The first projected polarization source vanishes at the linked radial configuration. Its second derivative does not vanish. It is determined by the moving normal curvature already obtained from the full spatial inverse. This gives another exact relation between the spectral and canonical-response calculations.

The adjacent hopping has additional second derivatives of its texture, connection and mode basis. Keeping those terms is necessary before using a linear matrix model to infer the stability of the actual spatial shape. The calculation uses the declared ordinary unbounded normal plane at depths 3, 4, 5, 6 and 10.

The second polarization follows from the same angular generator

For the common normal carrier, the exact compressed identity is

KP=−2PJP,J=L−d−12S,JΦl=jlΦl,jl=l−d−12.(1) K_P=-2P\mathcal JP,\qquad \mathcal J=L-\frac{d-1}{2}S, \qquad\mathcal J\Phi_l=j_l\Phi_l, \quad j_l=l-\frac{d-1}{2}. \tag{1}

Take one real polynomial coordinate xkx_k, with an orthonormal kernel basis transported so that Φ†Φ′=0\Phi^\dagger\Phi'=0 along this path. At the radial reference point,

Φ†Φ′′=−Φ′†Φ′,(KP′′)il=−4⟨Φi′,(J−ji+jl2)Φl′⟩.(2) \Phi^\dagger\Phi''=-\Phi'^\dagger\Phi',\qquad \boxed{(K_P'')_{il}=-4\left\langle\Phi_i', \left(\mathcal J-\frac{j_i+j_l}{2}\right)\Phi_l'\right\rangle.} \tag{2}

To derive (2), differentiate the matrix element in (1) twice. The complementary part of Φ′′\Phi'' drops out because J\mathcal J preserves the reference kernel. The in-kernel part is fixed by the normalization equation above. Thus no unspecified second basis derivative is needed for this identity.

Each Φl′\Phi_l' has angular labels l+kl+k and l−kl-k. Let their squared norms be Nl+,Nl−N_l^+,N_l^-, as in the moving normal basis. Equation (2) gives

(KP′′)ll=−4k(Nl+−Nl−)=−2k(Fxkyknormal)ll.(3) \boxed{(K_P'')_{ll}=-4k(N_l^+-N_l^-) =-2k(F^{\rm normal}_{x_k y_k})_{ll}.} \tag{3}

Possible off-diagonal terms require i−l=±2ki-l=\pm2k. Their shared intermediate angular label is exactly (i+l)/2(i+l)/2, so the factor in parentheses in (2) is zero. The complete second matrix is therefore diagonal in this transport convention. Both original and geometric carriers obey the result. The curvature is a canonical geometric quantity; this relation does not identify it with particle inertia.

The actual second geometric mean

Write the linked polynomial path as

P(z)=zd+αzd−k,u=u0+αfkcos⁡kθ+α2(g0,k+g2,kcos⁡2kθ)+O(α3).(4) P(z)=z^d+\alpha z^{d-k},\qquad u=u_0+\alpha f_k\cos k\theta +\alpha^2(g_{0,k}+g_{2,k}\cos2k\theta)+O(\alpha^3). \tag{4}

Here W=e−uPW=e^{-u}P, A=∇⊥uA=\nabla^\perp u, and Δu=2(1−∣W∣2)\Delta u=2(1-|W|^2). The radial second mean solves

(−Δ+4w2)g0,k=2Uk2,Uk=w(r−k−fk),(5) (-\Delta+4w^2)g_{0,k}=2U_k^2, \qquad U_k=w(r^{-k}-f_k), \tag{5}

with regular and decaying boundary conditions. Differentiating the actual coefficients twice gives their angular means

W0′′=w(fk22−2g0,k−r−kfk),Aθ,0′′=2g0,k′,ρ0′′=∂α2∣W∣2∣mean=2Uk2−4w2g0,k.(6) \begin{aligned} W''_0&=w\left(\frac{f_k^2}{2}-2g_{0,k}-r^{-k}f_k\right),\\ A''_{\theta,0}&=2g_{0,k}',\\ \rho''_0&=\partial_\alpha^2|W|^2\big|_{\rm mean} =2U_k^2-4w^2g_{0,k}. \end{aligned} \tag{6}

The first line is the radial coefficient multiplying eidθe^{id\theta}. For the density-gradient multiplier, h0′′=(ρ0′′)′eiθh''_0=(\rho''_0)'e^{i\theta}. The 2k2k harmonics do not contribute to a diagonal polarization or adjacent hopping second entry: their angular labels cannot match the two unperturbed external modes. They are excluded by this selection rule, not presumed absent from the full geometric field.

