A bounded curvature-moment carrier with positive averaged shape stiffness

A simple coefficient obtained from the existing linked geometry survives both the spectral and complete averaged-stiffness tests:

$$\boxed{h=z(1-|W|^2)=\frac{zB}{2}.} \tag{1}$$

It removes the independently prescribed radial multiplier. Unlike the density-gradient choice, it retains positive shape stiffness when the actual nonlinear geometry is included. The same rule is used at depths 2, 3, 4, 5, 6 and 10, with every maintained particle index and both surface signs.

The result is a stronger candidate spatial carrier. It is not yet a complete physical mass operator: the averaged supporting state has a large energy cost, and its coherent field still has time-dependent sources. Its physical normalization and sustaining motion remain to be derived from the one field.

One scalar profile determines the spatial coefficients

In the same ordinary dimensionless normal coordinates, use

$$\boxed{ \Delta u=2(1-e^{-2u}|P(z)|^2),\qquad W=e^{-u}P(z),\quad A=\nabla^\perp u,\quad B=\Delta u.} \tag{2}$$

The radial reference has $P=z^d$, $W=w(r)e^{id\theta}$, $A=a(r)e_\theta/r$, with

$$w'=\frac{d-a}{r}w,\qquad a'=2r(1-w^2). \tag{3}$$

These same fields determine the geometric normal operator, its bounded polarization and the new multiplier:

$$\begin{aligned} D_{\rm geo}&=-i\alpha\cdot\nabla+ \frac12\alpha\cdot A(\tau_3-I)+2\Gamma\cdot W,\\ K&=(d+1-a)S-aI,\qquad h=\frac{zB}{2}. \end{aligned} \tag{4}$$

Equation (1) is an explicit candidate choice using an existing geometric quantity. The physical master action has not uniquely selected it or fixed its overall coupling. There is no new fitted spatial profile in (4), and none of these normal coordinates is compactified.

At the radial reference, $h=r(1-w^2)e^{i\theta}$. Monotonicity of $w$ gives

$$a(r)=2\int_0^r s(1-w(s)^2)ds\ge r^2(1-w(r)^2),$$

so

$$\boxed{|h|\le\min(r,d/r)\le\sqrt d.} \tag{5}$$

The coefficient is regular at zero and decays at infinity. Bound (5) applies to this radial reference; arbitrary separated nonradial shapes require their own bound. A common translation carries the center and its relative coordinate $z$ together.

Spectrum and grouped response survive

For normalized normal modes $\Phi_l$, the adjacent integrals are

$$h_l=2\pi\int_0^\infty r^2(1-w^2) (U_{l+1}U_l+V_{l+1}V_l)dr>0. \tag{6}$$

The projected pair still has $H_P=\left(\begin{smallmatrix}0&T^\dagger\\T&0\end{smallmatrix}\right)$, with

$$T_{ll}=\beta+l-\frac{d-1}{2},\qquad T_{l+1,l}=\gamma h_l. \tag{7}$$

For $\beta=n+(d-1)/2$, the product of its diagonal entries is $d!\binom{n+d-1}{d}$. Therefore its small frequency remains

$$\boxed{\lambda_{n,d}= \frac{d!}{\gamma^{d-1}\prod_l h_l} \binom{n+d-1}{d}\,[1+O(\gamma^{-2})].} \tag{8}$$

The finite-hopping correction remains nonzero. The equality of the charged-lepton frequency units between depths six and ten is not proved by the numerator in (8).

For a real polynomial deformation, the density derivative is $\delta\rho=2wU_k\cos k\theta$. The coefficient now has

$$\delta h=-z\delta\rho, \qquad \delta_{x_k}h=-rwU_k \left(e^{i(1-k)\theta}+e^{i(1+k)\theta}\right). \tag{9}$$

For the imaginary coordinate, multiply the first angular coefficient by $i$ and the second by $-i$. The complete derivative also includes both moving normal modes. The polarization's full first derivative still cancels by its independent angular-generator identity.

The endpoint coefficient $\kappa_d$, normalized by the actual positive geometric metric $M_d$, is:

Its frequency response is $2\gamma\kappa_d\operatorname{Re}(v_0^*u_{d-1})$. For a fixed surface sign, the endpoint product approaches a common value across the assigned indices within each depth. This retains different frequencies with similar within-depth response. These dimensionless integrals do not themselves derive neutrino suppression, physical gauge couplings or encounter rates.

Retain the full nonlinear shape Hessian

Use the positive averaged populations $q_j=s_j(T_0)$, where $s_j$ are all positive singular values. Their complete centered source is zero by the weighted trace identity. Keep these canonical populations fixed during an energy variation. The canonical interpretation here uses the explicitly defined retained positive action. Projecting the full parent's square would add a complementary-mode term unless its invariant low subspace has first been derived.

