Curvature forces on the linked mass-carrier geometry

The actual spatial curvature supplies a positive restoring coefficient for the carrier's constant-polynomial deformation at depths 4, 5, 6 and 10. That contribution vanishes at depth 3. The ratio of its leading coefficient to the deformation metric is an exact polynomial in depth, with no independent numerical integral.

The same calculation also finds a force on a different centered shape coordinate. Consequently this restoring coefficient does not make the round reference a stationary occupied particle. A common-radius pressure balance must be accompanied by a balance of the internal shape forces.

These statements concern the declared geometric coefficient functional. Its physical energy and time normalization, its identification with sustaining modes of the one field, and the complete curved wave source remain open. Mass is the particle's selected frequency in equivalent units.

The ordinary spatial tube

Let a round field feature $S_R^m$ lie in ordinary $\mathbb R^{m+2}$, with $m=d-2$. Use the outward radial normal $y_1$ and the fixed binormal $y_2$. This normal frame is flat; no normal Chern class is inserted. In the nonsingular tube $r<R$,

$$r^2=y_1^2+y_2^2,\qquad J(y)=\left(1+\frac{y_1}{R}\right)^m. \tag{1}$$

The following expressions are per unit area of the reference feature. They retain normal derivatives of the linked coefficients and their exact tube volume factor. They restrict those coefficients to the planar first-order family; they are not the full Euler–Lagrange equations of a curved stationary solution.

The linked planar rule is

$$W=e^{-u}P(z),\quad A=\nabla^\perp u,\quad B=\Delta u=2(1-|W|^2),\qquad (D_1+iD_2)W=0,$$

where $D_i=\partial_i-iA_i$. Its positive geometric density and cost are

$$e=|DW|^2+\frac{B^2}{4}+(1-|W|^2)^2,\qquad G_J=\int Je\,d^2y. \tag{2}$$

At $J=1$, every degree-$d$ polynomial in this family has $G_1=2\pi d$, independently of its zero locations. The local identity

$$\boxed{e=B+\frac12\Delta |W|^2} \tag{3}$$

follows by applying the covariant Laplacian to $|W|^2$ and using the first-order rule. Changes of the localized density decay exponentially, which controls the boundary terms in the integrations below. The finite tube is used where $J$ is a physical Jacobian; replacing its localized integrals by full-plane integrals has an exponentially small tail error when the core is well inside the tube.

The constant-polynomial restoring coefficient

At $P=z^d$, write $w=|W|$, $a=ru_0'$. For $P=z^d+\alpha$, the expansion is

$$u=u_0+\alpha f_d\cos d\theta+ \alpha^2(g_0+g_2\cos2d\theta)+O(\alpha^3).$$

The metric calculation gives

$$\begin{aligned} (-\Delta+d^2/r^2+4w^2)f_d&=4w^2r^{-d},\\ (-\Delta+4w^2)g_0&=2w^2(r^{-d}-f_d)^2,\\ \int g_0\,d^2y&=\frac d2M_d. \end{aligned} \tag{4}$$

The angular mean of the second coefficient of $|W|^2$ is $-\Delta g_0/2$. The degree of $J$ is $m=d-2<d$, so its product with every nonzero angular harmonic of this deformation integrates to zero. Using (3) twice by parts yields

$$G_J(\alpha)-G_J(0)=G_2\alpha^2+O(\alpha^4),\qquad \boxed{G_2=\int\left(\Delta J-\frac14\Delta^2J\right)g_0\,d^2y.} \tag{5}$$

Rotation of the normal plane changes the phase of $\alpha$. Since $J$ cannot see the nonzero multiples of $d$, the restricted cost depends on $|\alpha|$; the real expansion is even. In particular,

$$G_2=\frac{m(m-1)}{R^2}\int g_0\,d^2y+O(R^{-4}), \qquad \boxed{\frac{G_2}{M_d} =\frac{d(d-2)(d-3)}{2R^2}+O(R^{-4}).} \tag{6}$$

At depths 4 and 5 the displayed leading term is exact for this restricted tube polynomial. At depth 3 it is identically zero. Higher inverse-radius terms remain at depths 6 and 10.

If a kinetic comparison has $M_d|\dot\alpha|^2$, its real kinetic and restoring Hessians both acquire a factor two, so the ratio in (6) is unchanged. It is nevertheless not yet a physical shape frequency: the complete reference is not stationary, and the curved kinetic metric also changes at higher orders. A physical geometric action would supply its energy and time units before this ratio could become a frequency.

