Coherent quartic variation

Summary: A smooth quartic action has exactly coherent extrema, an explicit complete mode Hessian and finite-mode orbital stability, with the smooth-contact frequency multiplier.

A specified action on the complete mode band

The cumulative tangent-mode carrier at physical depth $d$ is

$$\mathcal H_N=\operatorname{Sym}^N(\mathbb C^{d+1}), \qquad D_N=\binom{N+d}{d},\qquad N\ge1.$$

Its generating space consists of $d$ complexified spatial tangent directions and one homogenizing coefficient. Its projective ray space is an auxiliary mode fibre. It does not add physical spatial dimensions, or replace the actual electron spatial manifold by $\mathbb{CP}^6$. The self-contact module defines the physical base and mode fibre separately.

At a fixed spatial base point, or after a separately specified spatial profile reduction, retain every coefficient $c\in\mathcal H_N$. For a unit generating vector $u$, write $|u;N\rangle=u^{\otimes N}$ and define the normalized coherent transform

$$s_c([u])=\sqrt{D_N}\langle u;N|c\rangle, \qquad \int |s_c|^2\,d\nu_{\rm FS}=\|c\|^2=J, \tag{1}$$

where the Fubini--Study measure has total volume one. The phase of this transform depends on the lift of $[u]$; its absolute square is well defined.

Specify the canonical finite-mode action

$$S_N[c]=\int dt\left[ \frac i2(c^\dagger\dot c-\dot c^\dagger c)-E_N(c)\right], \qquad E_N(c)=\frac{\kappa}{2}\int |s_c|^4\,d\nu_{\rm FS}, \qquad \kappa\in\mathbb R. \tag{2}$$

The coefficient kinetic metric in (2) is part of this declared action. Equation (1) alone does not prove that a physical spatial Dirac reduction has this metric; its canonical normalization must be derived separately. No omitted spatial kinetic term or spinor component is silently identified with (2).

Exact symmetric product formula

Let $T:\mathcal H_N\otimes\mathcal H_N\to\mathcal H_{2N}$ be orthogonal symmetrization of all $2N$ tensor indices, with the target identified with its symmetric subspace. It satisfies $TT^\dagger=I$ and $T^\dagger T=\Pi_{2N}$, the projector onto that subspace. Put

$$\lambda_N=\frac{D_N^2}{D_{2N}},\qquad Q(c)=T(c\otimes c),\qquad A_c h=T(h\otimes c).$$

Applying the coherent resolution of identity at degree $2N$ gives

$$\boxed{\int |s_c|^4\,d\nu_{\rm FS} =\lambda_N\|Q(c)\|^2,\qquad i\dot c=\kappa\lambda_N A_c^\dagger Q(c).} \tag{3}$$

Indeed, the product of the two degree-$N$ coherent transforms is $(D_N/\sqrt{D_{2N}})s_{Q(c)}$, and $\delta Q=2A_c\delta c$. Varying (2) with respect to $\overline c$ produces the second identity, including its coefficient. It is the complete cubic vector field on $\mathcal H_N$, not its evaluation on a frozen ray.

For an orthonormal occupation basis $|\alpha\rangle$, $|\alpha|=N$, the map can be computed without a tensor of four mode indices:

$$T(|\alpha\rangle\otimes|\beta\rangle) =t_{\alpha\beta}|\alpha+\beta\rangle, \qquad t_{\alpha\beta} =\sqrt{\frac{\binom N\alpha\binom N\beta} {\binom{2N}{\alpha+\beta}}}. \tag{4}$$

The multi-index Vandermonde identity makes the sum of $t_{\alpha\beta}^2$ at any fixed $\alpha+\beta$ equal to one. This checks the coisometry normalization independently.

