The supporting multiplet's stiffness depends on the full spatial deformation

The complete positive-frequency multiplet can cancel every centered source after time averaging. That balance alone does not establish an energy minimum. The second variation must include the actual change of the spatial geometry, not only a linear perturbation of its projected matrix.

For the separate bounded multiplier $h=z/\sqrt{1+r^2}$, all tested coordinate curvatures of the balanced averaged energy are positive. For the density-gradient multiplier, the full spatial term produces negative curvatures, including the endpoint direction at every maintained depth. This distinction was absent from the linear matrix model.

These are energy curvatures at fixed positive canonical populations. They are not particle inertias or calculated growth rates. The averaged state has not been shown to be a pointwise coherent particle solution.

The positive linear-matrix result

Let $s_j(\alpha)$ be the positive singular values of $T(\alpha)$. Use the source-balancing populations $q_j=s_j(0)$, kept fixed during a variation. The averaged energy is

$$\mathcal E(\alpha)=\sum_jq_js_j(\alpha). \tag{1}$$

Dividing all populations by $\sum_js_j(0)$ gives unit canonical charge and divides every energy derivative by that same positive number. The complete trace identity gives $\mathcal E'(0)=0$ for each centered direction.

For a strictly linear matrix path $T(\alpha)=T_0+\alpha X$, twice differentiating $\sum_js_j^2=\operatorname{Tr}(T^\dagger T)$ gives

$$\boxed{\mathcal E''_{ \rm linear}=\|X\|_F^2-\sum_j(s_j')^2\ge0.} \tag{2}$$

To see the sign directly, take $T_0=U\operatorname{diag}(s_j)V^\dagger$ and $Z=U^\dagger XV$. Then $s_j'=\operatorname{Re}Z_{jj}$, so

$$\mathcal E''_{\rm linear} =\sum_{i\ne j}|Z_{ij}|^2+\sum_j(\operatorname{Im}Z_{jj})^2. \tag{3}$$

No positivity of a discarded signed wave response is being assumed. Equation (3) is an exact finite-dimensional identity for the stated positive populations.

The missing nonlinear spatial term

The actual carrier follows $T=T_0+\alpha T'+\alpha^2T''/2+\cdots$. The same trace calculation now gives

$$\boxed{\mathcal E''= \|T'\|_F^2-\sum_j(s_j')^2 +\operatorname{Re}\operatorname{Tr}(T_0^\dagger T'').} \tag{4}$$

The last term is not constrained by (3). The complete second spatial calculation computes it from the linked second geometry, moving normal modes and the coefficient's own derivative. At the radial configuration,

$$\operatorname{Re}\operatorname{Tr}(T_0^\dagger T'') =-\frac12\sum_l\left(\beta+l-\frac{d-1}{2}\right)(K_P'')_{ll} +\gamma^2\sum_lh_lh_l''. \tag{5}$$

The polarization part is fixed independently by $(K_P'')_{ll}=-2k(F^{\rm normal}_{x_k y_k})_{ll}$. The adjacent $h_l''$ requires all six terms of the full second hopping derivative. Omitting the nonlinear geometry or the explicit multiplier derivative changes (5).

The cross-depth comparison

The complete calculation keeps every real centered order $k=2,\ldots,d$, every maintained particle control and both surface signs. At $\gamma=10000$, representative endpoint values are:

The entries are $\mathcal E''/(\mathcal Q M_d)$, using the positive actual geometric mode metric. This normalization cannot change a sign. For example, the depth-three density-gradient endpoint has a positive linear contribution about 1.91, but a nonlinear spatial contribution about $-4.56\times10^7$ before division by $\mathcal Q M_d$. The omitted term determines the result.

All 96 real coordinate curvatures are positive for each fixed-multiplier carrier. The density-gradient case has 94 negative curvatures among the same 96 controls; its endpoint is negative in every control. These statements concern the actual finite hopping value, not an extrapolation from the leading simplex approximation.

A positive value in each coordinate direction does not prove that the full mixed Hessian is positive. A negative value in even one direction, however, is enough to exclude a local minimum of this averaged energy under the declared variations. The density-gradient replacement therefore preserves the earlier spectral numerator and first response pattern but loses ground in this supporting-state test. It cannot be treated as a complete simplification of a viable particle action.

The retained geometric cost is constant along its exact flat linked moduli. Adding only that constant or a radius-only term cannot change these fixed-radius shape curvatures. Other physical stresses, nonlinear binding terms or curved spatial configurations must be included explicitly before they can alter the result. No such contribution has been inferred from the sign in this table.

Verification

full_spatial_stiffness_checks.py compares (4) with independent 180-digit finite differences of the complete weighted singular-value energy. Separate finite spatial integrals verify selected adjacent hopping second derivatives, including their explicit coefficient changes. Radial-grid and outer-domain refinements test the sensitive low- and high-depth cases.

The 510-check report also verifies the second polarization identity and the full Cartesian second source across all five maintained depths. Raw second geometry and mode data are retained separately with the exact executed source hashes.

The depth-two extension repeats these spatial checks over all six active depths. The fixed-multiplier controls retain 102 positive real directions each; the density-gradient control has 100 negative directions among 102. These later totals include W, Z and Higgs controls at depth two.