The spatial complement and positive kinetic normalization

The small mass-frequency branch survives the leading response of the omitted normal modes. Their contribution includes a positive correction to kinetic normalization. It must be retained together with their change to the frequency operator.

This calculation uses the maintained full spatial parent. It supplies a specific improvement over a finite-band assumption, while leaving the infinite-complement remainder and the occupied supporting geometry open. The bounded angular completion then uses the same spatial structure to remove the leading mode-dependent departure from the simplex factors.

Eliminate the spatial modes before squaring

Let $P$ be the kernel of the linked geometric operator $D$, including its two canonical blocks, and $Q=I-P$. At a fixed surface parameter $\beta$, write the maintained first-order parent as

$$H=\Lambda D+M,\qquad M=-C\chi_1\beta-\frac12K\chi_1 -C\gamma(\chi_-h+\chi_+\bar h), \quad h=zB/2. \tag{1}$$

Here $\Lambda=R/L$, $\gamma=g\Lambda$, $K=(d+1-a)S-aI$, and $D$ is the geometric Jacobi operator. All its radial coefficients are shared by the particle indices at one depth. The normal plane remains an ordinary unbounded spatial plane.

Define

$$H_P=PMP,\qquad F=QMP,\qquad Y=D_Q^{-1}F. \tag{2}$$

The inverse exists because $|D_Q|\ge 2/\sqrt{d+1}$. Splitting an eigenfunction into $\psi=p+q$, the exact elimination is

$$\left[H_P-F^\dagger(\Lambda D_Q+QMQ-\omega)^{-1}F\right]p =\omega p. \tag{3}$$

The source $F$ has negative normal chirality, whereas $Y$ has positive chirality. This removes the term proportional to $\Lambda^{-1}$. Expansion through the first nonzero order gives

$$\boxed{ \left(H_P+\frac A{\Lambda^2}\right)p =\omega\left(I+\frac N{\Lambda^2}\right)p+\cdots, \qquad N=Y^\dagger Y\ge0,\quad A=Y^\dagger M_+Y.} \tag{4}$$

The same coefficient follows by normalizing the complete spatial wave: $q=-Yp/\Lambda+\cdots$, so $\|\psi\|^2=p^\dagger(I+N/\Lambda^2)p+\cdots$. The omitted field adds positive norm. It does not supply negative inertia.

Canonical normalization of (4) gives the Hermitian operator

$$\boxed{H_{\rm low}=H_P+\frac1{\Lambda^2} \left(A-\frac12\{N,H_P\}\right)+\cdots.} \tag{5}$$

The frequency shift computed from (4) agrees with (5), including both canonical blocks. The amplitude normalization fixes how this frequency enters the action; it is not a mass-fitting variable.

In contrast, direct compression of the full square gives

$$PH^2P=H_P^2+F^\dagger F. \tag{6}$$

The additional term is positive but does not describe the low spectral subspace after its normal dressing. Numerically, it overwhelms the tiny mass branch. Equation (6) therefore cannot replace (3).

The spatial inverse is explicit

The polarization part uses the exact identity

$$D(\mathcal R\Phi_l)=(K-k_l)\Phi_l, \qquad \mathcal R=-iS\alpha\cdot x,\quad k_l=d-1-2l. \tag{7}$$

The hopping part is obtained from the positive-chiral radial Green equations of $D^2$, retaining both radial components, their connection terms and the subtraction of the retained projection. This is an actual spatial inverse, rather than a scalar denominator assigned to an omitted mode.

The radial fields determine every entry of $N$ and $A$. Angular selection then restricts the off-diagonal block of (5): its row-minus-column shift lies between $-1$ and $2$. If an entry contains $\gamma^p$ and has shift $s$, it obeys $p-s\le2$.

Combining this rule with the small-branch endpoint vectors gives

$$\frac{\delta\omega_{n,d}}{\omega_{n,d}} =g^2\kappa_{n,d}(\gamma)+\text{higher orders}, \qquad g=\gamma/\Lambda, \tag{8}$$

where $\kappa_{n,d}$ has a finite large-$\gamma$ limit for each maintained index. The leading correction is not amplified by an inverse power of the tiny mass-frequency itself. A bound on the full operator norm would miss this angular information.

For example, at $\gamma=10000$, the positive surface branch gives:

The coefficients are not exactly common within a depth. The maintained parent therefore does not preserve the simplex law exactly after this correction. Its formal zero locations shift as well. This is a limitation of that full extension, not a change to the assigned particle indices.

What the larger spatial comparison establishes

A finite spatial Galerkin space containing $P$, the range of $F$, and the range of $Y$ reproduces the leading coefficient independently. It retains $6d$ complex basis vectors. The normal part has a positive gap, the complete matrix has the required chiral symmetry, and halving $g$ gives the expected fourth-order frequency remainder after (8).

The basis is deliberately sufficient to test this order. It is not an exhaustive discretization of the infinite normal plane. Independent radial refinement checks the spatial integrals used in the small eigenvalue; high-precision diagonalization alone would not establish that accuracy.

The result supports a positive, properly normalized reduced wave action under the stated parent. It does not supply the overall physical phase normalization, the same-field source balance, or the total inertia of a localized occupied particle. Those require the supporting action as well.

Verification

normal_complement_expansion.py constructs the spatial inverse matrices and the larger Galerkin comparison. normal_complement_expansion_checks.py passes 384 checks across all six depths, every maintained particle control and both surface signs. The report retains the formal-root shifts, independent radial refinements, exact executed source hashes and separate raw spatial arrays. The 384-check complementary-mode report contains the complete frequency, normalization and spatial refinement results.