Coherent peak contact and smooth response

Summary: Peak coherent intensity gives an exact local contact action near its coherent branch, while a smooth logarithmic replacement has calculable mass corrections, an energy-frequency defect and a full spatial spectral criterion.

The endpoint of the homogeneous construction

Use the full current $j_z=\Psi^\dagger(\beta\otimes\Lambda_z)\Psi$, the mode rank $D_N=\binom{N+d}{d}$, and the canonical spatial kinetic action defined in Homogeneous actions with coherent simplex masses. The mode rays are local coefficient data over the actual physical spatial point; they add no physical dimensions.

For fixed $N,d$, the even-moment potential has the limit

$$\lim_{p\to\infty}\mu\left[D_{pN}\int j_z^p d\nu\right]^{1/p} =\mu\max_z|j_z|, \tag{1}$$

because $D_{pN}$ grows polynomially in $p$. Near a coherent field $\Psi_*=\chi\otimes z_0^{\odot N}$ with $J=\chi^\dagger\beta\chi>0$, this is the same smooth branch as

$$U_{\rm peak}=\mu\max_z j_z. \tag{2}$$

At that field $j_z=D_NJ|z^\dagger z_0|^{2N}$ has a unique maximizer. In the affine ray chart $z=(z_0+w)/\sqrt{1+|w|^2}$ with $w\perp z_0$, its real Hessian at the maximum is $-2ND_NJ I$. The implicit function theorem therefore gives a smooth maximizing ray $z_*(\Psi)$ in a neighborhood of the coherent field.

Varying the optimized action gives

$$G_{\rm peak}=\mu\beta\Lambda_{z_*}\Psi, \qquad G_{\rm peak}[\Psi_*]=\mu D_N\beta\Psi_*. \tag{3}$$

The ray derivative drops out of the first variation because it is a stationary point. It must still be retained in the second variation. With $P_0$ the coherent-line projector and $P_1$ its complex mode tangent projector, that complete derivative is

$$\boxed{ \delta G_{\rm peak}=A_\infty h,\qquad B_\infty=0,\qquad A_\infty=\mu D_N\left[ \beta\otimes P_0+ \frac{\beta\chi\chi^\dagger\beta}{J}\otimes P_1\right]. } \tag{4}$$

This also follows from the fixed-$N$ limit of the finite-moment Hessian: $r_j^{(p)}\to0$ for $j\ge1$, while $(p-1)r_1^{(p)}\to1$ and $(p-1)r_j^{(p)}\to0$ for $j\ge2$. Equation (4) is Hermitian and complex-linear. The same nonconstant holomorphic spatial fields used in the homogeneous construction therefore solve the peak action with $\omega=\mu D_N$, $E=\mu D_NQ$, and full spatial linear norm conservation. The carrier and action remain inverse premises, rather than derived master dynamics.

Tied rays are a real limitation

For $N\ge2$, take a unit positive-chirality field with mode coefficient

$$c_\epsilon=\sqrt{\frac{1+\epsilon}{2}}e_0^{\odot N} +\sqrt{\frac{1-\epsilon}{2}}e_1^{\odot N}. \tag{5}$$

Since $|z_0|^N+|z_1|^N\le1$ on the unit sphere, its largest coherent overlap equals the larger coefficient magnitude. Hence

$$U_{\rm peak}(c_\epsilon)=\frac{\mu D_N}{2}(1+|\epsilon|). \tag{6}$$

The one-sided derivatives at zero differ. A smooth coherent neighborhood does not give a differentiable action for arbitrary encounters between competing patterns. The finite even-moment action removes this tied-ray cusp, but has its separate scalar-null singularity.

A smooth comparison using the same local field

For a fixed positive current-density scale $\sigma$, specify

$$Z_\Psi=\int e^{j_z/\sigma}d\nu(z),\qquad U_\sigma=\mu\sigma\log Z_\Psi,\qquad p_\Psi(z)=\frac{e^{j_z/\sigma}}{Z_\Psi}. \tag{7}$$

This is globally smooth for finite field components. It tends to the positive-peak functional (2), rather than to $\max|j_z|$ for arbitrary signed currents. The distinction disappears in the coherent neighborhood under consideration. The weight $p_\Psi$ is an algebraic function of the same field; it is not an additional mediator or a thermal assumption.

