Graph contact variation

Summary: A defined graph-measure pairing retains the simplex coefficient, but its full self-variation cancels ray stiffness and does not select a coherent pattern.

A well-defined graph pairing

At a point of the physical spatial manifold $Q_d$, use the homogenized mode carrier $\mathcal H_N=\operatorname{Sym}^N(\mathbb C^{d+1})$, $D_N=\binom{N+d}{d}$, with normalized coherent vectors $|z;N\rangle$. The projective mode fibre describes oscillator-symbol data; it adds no physical spatial dimensions. The physical base and fibre are distinguished in the self-contact module.

Write $P_z=|z;N\rangle\langle z;N|$ and define the single pairing kernel

$$B_N(z,w)=D_N\operatorname{tr}(P_zP_w) =D_N|z^\dagger w|^{2N},\qquad \int B_N(z,w)d\nu_{\rm FS}(w)=1. \tag{1}$$

For a real scalar current $J(q)$ and a specified field-dependent ray $z_\Psi(q)$, the finite signed measure $\mu_q=J(q)\delta_{[z_\Psi(q)]}$ is well defined. A candidate interaction is

$$E_{\rm graph} =\frac\kappa2\int_{Q_d} \int\!\int B_N(u,v)d\mu_q(u)d\mu_q(v)\,d\mu_{Q_d}(q). \tag{2}$$

Equation (2) pairs two measures against a smooth finite-band kernel. It never multiplies two delta distributions at the same integration variable. The kernel family, its normalization and this pairing are specified additions to the action; ordinary locality of $J^2$ does not derive them. This supplies a precise candidate completion of the graph-contact construction.

Exact total variation of one occupied graph

The coincident value $B_N(z,z)=D_N$ is constant. Therefore (2) is exactly

$$\boxed{E_{\rm graph} =\frac{\kappa D_N}{2}\int_{Q_d}J(q)^2d\mu_{Q_d}(q).} \tag{3}$$

Every variation of the ray cancels before any field equation is imposed. This remains true if $z_\Psi$ depends on derivatives of the physical field, provided it is a well-defined differentiable ray along the variation. The residual variation comes from $J$, the physical measure and any additional action terms. On a fixed physical geometry,

$$\delta E_{\rm graph}=\kappa D_N\int J\,\delta J\,d\mu_{Q_d}. \tag{4}$$

For $J=\bar\Psi\Psi$, an interaction Lagrangian equal to minus (3) contributes $-\kappa D_NJ\Psi$ to the Euler--Lagrange equation obtained by varying $\bar\Psi$. In Hamiltonian form its contribution is $\kappa D_N(\Psi^\dagger\beta\Psi)\beta\Psi$. The simplex factor survives as a nonlinear scalar coupling coefficient. A normalized stationary spatial profile, its kinetic contribution and its frequency must still be solved; (3) does not identify that frequency with a rest mass proportional to $D_N$.

Why the apparent self-orientation stiffness cancels

Hold $J$ fixed and let $P(\theta)$ be a moving coherent projector. At one physical point, (2) is $\kappa D_NJ^2\operatorname{tr}(P^2)/2$. Its second derivative is

$$\frac{d^2E}{d\theta^2} =\kappa D_NJ^2\left[ \operatorname{tr}\bigl((P')^2\bigr) +\operatorname{tr}(PP'')\right]=0. \tag{5}$$

For the unit generating curve $z(\theta)=(\cos\theta,\sin\theta,0,\ldots)$, at zero,

$$\begin{aligned} |z;N\rangle'&=\sqrt N\,|N-1,1\rangle,\\ |z;N\rangle''&=-N|N,0\rangle +\sqrt{2N(N-1)}\,|N-2,2\rangle. \end{aligned}$$

The last term is absent for $N=1$. Substitution gives

$$\operatorname{tr}((P')^2)=2N, \qquad \operatorname{tr}(PP'')=-2N. \tag{6}$$

The positive source-Gram contribution and the curvature of the moving source cancel exactly. Keeping only the first term in (5) would supply an artificial self-orientation stiffness. The weaker identity $P(\delta P)P=0$ already removes a diagonal first variation; (5) accounts for both terms at second order.

Occupied frequency does not determine the response operator

For a finite coefficient illustration only, choose $J=c^\dagger c$ and the canonical time-kinetic term. Equation (3) becomes

$$E(c)=\frac{\kappa D_N}{2}(c^\dagger c)^2, \qquad i\dot c=\kappa D_N(c^\dagger c)c. \tag{7}$$

Every coefficient pattern rotates by the same phase at a fixed norm. Every projective direction is stationary. In particular, this action does not distinguish the coherent tensors from other vectors in $\mathcal H_N$.

