The conformal Hessian of the whole-depth current carrier

The radial linked-carrier calculation and the whole-depth current carrier are distinct candidate actions. The latter has an explicit conformal metric dependence, so its geometric second variation can be calculated without borrowing the linked carrier's radial curvature. The calculation starts with uniform log radius and the same smooth shape direction $q=(x_1^2+x_2^2)/R^2$ for every occupied harmonic, then retains nine common zonal metric directions through scalar degree 16. For the spinor carrier, a pointwise criterion covers every smooth conformal direction. The round $d$-sphere is a stated candidate for the actual ordinary spatial geometry, not a compactified extra dimension.

One transported generating action

At the reference radius $R_0=1$, use the saturated negative-current branch of the homogeneous current action, with $\eta=0$. Its field current is strictly negative on the retained states, so the current root is differentiable along the small paths considered here. The common curvature law is the existing inverse trial

$$g(\kappa)=g_0\frac{\kappa^4}{1+\kappa^4},\qquad g(1)=g_0/2,\quad g'(1)=g_0,\quad g''(1)=-g_0. \tag{1}$$

No particle index receives its own $g_0$. The two tested values, 1000 and 10000, are dimensionless controls rather than measured physical strengths. At reference coupling $g_0/2$, let $u,v$ be the normalized left and right singular vectors of the finite component chain. Define

$$B=\sum_j u_jv_j,\qquad A=\sum_j(j-d/2)u_jv_j,\qquad s=-\sum_{j=1}^{d-1}u_jv_{j-1}>0,\qquad \beta=n+d/2. \tag{2}$$

For spinors at depths $d\ge3$, canonical field transport gives the fixed-field generating energy

$$E_1(\sigma)=N\left[\beta B\langle e^{-\sigma}\rangle_\rho +A\langle\kappa_\sigma\rangle_\rho -s\langle g(\kappa_\sigma)\rangle_\rho\right], \tag{3}$$

where $\rho$ is the normalized spatial density. For the actual depth-two electric form carrier, the conformal transport differs:

$$E_1^{(2)}(\sigma)=N\left[2nB\langle e^{-\sigma}\rho_e\rangle +\left\langle\kappa_\sigma \left((A+B/2)\rho+\frac B2(\rho_e-\rho_o)\right)\right\rangle -s\langle g(\kappa_\sigma)\rho\rangle\right], \quad\rho=\rho_e+\rho_o. \tag{4}$$

The form densities have total integral one. Equation (4) retains the even/odd form weights instead of assigning spinor transport to the photon sector. Here $N=\|\Psi\|^2$ is the wave norm. At $\sigma=0$, both expressions equal the positive smallest singular frequency $\omega_{n,d}$ times $N$. The full positive wave action gives energy $\mathcal Q\omega$ at fixed canonical phase charge; the derivatives here are those of its stated generating action on a fixed spatial profile. They are not a full positive-action Hessian.

Exact two-direction derivatives

Put $\sigma=t_0+t_1q$ and write $f_0=1$, $f_1=q$. On the unit sphere,

$$\widehat\Delta q=2(d+1)q-4,\qquad |\widehat\nabla q|^2=4q(1-q),\qquad \kappa_\sigma=e^{-\sigma} \sqrt{1+\frac2d\widehat\Delta\sigma -\frac{d-2}{d}|\widehat\nabla\sigma|^2}. \tag{5}$$

For either basis function let $k_i=-f_i+\widehat\Delta f_i/d$. The second derivative of the curvature scale at the reference is

$$k_{ij}=f_if_j- \frac{f_i\widehat\Delta f_j+f_j\widehat\Delta f_i}{d} -\frac{(\widehat\Delta f_i)(\widehat\Delta f_j)}{d^2} -\frac{d-2}{d}\widehat\nabla f_i\cdot\widehat\nabla f_j. \tag{6}$$

For spinors, the force vector and bare geometric Hessian per unit $N$ follow directly from (1), (3), (5) and (6):

$$F_i=-\beta B\langle f_i\rangle_\rho +(A-sg_0)\langle k_i\rangle_\rho, \tag{7}$$

$$G_{ij}=\beta B\langle f_if_j\rangle_\rho +(A-sg_0)\langle k_{ij}\rangle_\rho +s g_0\langle k_ik_j\rangle_\rho. \tag{8}$$

The script evaluates the depth-two derivatives from (4), using its actual even and odd weights. All angular averages reduce to moments of $q\sim\operatorname{Beta}(n+1,(d-1)/2)$, so the entries are exact finite-moment evaluations after the numerical singular vectors are fixed. The same script also differentiates the unexpanded energy by independent spatial quadrature for one maintained depth-two form and one depth-six spinor. This tests the curvature expansion and the distinct form transport.

