Nonlinear response of the bending modes

Summary: The same surface curvature energy that gives simplex frequencies has nonzero canonical mode couplings, an exact quadratic resonance and a scale response that must enter any nonlinear particle fixed point.

A frequency calculation also needs its interaction

The bending construction gives the exact frequency sequence under a declared geometric action. Its nonlinear curvature energy supplies a way to test interactions without inserting a separate coupling matrix for each pair of modes.

This note uses the four-dimensional round surface in ordinary six-dimensional space. It is a conditional feature of a field, not a replacement of that physical space. The calculation retains both normal directions. Individual zonal harmonics are specified spatial polarizations within the complete harmonic multiplets; they do not exhaust possible particle patterns.

The mass-frequency targets are unchanged. The small displacements below differentiate the geometry and test its nonlinear response. They are not variables used to fit particle masses. Physical magnitude may be needed in a full nonlinear fixed-point solution, after specifying its frequency.

Canonical coordinates on the actual graph

Let $\theta\in S^m$ be a unit radial vector and describe the surface by

$$\begin{aligned} F(\theta)&=(r(\theta)\theta,z(\theta)),\\ F(\theta)&\in\mathbb R^{m+1}\times\mathbb R. \end{aligned} \tag{1}$$

The last coordinate specifies the chosen axis of this round comparison. Rotating the complete geometry changes that axis; it is not a distinguished spatial dimension of a physical sector.

The geometric symplectic form on this chart is exactly

$$\Omega=\Gamma\int_{S^m}r^m\,\delta r\wedge\delta z\,d\Omega_m. \tag{2}$$

To obtain (2), contract the containing volume form with the two variations and the $m$ graph tangents. Terms involving tangential derivatives of $r$ or $z$ repeat one of the variation directions and drop out of the determinant. Each remaining spherical tangent contributes one factor $r$.

Consequently a canonical radial graph coordinate is

$$q=\frac{r^{m+1}-R^{m+1}}{(m+1)R^m},\qquad v=z, \qquad \Omega=\Gamma R^m\int\delta q\wedge\delta v\,d\Omega_m. \tag{3}$$

Linear radial displacement equals $q$ at $r=R$, but their higher variations differ. That difference matters for nonlinear coupling. This is canonical normalization, obtained from the kinetic form of the same action.

For $m=4$ and $R=1$,

$$r=(1+5q)^{1/5}=1+q-2q^2+6q^3+O(q^4). \tag{4}$$

The following numerical coefficients use $R=\Gamma=\kappa=1$ and real harmonics $Y_\ell$ normalized by $\int_{S^4}Y_\ell^2dA=1$. The associated mode coordinates are $q_\ell,v_\ell$. Their quadratic Hamiltonian is

$$H_2=\frac12\sum_\ell\omega_\ell(q_\ell^2+v_\ell^2),\qquad \omega_\ell=\frac14(\ell-1)\ell\cdots(\ell+4). \tag{5}$$

Differentiate the curvature of the spatial surface

There is an analytic expression for the full cubic interaction. Let $A,B$ be geodesic normal coordinates about the equator in $S^{m+2}$, with $B$ pointing toward the stereographic pole. Reversing both normal coordinates is an ambient isometry fixing the equator. Consequently the conformal bending energy has no odd terms in $(A,B)$ at that equator.

Stereographic projection to the Euclidean graph gives, to second order,

$$r=1+B+\frac12(B^2-A^2)+O(3),\qquad z=A+AB+O(3). \tag{5a}$$

Combining this with the exact canonical coordinate (3) gives

$$B=q-\frac{m+1}{2}q^2+\frac12v^2+O(3),\qquad A=v-qv+O(3). \tag{5b}$$

Let $K=\mathcal J/m$ be the frequency operator in the common unit normalization. The spherical quadratic energy is $\tfrac12\int(AKA+BKB)dA$. Its cubic contribution after (5b) is exactly

$$\boxed{ H_3=-\frac{m+1}{2}\int q^2Kq\,dA -\int qvKv\,dA+\frac12\int v^2Kq\,dA. } \tag{5c}$$

There is no omitted spherical cubic term, by the normal-reflection symmetry. If $G_{abc}=\int Y_aY_bY_c\,dA$, (5c) therefore gives

$$\begin{aligned} \partial_{q_a}\partial_{q_b}\partial_{q_c}H &=-(m+1)(\omega_a+\omega_b+\omega_c)G_{abc},\\ \partial_{q_a}\partial_{v_b}\partial_{v_c}H &=(\omega_a-\omega_b-\omega_c)G_{abc}. \end{aligned} \tag{5d}$$

These identities apply to the full real harmonic multiplets, including mixed spatial polarizations. The following direct curvature calculation provides an independent check at the specified zonal modes.

