Obtain the hopping coefficient from the linked density

The common normal operator previously used a separately specified bounded function to couple adjacent normal modes. That function can instead be obtained from the linked geometry itself:

$$\boxed{h_{\rm geom}=(\partial_1+i\partial_2)|W|^2.} \tag{1}$$

The same rule works at depths 3, 4, 5, 6 and 10. It removes one independent spatial function. The simplex numerator and grouping of responses within a depth survive; the spatial integrals and response between depths change. Equation (1) is an explicit candidate choice, not a uniquely derived master interaction. Its dimensional coefficient remains to be fixed.

All coordinates here are the same dimensionless ordinary normal coordinates used for the linked profile. Restoring physical length gives a derivative an inverse-length factor. That factor must enter the common operator normalization, rather than becoming a separate particle fit.

Radial form, boundedness and positive hopping

For $W=w(r)e^{id\theta}$, the linked equations give

$$h_{\rm geom}=f(r)e^{i\theta},\qquad f=2ww'=\frac{2(d-a)w^2}{r}>0\quad(r>0). \tag{2}$$

The coefficient is regular at the center and decays at infinity. It is invariant under changes of the phase representative of $W$. The profile bounds also give an explicit, conservative estimate

$$\boxed{|h_{\rm geom}|\le2\sqrt{e d}.} \tag{3}$$

For $r\ge\sqrt d$, use $w\le1$ and $d-a\le d$. For $r\le\sqrt d$, integration of $w'/w\ge d/r-r$ gives $w(r)\le(r/\sqrt d)^d\exp[(d-r^2)/2]$. Maximizing the resulting $r^{2d-1}e^{-r^2}$ bound proves (3). The number $e$ in (3) is Euler's number, not the connection coefficient in $D_{t,e}$.

With the normalized positive radial carrier components, the adjacent entries become

$$\boxed{h_l=2\pi\int_0^\infty r f(r) (U_{l+1}U_l+V_{l+1}V_l)dr>0.} \tag{4}$$

The geometric carrier gives, for example, $(0.39086323,0.46980658)$ at depth three and $(0.12082700,0.26469052,0.38308533,0.43768548,0.39529922)$ at depth six. These integrals are computed once per depth from the same spatial rule.

The projected operator still has $T_{ll}=n+l$, $T_{l+1,l}=\gamma h_l$. Consequently

$$\prod_lT_{ll}=d!\binom{n+d-1}{d},\qquad \lambda_{n,d}=\frac{d!}{\gamma^{d-1}\prod_lh_l} \binom{n+d-1}{d}[1+O(\gamma^{-2})]. \tag{5}$$

The finite-hopping correction is still present. A shared charged-lepton frequency unit between depths six and ten is not implied by (5): the products of spatial integrals and the dimensional factors must produce that relation. This calculation preserves the numerator, rather than deriving all sector scales.

The full normal complement estimate in the common carrier remains applicable after replacing its perturbation norm by $v=\gamma\|h_{\rm geom}\|_\infty+(d+1)/2$, bounded using (3). Using the old $v=\gamma+(d+1)/2$ without checking the new coefficient would be unjustified.

The coefficient's own shape derivative must be retained

Let $P(z)=z^d+\alpha_kz^{d-k}$. In the horizontal representation, the real geometric zero mode has $\delta W=U_ke^{i(d-k)\theta}$. The density variation is $\delta|W|^2=s_k(r)\cos k\theta$, with $s_k=2wU_k$. Differentiating (1) gives

$$\boxed{ \delta_{x_k}h= \frac{s_k'+ks_k/r}{2}e^{i(1-k)\theta} +\frac{s_k'-ks_k/r}{2}e^{i(1+k)\theta}.} \tag{6}$$

For the imaginary coordinate, multiply the first coefficient by $i$ and the second by $-i$. Because the density is phase invariant, this is also its derivative in the polynomial gauge. The full hopping matrix therefore has three terms:

$$\boxed{\delta h_{ij}= \langle\delta\Phi_i,h\Phi_j\rangle+ \langle\Phi_i,h\delta\Phi_j\rangle+ \langle\Phi_i,(\delta h)\Phi_j\rangle.} \tag{7}$$

The last term is substantial. Keeping the multiplier fixed while varying the geometry would not test (1). The complete first polarization derivative still vanishes by the independent anticommutator identity, so the complete first projected source is $\delta T=\gamma\delta h$.

For reproducing the complementary calculation, a general multiplier $f(r)e^{i\theta}$ has positive-chiral Green sources

$$\begin{aligned} h\Phi_l:&\quad ((f'+f/r)V_l,(f/r-f')U_l),\quad j=l+1,\\ \bar h\Phi_l:&\quad ((f'-f/r)V_l,-(f'+f/r)U_l),\quad j=l-1. \end{aligned} \tag{8}$$

They follow by applying the full $D$ to the multiplied zero mode. The independent computation instead differentiates the modes using $-D L_+^{-1}\delta D\Phi_l$ and contracts them directly in (7).

Responses within and between depths

The endpoint still has a single matrix entry $\delta h=\kappa_d\alpha_d|0\rangle\langle d-1|$. Its positive-frequency derivative is $2\gamma\kappa_d\operatorname{Re}(v_0^*u_{d-1})$. For a fixed surface sign, the endpoint product tends to a common value across the assigned indices at each depth. Different masses can therefore retain similar geometric response under this simplified rule.

Using the actual geometric metric $M_d$, the endpoint values are:

Each entry is $\kappa_d/\sqrt{M_d}$. These values are spatial response coefficients, not encounter rates. They do not derive special neutrino suppression or a physical normalization across sectors.

All centered sources were also checked under mixtures of the two positive surface-sign branches. To avoid dependence on rescaling a coordinate, the test minimizes

$$\min_{0\le q\le1}\max_k \frac{|qF_k^++(1-q)F_k^-|}{|F_k^+|+|F_k^-|}. \tag{9}$$

Every tested mixture leaves a nonzero residual. The smallest residual in the geometric-carrier controls is about 0.0230 at depth three; other depths remain near 0.5. The fixed-multiplier source sign theorem must not be copied to this new calculation: the explicit term (6) changes some penultimate signs. Equation (9) records the newly tested scope.

The later second-variation test

The preserved spectral numerator and first response are insufficient for the supporting-state calculation. The complete nonlinear spatial Hessian has negative energy curvature in the endpoint direction at every tested depth for this gradient rule. It therefore fails that averaged local-minimum test. These energy curvatures are not negative particle inertia. The curvature-moment rule removes the same independent spatial function and passes the complete centered mixed-Hessian test under its stated retained-action assumptions.

Verification

profile_gradient_hopping.py implements the density-derived multiplier and its full derivative. profile_gradient_checks.py checks Cartesian differentiation, two independent Green contractions, all maintained controls and surface signs, positive hopping, the exact polarization spacing and the complete source mixtures. Its 362-check report also includes radial-grid and outer-domain refinements. Raw arrays are preserved separately with the exact executed source versions.