The same derivatives apply to an arbitrary smooth real conformal function $f$. Write $T=\beta B$, and $\Delta f=\widehat\Delta f$. Equation (8) has the pointwise form

$$\begin{aligned} G[f,f]=\int\rho\Big[& (T+A)f^2-\frac{2A}{d}f\Delta f +\frac{2sg_0-A}{d^2}(\Delta f)^2\\ &+\frac{(sg_0-A)(d-2)}d|\widehat\nabla f|^2 \Big]dV. \end{aligned} \tag{8a}$$

The $(f,\Delta f/d)$ quadratic matrix is positive when

$$T+A>0,\quad 2sg_0-A>0,\quad (T+A)(2sg_0-A)-A^2>0. \tag{8b}$$

All three inequalities, and the positive gradient coefficient, hold at every maintained spinor mode and both common strengths. Since $\rho>0$ almost everywhere, (8a) proves positive bare fixed-profile conformal curvature for every nonzero smooth $f$, including non-zonal directions. It does not cover traceless metric changes or the depth-two form carrier.

One inverse background and its common threshold

A retained same-field background coordinate $x=v+\xi$ can be coupled to the complete reference metric derivative of this same conformal current action. Distinguish the generating functional $W$ from the complete positive wave action. For a nearby metric $g$, define

$$W[\Psi,g,\xi] =E_1[\Psi,g]-\frac{v+\xi}{v} D_gE_1[\Psi,g_*]\bigl(g-g_*\bigr). \tag{9}$$

Here the last term is the linear functional of the metric displacement at the round reference $g_*$; its reference operator is common to every harmonic. At $g=g_*$, $\xi=0$, it vanishes along with its field variation. Its metric force cancels $D_gE_1$, and the field–metric mixed block cancels as an operator identity wherever the current branch is differentiable. The reference frequencies remain $\omega_{n,d}$.

Take $F_W=\delta W/\delta\Psi^*$ and the candidate complete action $L_+=\|\dot\Psi\|^2-\|F_W\|^2+L_{g,\xi}^{\rm kinetic}-a\xi^2/2$, with positive geometric and background kinetic terms left as inputs. On a positive standing branch, the phase charge is $\mathcal Q=2\omega N$, and its wave energy at the reference is $\mathcal Q\omega$. Because $D_gF_W=0$ at the reference for every field, the fixed-profile geometric second derivative of the squared wave potential is $\mathcal Q G$, while its geometry–background derivative is $-\mathcal Q F/v$. The quadratic field–geometry block also vanishes. This uses the degree-two homogeneity of $W$ on the specified current branch; it does not assume that the generating energy itself is the physical positive-wave potential.

At unit phase charge $\mathcal Q=1$, the geometric and background potential block is

$$\begin{pmatrix}G&-F/v\\-F^T/v&a\end{pmatrix}. \tag{10}$$

When $G$ is positive, this retained block is positive exactly when

$$\boxed{a v^2>F^TG^{-1}F.} \tag{11}$$

At general phase charge the threshold is $a v^2>\mathcal Q F^TG^{-1}F$. At unit wave norm $N=1$ instead, it is $2\omega F^TG^{-1}F$. One common $a,v$ within a depth must exceed the largest threshold among its occupied modes. The background coordinate is a retained amplitude of the same field, not an additional particle. Equation (9) is an inverse action: it does not derive the background potential, its kinetic coefficient, the reference geometry, or this derivative coupling from the original master field.

Maintained-mode calculation

Both eigenvalues of the original two-direction $G$ are positive for every maintained mode at all six depths and both common strengths. The table lists the largest unit-phase-charge threshold in (11) within each depth, and the largest-to-smallest spread of $\omega_{n,d}/\mathcal S(n,d)$. Depth ten has only one maintained mode, so its within-depth spread is undefined. These are normalized candidate values, not stiffnesses or masses inferred from experiment.

The finite-strength current carrier approaches the simplex ratios as the strength increases but does not give exact count proportionality in these controls. The absolute frequency unit and common physical radius remain undetermined. The extended calculation tests the joint bare Hessian on the nine unit-sphere-normalized zonal functions of degrees 0, 2, ..., 16. Every maintained mode remains positive. The common unit-phase-charge thresholds change only modestly:

The spinor pointwise result is stronger than this finite scan for the bare conformal block. The depth-two form has only the finite zonal result; its complete conformal Hessian remains open. All sectors still lack a test of traceless metric changes, the complete wave Hessian, the background's causal motion, nonlinear profile relaxation, and compatibility of all nested sector geometries. The operator-level cancellation removes the quadratic field–geometry mixing, but it does not prove that the field block itself is stable. In particular, a positive conformal Hessian does not overturn the independent angular-stress obstruction for a scalar local curvature law plus isotropic support; (9) assumes a patterned same-field response whose origin remains to be supplied.

The unified calculation passes 988 conditions. Its report contains every maintained mode, both common strengths, all Hessian entries, the common thresholds, finite-strength frequency spreads, pointwise spinor margins, and independent quadrature controls for both $q$ and $q^2$ deformations. The next discriminating calculation needs a derived background potential and metric response from one spatial field law, followed by the tensor-geometry and complete wave Hessians and the fixed mass-frequency constraints.