Coherent states are precisely the sharp extrema

Orthogonal projection cannot increase a norm. Consequently

$$\boxed{\int|s_c|^4\,d\nu_{\rm FS} \le\lambda_N\|c\|^4.} \tag{5}$$

For nonzero $c$, equality holds exactly when $c$ is a coherent tensor, $c=\sqrt J e^{i\phi}z^{\otimes N}$ with $\|z\|=1$. Here is a direct proof of the equality condition. Normalize $c$ and suppose $c\otimes c$ lies in the fully symmetric $2N$-index subspace. Exchanging one index between the two copies then has expectation one. That expectation is $\operatorname{tr}(\rho_1^2)$, where $\rho_1$ is the one-index contraction of $|c\rangle\langle c|$. Since $\rho_1$ is positive with trace one, it must have rank one. A tensor with a pure one-index contraction factors across that index; symmetry forces all remaining indices to carry the same vector. The converse follows immediately from $z^{\otimes N}\otimes z^{\otimes N}=z^{\otimes2N}$. For $N=1$ every vector is already coherent.

This is the fourth-moment instance of coherent-state concentration. The broader convex-function inequality is due to Lieb and Solovej, with equality and quantitative results developed by Nicola, Riccardi and Tilli. The elementary projector proof above supplies the particular result needed for the action without an entropy argument.

At fixed $J$, the coherent orbit is therefore the complete set of global energy maxima for $\kappa>0$, and global minima for $\kappa<0$. This is variational selection in the specified finite coefficient space; it does not establish the preferred sign of a physical scalar interaction.

Exact coherent branch and nonlinear frequency

For $c_0=\sqrt J z^{\otimes N}$, (3) gives

$$\boxed{c(t)=e^{-i\Omega_N t}c_0,\qquad \Omega_N=\kappa J\lambda_N =\kappa J\frac{D_N^2}{D_{2N}}.} \tag{6}$$

Every coefficient equation is satisfied. The ray $[z]$ is stationary and only its common phase evolves. Coherence therefore supplies an exact stationary branch of the smooth action, without replacing its spectrum by that of a point insertion.

The norm $J$ is conserved by the phase symmetry. All mode-frame rotation moments are also conserved. In tensor notation,

$$\rho_1=\frac1N\bigl(\langle c,a_j^\dagger a_i c\rangle\bigr)_{ij}, \qquad \operatorname{tr}\rho_1=J, \qquad \dot\rho_1=0. \tag{7}$$

The operators $a_j^\dagger a_i$ act on the symmetric occupation indices. Equation (7) follows from the $U(d+1)$ invariance of (2). This mode-fibre symmetry is an assumption of the isotropic completion, not an additional set of physical spatial rotations. In particular, conservative evolution does not generally capture an arbitrary pattern into a coherent minimum: its energy and the one-index contraction are fixed.

Complete perturbation equation

Choose $z=e_0$ by mode-frame covariance and use a frame rotating with $\Omega_N$. For $c=e^{-i\Omega_Nt}(c_0+\eta)$, define the normalized Toeplitz eigenvalue ratio

$$r_j=\frac{N(N-1)\cdots(N-j+1)} {2N(2N-1)\cdots(2N-j+1)},\qquad r_0=1, \qquad 0\le j\le N. \tag{8}$$

Here $j$ counts occupations outside coordinate zero. Its complex multiplicity is $\binom{j+d-1}{d-1}$. The full linearized equation is

$$\boxed{ \begin{aligned} i\dot\eta_0&=\kappa J\lambda_N(\eta_0+\overline\eta_0),\\ i\dot\eta_j&=\kappa J\lambda_N(2r_j-1)\eta_j, \qquad 1\le j\le N. \end{aligned}} \tag{9}$$

To obtain it directly, write $V$ for multiplication by $|s_{c_0}|^2$ followed by projection. Its eigenvalue in sector $j$ is $J\lambda_Nr_j$, as follows from the exact FS monomial integral. The variation of $|s|^2s$ has both terms

$$\delta(|s|^2s)=2|s|^2\delta s+s^2\delta\overline s.$$

The second term has a nonzero projected matrix entry only when the two occupation multi-indices sum to $2Ne_0$, so only the radial entry survives. Subtracting the rotating-frame frequency gives (9). Dropping this conjugate term would give an incorrect radial equation.