The mass endomorphism and full response are

$$\begin{aligned} \mathcal M_\sigma&=\mu\langle\Lambda\rangle_p, &G_\sigma&=\beta\mathcal M_\sigma\Psi,\\ \delta G_\sigma &=\beta\mathcal M_\sigma h+ \frac{\mu}{\sigma}\left[ \langle\beta\Lambda\Psi\,\delta j\rangle_p -\langle\beta\Lambda\Psi\rangle_p\langle\delta j\rangle_p\right], &\delta j_z&=2\operatorname{Re}\langle\beta\Lambda_z\Psi,h\rangle. \end{aligned} \tag{8}$$

Thus the response includes the variation of the ray weights. It is reciprocal because it is the Hessian of one real action. Also $\operatorname{tr}_{E_N}\mathcal M_\sigma=\mu D_N$ and $\mathcal M_\sigma(0)=\mu I$. The vacuum coefficient is independent of the band. Spatial propagation retains the principal Dirac symbol for this independent canonical carrier; global nonlinear existence is a separate question.

Exact coherent reduction and mass corrections

On the coherent positive-current branch put $a=D_NJ/\sigma$. Then

$$\begin{aligned} Z_N(a)&=d\int_0^1(1-u)^{d-1}e^{au^N}du,\\ F_N(J)&=\mu D_N\frac{Z_N'(a)}{Z_N(a)},\\ F_N'(J)&=\frac{\mu D_N^2}{\sigma}\operatorname{Var}_p(u^N). \end{aligned} \tag{9}$$

At finite $a>0$, $0<F_N(J)<\mu D_N$. At large $a$ a resolved endpoint expansion gives

$$\begin{aligned} \log Z_N(a)&=a-d\log(aN)+\log(d!)+\frac{c_1}{a}+O(a^{-2}),\\ c_1&=\frac{d(d+1)(N-1)}{2N},\\ F_N(J)&=\mu\left[D_N-\frac{d\sigma}{J} -\frac{c_1\sigma^2}{D_NJ^2}+O\left(\frac{\sigma^3}{D_N^2J^3}\right)\right]. \end{aligned} \tag{10}$$

The change of variable $y=a(1-u^N)$ proves the expansion: the leading integral has Gamma weight $y^{d-1}e^{-y}$, and its first correction is $(d+1)(N-1)y/(2Na)$. Its mean $d$ gives $c_1$. This change also resolves the narrow peak in numerical quadrature.

The energy is not simply frequency times norm. On any regular stationary field of (7), the exact defect is

$$\boxed{\omega Q-E =\mu\sigma\int D_{\rm KL}(p_\Psi\|\nu)\,dV\ge0,} \tag{11}$$

where $D_{\rm KL}(p\|\nu)=\int p\log p\,d\nu$ for the probability density relative to unit invariant measure. Equation (11) follows directly from $\operatorname{Re}\langle\Psi,G_\sigma\rangle =\mu\langle j\rangle_p$ and (7). It does not assign a thermodynamic meaning to $\sigma$.

The general energy-frequency identity therefore has an explicit positive defect for this action.

A common-scale comparison

Let $\eta=\sigma\mathcal V/Q$. On the slack control, $\omega_N/\mu=D_NZ_N'(D_N/\eta)/Z_N(D_N/\eta)$ and $E_N/(\mu Q)=\eta\log Z_N(D_N/\eta)$. Independent resolved quadratures give the following ratios at $N=12,34$, without changing their assignments.

The tabulated controls satisfy $b/M<1$. Their finite-$\sigma$ corrections are physical consequences of the specified action, not numerical errors to remove. A master law would have to select its density scale and meet the actual mass and coupling constraints.

Combined bands and spatial continuation

The joint current calculation tests the same smooth action on unequal mass branches. The combined peak has width controlled by $ND_NA+MD_MB$, so the leading mixed susceptibility does not inherit the isolated band's apparent universality. Those particular finite-shape members fail that condition; the parallel slack control remains valid.

Verification

coherent_peak_action_checks.py independently optimizes the moving ray, checks complete complex spinor variations and the covariance response, verifies the exact spectral-block algebra, and resolves the assigned large-band moments at two quadrature orders. It also checks the trace and ray-tangent identities of the full mode response, the tied-ray cusp, spatial normalization and (11).