For zonal modes, use a surface of revolution

$$F(\vartheta,\eta) =(a(\vartheta)\eta,b(\vartheta),z(\vartheta)),\qquad \eta\in S^3,\quad a=r\sin\vartheta,\quad b=r\cos\vartheta. \tag{6}$$

Write $C=(a,b,z)$, $s=|C'|$, $T=C'/s$ and $e_a=(1,0,0)$. The two principal curvature vectors, with the spherical one repeated three times, are

$$K_\vartheta=\frac{T'}s,\qquad K_\eta=\frac{-e_a+(e_a\cdot T)T}{a},\qquad H=K_\vartheta+3K_\eta. \tag{7}$$

The required contractions are

$$\begin{aligned} |\nabla^\perp H|^2&= \frac{|H'|^2-(H'\cdot T)^2}{s^2},\\ |A_H|^2&=(H\cdot K_\vartheta)^2+3(H\cdot K_\eta)^2,\\ dA&=s a^3\,d\vartheta\,dA_{S^3}. \end{aligned} \tag{8}$$

The angular derivatives of $H$ are tangent to the surface and have no normal component. Equations (7)–(8), inserted into the explicit curvature energy, differentiate the actual spatial geometry. No spectral polynomial is substituted for the nonlinear energy.

The implementation uses two independent methods: symmetric variations of the nonlinear integral, and third-order Taylor arithmetic applied to its curvature expressions. Angular quadrature is independently refined.

An exact resonance has a nonzero interaction

The simplex recurrence implies

$$\frac{S(n+1,d)}{S(n,d)}=\frac{n+d}{n}. \tag{9}$$

At $n=d$, the ratio is exactly two. For the six-dimensional comparison, the corresponding surface degrees are $\ell_a=7$ and $\ell_c=8$:

$$\omega_a=83\,160,\qquad\omega_c=166\,320=2\omega_a. \tag{10}$$

These are available modes of the trial operator, not new particle assignments. Their real zonal harmonics have nonzero spatial overlap

$$G=\int_{S^4}Y_7^2Y_8dA=0.368980006548\ldots. \tag{11}$$

The diagnostic evaluates this integral exactly using Gegenbauer polynomials and the measure $2\pi^2(1-t^2)dt$, then compares the value with the curvature quadrature.

Define three canonical third derivatives of the same energy:

$$C_q=\partial_{q_a}^2\partial_{q_c}H,\qquad C_v=\partial_{v_a}^2\partial_{q_c}H,\qquad C_\times=\partial_{q_a}\partial_{v_a}\partial_{v_c}H. \tag{12}$$

At the round surface their analytic and directly computed values agree:

The positive-frequency equation for $w_c=q_c-iv_c$ contains a doubled frequency source from the rotating $a$ mode. Its coefficient is

$$F_{caa}=\frac{C_q-C_v+2C_\times}{8} =-92053.1320336\ldots. \tag{13}$$

In fact, (5d) and $\omega_c=2\omega_a$ give the general exact expression $F_{caa}=-(m+2)\omega_aG_{aac}/2=-d\omega_aG_{aac}/2$.

Thus the temporal resonance in (10) is enabled by a nonzero spatial interaction. An eigenvalue coincidence alone would not establish that.