For the well-conditioned coordinate $\tau=c_0\alpha$, with $w\sim c_0r^d$, numerical coefficients at $R=40$ are:

The coordinate factor changes both $G_2$ and $M_d$ equally. These entries are geometric comparisons, not measured couplings.

The other polynomial directions carry forces

For $P=z^d+\alpha_kz^{d-k}$, let $U_k=w(r^{-k}-f_k)$. The first density variation is $\delta|W|^2=2wU_k\cos k\theta$. Equation (3) gives the finite formula

$$F_k=\partial_{\alpha_k}G_J =-2\int\left(2J-\frac12\Delta J\right)wU_k\cos k\theta\,d^2y. \tag{7}$$

The holomorphic moment identity $\int_0^\infty r^{k+1}wU_k\,dr=k/2$ fixes its leading term:

$$\boxed{F_k=-\frac{2^{2-k}\pi k}{R^k}\binom{m}{k} +O(R^{-k-2}),\qquad k\le m.} \tag{8}$$

For $k>m$, the first derivative vanishes by angular selection. When $m-k<2$, no higher tube-polynomial term contributes to (8). In particular,

$$F_1=-\frac{2\pi m}{R}+O(R^{-3}),\qquad F_2=-\frac{\pi m(m-1)}{R^2}+O(R^{-4}). \tag{9}$$

The first coordinate shifts the center by $-\alpha_1/d$, giving the usual mean-curvature tension force of a cost $2\pi d$. The second coordinate keeps the center fixed. Positive $\alpha_2$ splits a pair of zeros along the binormal and lowers the weighted cost at leading order. Therefore fixing the center does not hold the coincident polynomial stationary for $d\ge4$.

For $d=3$, $J$ is linear. Every centered planar first-order configuration has the same restricted cost. This follows from (3) and the first holomorphic moment; it is not merely a numerical zero in one quadratic coefficient.

The full first variation and complementary fields

It is useful to check the full energy rather than just its restriction. At any planar first-order solution, varying arbitrary $W,A$ gives

$$\begin{aligned} \frac{\delta G_J}{\delta W^*}&=-(\partial_iJ)D_iW,\\ \frac{\delta G_J}{\delta A}&=-\frac12B\nabla^\perp J. \end{aligned} \tag{10}$$

The constant-weight Euler–Lagrange terms cancel, leaving derivatives of $J$. For a linear weight $J=1+\varkappa y_1$, define the horizontal translation vector

$$Z_1=(-D_1W,-B e_2),\qquad \|\delta\|_M^2=\int(|\delta W|^2+|\delta A|^2/4)d^2y.$$

Then

$$\boxed{dG_J[\delta]=2\varkappa\langle Z_1,\delta\rangle_M, \qquad \|Z_1\|_M^2=\pi d.} \tag{11}$$

Thus the order-$R^{-1}$ source is purely a translation in this specified metric. If translation is removed by the corresponding orthogonal projection, there is no order-$R^{-1}$ force on complementary fields. The usual second-order energy lowering from relaxing such a forced complement therefore vanishes in this particular projected problem.

That projection does not derive physical pressure. A different imposed center constraint, or the occupied wave's actual support stress, can act on the complementary fields differently. The full physical source still needs its own equation. At order $R^{-2}$, the nonzero quadrupole force already lies within the geometric zero-mode space and remains explicit.

The existing occupied wave also has a nonzero constant-polynomial force, as computed in the reciprocal source calculation. A positive coefficient in (6), by itself, does not cancel that force at $\alpha=0$. Both the changed equilibrium and the other shape directions must be included before claiming a stationary mass carrier.

Independent spatial verification

nonlinear_moduli_plane.py solves the nonlinear planar equation for both signs of the constant and quadrupole coefficients. Fourier collocation retains their complete angular dependence. The tube weight is folded over the exact symmetry sectors, preserving every angular component that can contribute.

The geometric energy is evaluated directly from covariant spatial kinetic, curvature and potential terms. An independent integral using (3) agrees and improves under radial refinement. At depth ten and $R=40$, the finite quadrupole derivative is approximately $-0.111678$, including the higher inverse-radius terms; at depth three it is zero.

The 357-check report includes every polynomial zero mode at the five depths, arbitrary scalar and connection variations in (10), the translation identity, and nonlinear solutions on 1,201- and 2,401-point radial grids. Exact source hashes and separately retained spatial arrays make the calculation reproducible. It verifies these conditional geometric statements, not the missing physical normalization or a completed nonlinear particle solution.