bJf-JH_af=\kappa xyf(x),$$

which depends on $y$ and leaves the cylindrical class. Coordinate nesting survives, but this exact autonomous lower-sector evolution does not.

A constant extension is not square-integrable over a noncompact fibre of infinite volume. In that case the preceding calculation is a statement about fields or generalized modes, not a normalized Hilbert-space embedding. For a Hilbert-space version one can instead specify a normalized transverse eigenfunction $\chi$ and use $J_\chi f=f\chi$; a separated Hamiltonian then gives

$$H_bJ_\chi f=J_\chi(H_a+\epsilon)f,\qquad H_y\chi=\epsilon\chi.$$

Such an eigenfunction and its boundary conditions must actually exist. They are extra model data, not a derivation of cylindrical continuation.

Consequence for interactions

Nesting allows a shared-coordinate description; it does not guarantee that every interacting sector retains its isolated spectrum. A nonzero coupling can introduce backreaction, mix states, or shift energies. Use the explicit intertwining condition when exact inheritance is needed, and an effective reduction when it fails.

The spatial structure checks verify the separated and mixed differential examples.

Related

• Compression versus effective dynamics — explains why eliminating coupled variables differs from simply restricting an operator.
• Cylindrical barrier separation — applies the separated case to a concrete interaction question.
• Nested spatial tools — companion tools for spatial evolution and normalization.