Retain the reciprocal response

In the canonical complex coordinates $a=(q_a-iv_a)/\sqrt2$ and $c=(q_c-iv_c)/\sqrt2$, the resonant cubic Hamiltonian is

$$H_{\mathrm{res}}=\omega_a|a|^2+\omega_c|c|^2 +g(c^*a^2+c(a^*)^2),\qquad g=\sqrt2 F_{caa}. \tag{14}$$

Both interaction directions come from (14). After removing the two linear phases, its complete complex equations are

$$i\dot A=2gA^*C,\qquad i\dot C=gA^2. \tag{15}$$

They conserve $|A|^2+2|C|^2$ and the interaction energy. With a small test displacement $A(0)=a_0$ real and $C(0)=0$, the exact solution is

$$A(t)=a_0\operatorname{sech}(\sqrt2ga_0t),\qquad C(t)=-\frac{ia_0}{\sqrt2}\tanh(\sqrt2ga_0t). \tag{16}$$

Numerical integration checks both complex modes and their conserved quantities against (16). The initial displacement tests energy transfer; it does not set either mass-frequency in (10).

This is the cubic resonant truncation. It has not been established as a controlled approximation to the complete nonlinear spatial equation over arbitrary times. Complementary harmonics, the slow background response and higher-order frequency shifts still require their own equations.

The assigned modes also force the background

At each of the unchanged assignments $(d,n)=(6,13),(6,35),(10,23)$, there is no available degree with exactly twice its frequency inside the angular product range $2\le\ell_c\le2(n+1)$. This checks one specified quadratic self-resonance. It neither proves nonlinear stability nor selects these particle indices uniquely.

There is a separate zero-frequency response. Let $Y_0=1/\sqrt{A_m}$ be the normalized constant harmonic. Scale invariance fixes its third derivatives without fitting a coefficient:

$$\begin{aligned} \partial_{q_0}\partial_{q_\ell}^2H &=-\frac{(2m+2)\omega_\ell}{\sqrt{A_m}},\\ \partial_{q_0}\partial_{v_\ell}^2H &=-\frac{2\omega_\ell}{\sqrt{A_m}},\\ \partial_{v_0}\partial_{q_\ell}\partial_{v_\ell}H&=0. \end{aligned} \tag{17}$$

Here the common units are again $R=\Gamma=\kappa=1$. For the derivation, move the constant-radius background to $r$. At fixed physical normal displacement, its quadratic bending energy scales as $r^{-2}$. The canonical radial displacement in (3) additionally satisfies $\delta r=r^{-m}\delta q$. Thus the radial quadratic coefficient scales as $r^{-2m-2}$ and the axial coefficient as $r^{-2}$. Differentiating with respect to $q_0$, for which $\delta r=\delta q_0/\sqrt{A_m}$ at unit radius, gives the first two identities. Independence of the energy under uniform axial translation gives the third.

Direct third variations verify (17) at surface degrees $7,14,36$. For a circular test motion $q_\ell=b\cos\omega_\ell t$, $v_\ell=b\sin\omega_\ell t$, the constant part of its force is

$$\left\langle\partial_{q_0}H_3\right\rangle =-\frac{(m+2)\omega_\ell b^2}{2\sqrt{A_m}}. \tag{18}$$

Since $q_0$ and $v_0$ are conjugate, this term produces axial motion of the background. A circular linear mode on an unchanged stationary round surface therefore omits a nonlinear equation. It must be included when finding a complete oscillating fixed point or its moving-frame version. Equation (18) is a response to an oscillation of specified frequency; it is not a second prescription for that mode's mass.

Consequences for a shared coupling response

The same action fixes both the simplex spectrum and the geometric coupling. At a common radius and common canonical normalization, the scale vertices in (17) are proportional to $\omega_\ell$. Their muon-to-electron ratio is therefore the simplex ratio, approximately $206.7647$, in this particular channel.

The fractional frequency response to a radius change is common, $\delta\omega/\omega=-d\,\delta R/R$, but the absolute canonical vertex is not. This reproduces the distinction emphasized in the common-cavity coupling calculation. A universal fractional shift is insufficient to establish the required grouping of physical particle couplings by depth.

These computations identify a concrete nonlinear response of a natural geometric frequency operator. Its realization in the master field, the particle fixed points, their complete inertial response and the physical pair-interaction channel remain open. The time law also retains the finite-propagation limitation stated in the bending construction.

Verification

bending_surface_checks.py checks the canonical curvature derivatives, their independent finite variations and quadrature refinements, the exact Gaunt integral, the unchanged particle-index resonance test and the scale identities. bending_curvature_jets.py provides the direct Taylor differentiation of the actual spatial energy. The full reciprocal complex trajectories and their exact controls are preserved with the numerical report.