Exact Sech-Power Normal Kernels

Profile, Well And Curvature

Let $a=\hbar^2/(2m)>0$, with $m$ the normal kinetic coefficient, and consider the classical field functional

$$E=a\int|\psi'|^2dy-\frac g2\iint\rho(y)K(y-y')\rho(y')dy\,dy',\qquad \rho=|\psi|^2.$$

The kinetic action is $i\hbar(\psi^*\dot\psi-\dot\psi^*\psi)/2$. Require a real even normalized convolution kernel, $\int K=1$, and the stationary branch

$$\phi=B\operatorname{sech}^{p}(\kappa y),\quad gK*\phi^2=A\operatorname{sech}^2(\kappa y),\quad p,\kappa>0.$$

Differentiating the profile gives

$$\phi''/\phi=\kappa^2[p^2-p(p+1)\operatorname{sech}^2(\kappa y)].$$

Thus the full stationary equation is satisfied precisely when

$$A=ap(p+1)\kappa^2,\qquad \mu=-ap^2\kappa^2.$$

The frozen-well curvature diagnostic is

$$R=\frac{\hbar\Omega_{\rm curv}}A=\frac2{\sqrt{p(p+1)}}.$$

It is different from the carrier frequency $\mu/\hbar$, frozen bound spacings, or the responding field's fluctuation frequencies.

Necessary Kernel And Exact Existence Classification

Use $\widehat f(k)=\int e^{-iky}f(y)dy$ and $t=k/(2\kappa)$. Substitution $u=e^{2\kappa y}$ in the integral gives

$$\widehat{\operatorname{sech}^{\alpha}(\kappa\cdot)}(k) =\frac{2^{\alpha-1}}{\kappa\Gamma(\alpha)} \Gamma(\alpha/2+it)\Gamma(\alpha/2-it).$$

This follows directly from Euler's beta integral, recorded in DLMF 5.12.3. Dividing the normalized transforms of the well and density fixes the unique possible kernel multiplier:

$$\boxed{\widehat K_p(k)=\frac{\Gamma(p)^2|\Gamma(1+it)|^2}{|\Gamma(p+it)|^2}.}$$

The denominator never vanishes for real $k$. The gamma ratio asymptotics in DLMF 5.11 give $\widehat K_p(k)\sim\Gamma(p)^2|t|^{2-2p}$. A finite signed measure has a bounded Fourier transform, bounded by its total variation. Consequently no finite signed measure kernel produces this exact branch for $0<p<1$. This includes all integrable finite-width kernels, without requiring positivity.

For every $p>1$ a positive integrable kernel does exist. Define the positive normalized integration weight

$$w_p(u)=(p-1)(1-u)^{p-2},\qquad 0<u<1.$$

Its logarithmic transform obeys

$$\int_0^1 w_p(u)u^{it}du =\frac{\Gamma(p)\Gamma(1+it)}{\Gamma(p+it)}.$$

Push the product weight $w_p(u)w_p(v)du\,dv$ to the real coordinate $y=(\log u-\log v)/(2\kappa)$. The resulting positive even convolution density has transform $\widehat K_p$, proving existence, normalization and positivity. These weights construct a deterministic integral kernel; they do not introduce a probability interpretation of electrons. Its second moment is finite:

$$\int y^2K_p(y)dy=\frac{\psi_1(1)-\psi_1(p)}{2\kappa^2},$$

where $\psi_1$ is the trigamma function. The kernel may be unbounded at its origin when $1<p\le3/2$, but it remains integrable and its Fourier multiplier is bounded. Here finite width means finite second moment, not compact support or a bounded central value.

At $p=1$, $K_1=\delta_0$, the local cubic case. For integer $p\ge2$, the gamma recurrence supplies an especially simple realization:

$$\widehat K_p(k)=\prod_{j=1}^{p-1}\frac{j^2}{j^2+t^2},\qquad K_p=\mathop{*}_{j=1}^{p-1}\left[j\kappa e^{-2j\kappa|y|}\right].$$

Each factor is a normalized positive Helmholtz response. The case $p=2$, $\ell=1/(2\kappa)$, is exactly the exact normal-response calculation's solution. Integer labels in this formula index factors of a normal response kernel; they are not particle indices or physical depths.

Complete Branch Normalization And Energy

Integrating the well equation fixes $gQ=2A/\kappa$, where $Q=\int\phi^2dy$. Therefore

$$Q=\frac{2ap(p+1)\kappa}{g},\qquad B^2=\frac{2A\Gamma(p+1/2)}{g\sqrt\pi\Gamma(p)}.$$

The kernel depends on both p and kappa. Varying kappa changes the action; the displayed Q(kappa) is not an arbitrary-norm family at one fixed kernel. These are stationary consistency relations for the stated action, not a fit of amplitude to a particle mass. The exact kinetic and total energies are

$$T=\frac{ap^2\kappa^2Q}{2p+1},\quad \int\rho W=\frac{2pAQ}{2p+1},\quad \boxed{E=-\frac{ap^3\kappa^2Q}{2p+1}.}$$

Hence $E/Q\ne\mu$, as expected for a quartic mean-field functional. The well appears only once with the factor one half in the conserved energy.

For these positive normalized kernels, the fixed-$Q$ energy is bounded below on the one-dimensional $H^1$ domain. Young's inequality bounds the attraction by $g\|\psi\|_4^4/2$; the elementary one-dimensional interpolation bound $\|\psi\|_4^4\le C Q^{3/2}\|\psi'\|_2$ then bounds the energy below by a quadratic minus a linear function of $\|\psi'\|_2$. This establishes boundedness, not that the displayed branch minimizes energy or is dynamically stable. It makes no claim for a full higher-dimensional radial extension.

Consequence For The Ratio-Two Target

Within the finite-measure convolution family, $p\ge1$, and therefore $R\le\sqrt2$. Ratio two requires

$$p=(\sqrt5-1)/2<1.$$

The required multiplier then grows as $|k|$ to power $3-\sqrt5$. A distributional derivative response may have that order, but it is outside the finite-measure family and needs its own action-domain and stability analysis. This is a precise change of premise, not an exclusion of all same-field dynamics.

The exact normal-response calculation's local-power action instead achieves this same profile with exponent $\sigma=1/p=(1+\sqrt5)/2$. That action is nonlinear in density and is not a linear convolution of density, so it is not excluded. Its frozen potential has only one bound level. In the general frozen well, levels are $-a\kappa^2(p-j)^2$, $j=0,1,\ldots<p$; a bound-to-bound gap exists only for $p>1$, and its first value is $a\kappa^2(2p-1)$. Curvature two consequently supplies no such gap in either realization of the $p<1$ profile.

The same one-dimensional construction can occupy a normal factor at depths 2, 3, 4, 5, 6 and 10 under the exact normal-response construction's explicit product metric and tangential-marginal action. It is identical at each depth; it selects neither depth five nor the quark depths. Its main contribution is a reproducible class of positive response kernels and a sharpened spectral target. Deriving a kernel and product structure from the master field remains open.

Reproduction

The beta-integral substitution, normalized gamma-ratio multiplier, positive pushforward construction and energy integrals displayed above give the analytic reproduction. For integer exponents, successive factors can be checked directly with the gamma recurrence; the p=2 profile, energy and frozen gap are also checked by the normal-response checker. The finite-measure obstruction uses boundedness of Fourier transforms of finite measures and the stated gamma asymptotics; finite numerical checks do not replace that proof.