The complete derivative of the right-hand side is

$$\delta F=A\eta+B\bar\eta,\qquad A=\kappa D_N(JI+cc^\dagger),\qquad B=\kappa D_Ncc^{\mathsf T}. \tag{8}$$

After subtracting the carrier frequency $\Omega=\kappa D_NJ$, all directions orthogonal to $c$ have zero frequency. By contrast, freezing the rank-one matrix $\kappa D_Ncc^\dagger$ would give the same occupied frequency but a transverse rotating-frame frequency $-\Omega$. When that matrix depends on the evolving $c$, its derivative restores exactly (8). Its instantaneous rank does not determine the nonlinear response spectrum.

Here the arbitrary coefficient ray $[c]\in\mathbb P(\mathcal H_N)$ must not be confused with the smaller coherent subset $[z^{\otimes N}]$. Nor is the positive coefficient norm in (7) a replacement for the scalar Dirac current used in (4).

For another distinct prescription, fix $z$ and vary $E=\kappa|s_c(z)|^4/2$, where $s_c(z)=\sqrt{D_N}\langle z;N|c\rangle$. On $c=\sqrt J|z;N\rangle$, its frequency is $\kappa D_N^2J$. This quartic evaluated density is different from the graph current $J\delta_z$ in (2). The number of evaluations matters.

The smooth quartic action, in contrast, has nonzero restoring frequencies for departures from the coherent set. Its occupied multiplier is $D_N^2/D_{2N}$. Matching one occupied eigenvalue is therefore insufficient to identify either action's complete response or its pattern-selection properties.

Two graph rays retain a reciprocal orientation interaction

At one common mode fibre, replace the single measure by $\mu=J_a\delta_{z_a}+J_b\delta_{z_b}$. For fixed positive coefficient norms $J_a,J_b$, (2) gives

$$E=\frac{\kappa D_N}{2} \left[J_a^2+J_b^2+2J_aJ_b\tau^N\right], \qquad \tau=|z_a^\dagger z_b|^2. \tag{9}$$

There is a cross-ray force although the isolated self-ray force vanishes. To make its time evolution explicit, restrict both degree-$N$ coefficient patterns to coherent tensors and pull back their canonical kinetic form. The two ray symplectic weights are $NJ_a$ and $NJ_b$. In a horizontal phase gauge, with $s=z_a^\dagger z_b$, the equations are

$$\begin{aligned} i\dot z_a&=\kappa D_NJ_b\tau^{N-1}\bar s\,(z_b-sz_a),\\ i\dot z_b&=\kappa D_NJ_a\tau^{N-1}s\,(z_a-\bar s z_b). \end{aligned} \tag{10}$$

Both objects respond to the same energy. Direct differentiation gives

$$\dot\tau=0,\qquad \frac d{dt}\left(J_a z_az_a^\dagger+J_bz_bz_b^\dagger\right)=0. \tag{11}$$

Indeed, put $S=J_a z_az_a^\dagger+J_bz_bz_b^\dagger$ and $\omega=\kappa D_N\tau^{N-1}$. The ray projectors obey $\dot P_a=-i\omega[S,P_a]$ and the same equation for $P_b$; their exact solution is common conjugation by $e^{-i\omega St}$. These $P_a,P_b$ act on the generating vectors, rather than on $\mathcal H_N$. A common unitary change of mode frame preserves all equations. Its physical rotation subgroup also preserves them.

Thus this specified first-order two-ray reduction produces reciprocal precession, with constant relative overlap. It does not by itself implement changing compatibility of complete physical spatial patterns. The bilateral orientation benchmark has a related conservation law. Full coefficient deformations, spatial transport, spinor structure and kinetic normalization remain separate parts of a particle-pair calculation in the one field $\Psi_\infty$.

Physical spatial profile

Complete local scalar response also separates rank from occupied-ray self-contact at an actual physical point. Its scalar-null source derivative has one real responsive direction, but that direction is orthogonal to the occupied complex ray and gives zero after projection onto it. This does not derive the graph kernel or its simplex normalization.

Local scalar normalization tracks the remaining spatial equation: virial and fixed-norm constraints make the nonlinear coefficient and its frequency distinct. In the flat massless local reduction, the standard regular radial ansatz has no nonzero finite-norm stationary solution. The actual physical geometric and bundle terms must be retained when testing a particle profile.

The direct derivative calculation gives the local scalar term before adding the graph pairing. Its coefficient count repeats local source eigenvalues; it does not generate the factor in (3). This makes the extra interaction premise explicit even when the graph ray is described by field derivatives.

Verification

graph_contact_variation_checks.py checks kernel normalization, complete moving-projector curvature, both coefficient Hessian blocks, fixed and varying rank-one matrices, reciprocal ray derivatives and the exact pair trajectory. It includes depths six and ten without constructing their large occupied-band matrices. Canonical normalization explains the additional physical metric needed to interpret the finite coefficient frequencies.