For real coordinates $x_k$, define

$$\mathcal A_{kh}=\operatorname{Re}\operatorname{Tr} (T_k^\dagger T_h+T_0^\dagger T_{kh}),\qquad F_{jk}=\partial_{x_k}s_j. \tag{10}$$

The complete static energy Hessian is

$$\boxed{\mathcal H_{xx}=\mathcal A-F^TF,\qquad \mathcal H_{yy}=\mathcal A,\qquad \mathcal H_{xy}=0.} \tag{11}$$

The zero mixed real–imaginary block follows from complex conjugation at the real reference. It does not set the dynamical Berry curvature to zero. The imaginary first frequency gradients vanish there, and the first matrix norms and relevant second spatial traces agree on the two quadratures. Thus $\mathcal H_{yy}-\mathcal H_{xx}=F^TF\ge0$.

Angular selection makes $\mathcal A$ tridiagonal in the centered orders $k=2,\ldots,d$. The full real Hessian is nevertheless generally dense because of the frequency-gradient Gram matrix. Discarding its mixed entries would not test (11).

The diagonal nonlinear terms use the complete second geometry, with $h''_0=-z\rho''_0$. For adjacent orders $h=k+1$, the required mixed geometric harmonic solves

$$\boxed{(-\Delta_1+4w^2)g_{k,k+1}=4U_kU_{k+1}.} \tag{12}$$

It follows by taking the mixed derivative of (2) and retaining the angular difference harmonic. Its density coefficient is $2U_kU_{k+1}-2w^2g_{k,k+1}$. The second moving mode also includes $D_k\Phi_h+D_h\Phi_k$, the explicit $D_{kh}$, and its symmetric in-kernel normalization. All these terms are retained. They supply

$$\mathcal A_{k,k+1}= \gamma\sum_l\left(\beta+l-\frac{d-1}{2}\right)(h_{kh})_{ll} -\frac\gamma2\sum_lh_l(K_{kh})_{l+1,l}. \tag{13}$$

Orders separated by more than one cannot contribute to this nonlinear trace. Equation (12) is checked independently with a scalar boundary value solver, and (13) with direct finite spatial integrals.

The complete centered Hessian is positive in the tested controls

Normalize to unit canonical charge $\mathcal Q=\sum_js_j(T_0)$ and use the diagonal positive geometric metric $M_k$. At $\gamma=10000$, the smallest eigenvalues of $\mathcal H_{xx,kh}/(\mathcal Q\sqrt{M_kM_h})$ include:

The complete matrices are positive for every maintained control and both surface signs. Equation (11) then also gives positivity of the entire imaginary block. This establishes a local minimum of the stated averaged energy over all centered complex polynomial deformations at fixed populations, within the declared spatial reduction. It is stronger than the earlier separate-coordinate test.

It does not establish a pointwise coherent equilibrium or dynamical stability of the full master field. The positive-band energy bound still places the supporting state's energy per canonical charge well above the small frequency. A coherent occupation also has beat-frequency sources that averaging removes, as quantified in the coherent polarization test. Those two issues remain physical work for the same-field action; they are not repaired by calling a static Hessian an inertia.

Dimensional normalization remains explicit

Let $R_d$ be the tangential length and $L_d$ the normal profile's length unit. Both are ordinary spatial lengths shared by the particles at one depth. With a common wave speed $c$, the physical coefficient $z_{\rm phys}B_{\rm phys}/2$ has units of inverse length. If its dimensionless physical multiplier is $g_d$, then

$$\gamma_d=g_d\frac{R_d}{L_d},\qquad \omega_{n,d}=\frac c{R_d}\lambda_{n,d},$$

and (8) gives

$$\boxed{\omega_{0,d}=\frac{c\,d!}{g_d^{d-1}\prod_lh_l} \frac{L_d^{d-1}}{R_d^d}.} \tag{14}$$

Equation (14) identifies the combinations that must be supplied by the physical geometry and coupling law. It does not assign independent lengths to different particle modes, derive a common size across all depths, or introduce compactified dimensions. Mass is already frequency in equivalent units; the unresolved step is the dynamics that fixes these shared physical coefficients. The dimensional comparison tests those remaining parameters explicitly, including all surface-sign choices in the six/ten equal-parameter trial.

The depth-two extension reruns the common calculation at all six depths, with 102 positive real coordinate controls and the complete complex Hessians. It also identifies an exact two-band coherent source cancellation. The polarization-dimension test examines its generalization and distinguishes it from cancellation of inertia.

Verification

flux_moment_hopping.py implements the coefficient and its explicit first and second responses. flux_moment_checks.py verifies the bounded profile, independent first spatial contractions, finite second spatial differences on both quadratures, the common spectral numerator, grouped response and positive-support energy bound. Its 140-check report includes all 96 real coordinate-curvature controls.

curvature_moment_mixed.py and curvature_moment_mixed_checks.py retain the full mixed nonlinear geometry and every centered coordinate. The 132-check report checks the complete matrices, the independent mixed scalar profile, finite spatial derivatives and low/high-depth refinements. Raw geometry, matrices and exact executed source versions are stored separately.