The second moving mode and complete hopping derivative

Differentiate DΦl=0D\Phi_l=0 twice:

DΦl′′=−D′′Φl−2D′Φl′,PΦl′′=−∑iΦi⟨Φi′,Φl′⟩.(7) D\Phi_l''=-D''\Phi_l-2D'\Phi_l',\qquad P\Phi_l''=-\sum_i\Phi_i\langle\Phi_i',\Phi_l'\rangle. \tag{7}

The complementary solution uses the same gapped spatial inverse as the first derivative. Its radial mean is

(Φl′′)mean=−D−1(D′′Φl+2D′Φl′)mean−(Nl++Nl−)Φl.(8) (\Phi_l'')_{\rm mean} =-D^{-1}(D''\Phi_l+2D'\Phi_l')_{\rm mean} -(N_l^++N_l^-)\Phi_l. \tag{8}

The second source retains both first angular channels, which return to the original label under the opposite first variation. In the positive radial component convention of the moving-basis note, the direct D′′ΦlD''\Phi_l contribution is

(tW0′′Ul+(e+1/2)Aθ,0′′Vl,−tW0′′Vl+(e−1/2)Aθ,0′′Ul).(9) \left(tW''_0U_l+(e+1/2)A''_{\theta,0}V_l, -tW''_0V_l+(e-1/2)A''_{\theta,0}U_l\right). \tag{9}

The twice-iterated first source 2D′Φl′2D'\Phi_l' must be added to (9). The in-kernel term in (8) is equally essential for normalization.

For adjacent entries hl=⟨Φl+1,hΦl⟩h_l=\langle\Phi_{l+1},h\Phi_l\rangle, the complete result is

hl′′=⟨Φl+1′′,hΦl⟩+⟨Φl+1,hΦl′′⟩+2⟨Φl+1′,hΦl′⟩+2⟨Φl+1′,h′Φl⟩+2⟨Φl+1,h′Φl′⟩+⟨Φl+1,h′′Φl⟩.(10) \begin{aligned} h_l''={}&\langle\Phi_{l+1}'',h\Phi_l\rangle +\langle\Phi_{l+1},h\Phi_l''\rangle +2\langle\Phi_{l+1}',h\Phi_l'\rangle\\ &+2\langle\Phi_{l+1}',h'\Phi_l\rangle +2\langle\Phi_{l+1},h'\Phi_l'\rangle +\langle\Phi_{l+1},h''\Phi_l\rangle. \end{aligned} \tag{10}

The last three terms are present for the density-gradient rule. They vanish only for the separate fixed-multiplier control. Formula (10) retains the second geometry, two first basis changes, the normalization of the second modes and the coefficient's own response.

At the radial configuration the parts needed for the spectral energy trace consequently have

(T′′)ll=−12(KP′′)ll,(T′′)l+1,l=γhl′′.(11) (T'')_{ll}=-\frac12(K_P'')_{ll},\qquad (T'')_{l+1,l}=\gamma h_l''. \tag{11}

Other angular entries cannot contribute to Re⁡Tr⁡(T†T′′)\operatorname{Re}\operatorname{Tr}(T^\dagger T'') there. This is the second spatial term needed when testing an occupied frequency multiplet's averaged energy.

Verification

full_spatial_stiffness.py uses the independently computed linked second profile and full positive Green inverses. The explicit second polarization variation is compared with (3); the complementary second modes satisfy their normalization condition. Independent Cartesian matrices and angular projection check the complete source in (7).

full_spatial_stiffness_checks.py also evaluates finite differences of the actual spatial field expansion for selected adjacent entries in (10), and refines radial grids and outer domains. The omitted second angular harmonics cannot affect these entries at second order. The associated averaged energy test retains the full nonlinear contribution (11), rather than replacing it by a fixed matrix coefficient. The 510-check report passes all five maintained depths, both original and geometric carriers, and the density-gradient multiplier control. Its energy consequence is described in the supporting multiplet's stiffness.

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.