Since $r_1=1/2$, the $d$ complex tangent directions to the coherent ray are zero modes. For $j\ge2$, $0<r_j<1/2$, and all remaining frequencies are real and have the same sign opposite to $\kappa$. When $N\ge2$, the smallest nonzero absolute frequency is

$$|\Omega_{\rm transverse,min}| =|\kappa|J\lambda_N\frac{N}{2N-1}. \tag{10}$$

The radial equation contains the phase zero mode and the phase drift caused by changing the conserved norm. On a fixed-norm space modulo phase this does not supply an exponentially growing perturbation.

Finite-mode nonlinear orbital stability

At fixed nonzero $J$, the state sphere is compact. For $N\ge2$ the nonnegative deficit

$$\mathcal D(c)=\lambda_NJ^2-\int|s_c|^4\,d\nu_{\rm FS}$$

vanishes exactly on the compact coherent orbit. It has a strictly positive minimum on every closed subset a positive distance from that orbit. Norm and energy conservation therefore imply nonlinear orbital stability of the coherent set under (3), for either nonzero sign of $\kappa$. The polynomial vector field exists globally because the norm remains bounded. This argument supplies finite-mode nonlinear stability, rather than inferring it solely from real linearized frequencies.

The conclusion concerns the whole coherent set, including phase and ray rotations. It is not attraction to one prepared ray. Physical spatial variation, omitted bands, Dirac components, nuclear sources and additional interaction terms require their own stability analysis.

Consequence for the mass construction

For unit coefficient norm and a band-independent $\kappa$, the derived frequency multiplier is $D_N^2/D_{2N}$. The occupied point-contact multiplier is $D_N$. Their ratio is

$$\frac{\lambda_N}{D_N} =\frac{D_N}{D_{2N}} =\prod_{r=1}^d\frac{N+r}{2N+r},$$

which depends on $N$ and tends to $2^{-d}$ for large $N$ at fixed $d$. At the specified electron and muon bands $(12,34)$ in depth six, (6) gives the smooth-action ratio $137.029686760779$, while the point count gives $206.764705882353$. These are the distinct prescriptions compared in the contact-kernel audit.

Thus coherent extrema and finite-mode stability can be established for a specific action while its mass-sequence multiplier remains different. Converting (6) into a physical rest mass additionally requires the spatial Dirac equation, its normalized profile and the identification of its pole.

The homogeneous coherent action uses a normalized square root of the quartic current integral to obtain the occupied coefficient $D_N$. Its full variation has a different response, with a cancellation of the radial conjugate term. It supplies exact nonconstant spatial solutions for the specified independent bundle, while the changed functional and its normalization require a master origin. This does not alter the quartic action's result (6).

Reciprocal extension

The complete pair action retains both full mode families. A generic pair leaves the coherent subset, and conservation relates its shape deformation to changing density contact. The aligned stationary set has a separate complete stability calculation.

Verification and related

The current projection identity now expresses this same smooth quartic as an exact weighted sum of all reduced tensor-current purities. It identifies the coefficients supplied by the projection, including signed full-spinor currents and different grades. This gives a direct operator comparison with the shell-weighted self-mass action, rather than inferring that action from the coherent count alone.

coherent_contact_variation_checks.py checks independent FS quadrature, the full nonlinear gradient and both Hessian blocks, symmetric-product normalization, the coherent stationary equation, all perturbation sectors and complete finite-mode evolution. Exact rational formulas also cover large bands without allocating their matrices. The supporting algebra is in coherent_mode_algebra.py.

• Contact-kernel audit — distinguishes the point and smooth spectral prescriptions used here.
• Canonical coupling normalization — specifies the physical kinetic metric needed before interpreting the coefficient frequency.
• Bilateral polarization coupling — separates conservative orientation evolution from encounter-driven changes in compatibility.