Dimensional Constriction and Hadronic Mass — Part 9

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38. Exact self-consistent cubic profile test: the Gaussian factor fails

The Gaussian curvature closure must be tested against the exact stationary profile of the local attractive scalar-density mean field rather than assumed.

Consider the one-dimensional nonrelativistic normal equation

$$\left[-\frac{1}{2m_\perp}\frac{d^2}{dy^2}-G|\psi|^2\right]\psi=E\psi, \qquad G>0.$$

Its normalized localized solution is

$$\boxed{\psi(y)=\sqrt{\frac{k}{2}}\,\operatorname{sech}(ky)}$$

with

$$\boxed{G=\frac{2k}{m_\perp}}$$

and

$$E=-\frac{k^2}{2m_\perp}.$$

The self-consistent potential is

$$V(y)=-G|\psi(y)|^2=-A\,\operatorname{sech}^2(ky),$$

where

$$A=G\frac{k}{2}=\frac{k^2}{m_\perp}.$$

Near the minimum,

$$\operatorname{sech}^2(ky)=1-k^2y^2+O(y^4),$$

so

$$V(y)=-A+A k^2y^2+O(y^4).$$

Matching to

$$V(y)=-A+\frac12m_\perp\omega_{\rm curv}^2y^2+\cdots$$

gives

$$\frac12m_\perp\omega_{\rm curv}^2=A k^2=\frac{k^4}{m_\perp}.$$

Therefore

$$\boxed{\omega_{\rm curv}=\sqrt2\,\frac{k^2}{m_\perp}=\sqrt2\,A.}$$

This is the decisive correction to the Gaussian approximation. The earlier relation

$$\omega=2A$$

is not a universal consequence of self-consistency; it follows only when the mode amplitude is assumed Gaussian and its oscillator-width relation is imposed. The exact cubic self-contact solution instead gives

$$\boxed{\omega=\sqrt2\,A.}$$

38.1 Consequence for the hadronic candidate

Keeping the reciprocal normalization

$$A_{34}=\frac76\mu_3g_{33}$$

and the assumed frequency-rescaling $4\to3$ factor gives

$$\hbar\omega_3=\frac43\sqrt2\,A_{34}=\boxed{\frac{14\sqrt2}{9}\mu_3g_{33}}.$$

Numerically,

$$\frac{14\sqrt2}{9}\approx2.199887764,$$

so with

$$\mu_3g_{33}=99.5195753\ {\rm MeV},$$

one obtains

$$\boxed{\hbar\omega_3\approx218.932\ {\rm MeV}.}$$

This is far below the nucleon bulk target near $309.5$ MeV.

Therefore the simple local cubic self-contact model does not close the hadronic mass scale.

The earlier numerical success of

$$\frac{28}{9}\mu_3g_{33}$$

depends critically on the Gaussian profile assumption and cannot be promoted to a V2 prediction from the local quartic scalar interaction alone.

39. Revised status after the exact-profile test

Several parts of the constriction mechanism remain intact:

• the Schur-complement identity $g_{34}^2/g_{44}=g_{33}$ under the rank-one coupling hypothesis;
• the distinction between relaxed higher-dimensional motion and forced dimensional constriction;
• the canonical reciprocal factor $7/6$;
• the conditional assumed frequency-rescaling factor $4/3$.

What fails is specifically the Gaussian curvature factor $2$ for the local cubic self-consistent normal profile.

A successful first-principles closure must therefore derive a different normal kernel or relativistic normal equation whose exact localized profile has curvature-to-depth ratio

$$\boxed{\frac{\omega_{\rm curv}}{A}=2}$$

or otherwise supplies the missing factor

$$\frac{2}{\sqrt2}=\sqrt2$$

relative to the cubic $\operatorname{sech}$ solution.

This becomes a new falsifiable operator target rather than an adjustable coefficient.

40. Exact relativistic scalar-Soler profile test

The preceding $\operatorname{sech}$ calculation used the nonrelativistic cubic reduction. The actual V2 scalar self-contact is Dirac-like, so the relevant comparison is the exact one-dimensional scalar-Soler solitary wave.

For the cubic scalar-scalar nonlinear Dirac equation

$$(i\gamma^\mu\partial_\mu-m)\Psi +g^2(\bar\Psi\Psi)\Psi=0,$$

write the stationary frequency as $0<\omega<m$ and

$$\beta^2=m^2-\omega^2.$$

The exact scalar density follows from the known solitary-wave solution and can be reduced to

$$\boxed{ \bar\psi\psi =\frac{2\beta^2} {g^2\left(m+\omega\cosh 2\beta y\right)}. }$$

Hence the attractive scalar mean field is

$$U(y) =-g^2\bar\psi\psi =-\frac{2\beta^2} {m+\omega\cosh 2\beta y}.$$

At the center,

$$U(0)=-\frac{2(m^2-\omega^2)}{m+\omega} =-2(m-\omega).$$

Thus its central depth is

$$\boxed{A=2(m-\omega).}$$

The exact second derivative at the minimum is

$$\boxed{ U''(0)=8\omega(m-\omega)^2. }$$

If the local curvature is converted to a Schrödinger-type oscillator frequency using the standard kinetic coefficient $1/(2m)$,

$$\Omega_{\rm loc}^2=\frac{U''(0)}{m},$$

then

$$\Omega_{\rm loc} =2(m-\omega)\sqrt{\frac{2\omega}{m}},$$

and therefore

$$\boxed{ \frac{\Omega_{\rm loc}}{A} =\sqrt{\frac{2\omega}{m}}. }$$

Because $0<\omega<m$ for a bound Soler mode,

$$\boxed{ 0<\frac{\Omega_{\rm loc}}{A}<\sqrt2. }$$

In particular the ratio can never reach

$$\frac{\Omega_{\rm loc}}{A}=2.$$

So the exact relativistic scalar-Soler profile does not rescue the Gaussian curvature factor. In its nonrelativistic limit $\omega\to m$, the ratio approaches $\sqrt2$, reproducing the cubic $\operatorname{sech}$ result obtained above.

This makes the negative result considerably stronger: neither the exact local cubic Schrödinger self-contact nor the exact local cubic scalar-Dirac/Soler self-contact supplies the factor of two needed by the $309.6$ MeV closure.

The only ways left to recover that factor from first principles are therefore structurally different normal dynamics, such as

• a genuinely harmonic coordinate kernel rather than a local cubic self-density well;
• a finite-width/nonlocal contact kernel whose exact curvature-to-depth ratio is two;
• a derivative or tensor normal interaction;
• or an operator relation in which the physical confinement energy is not identified with the local curvature frequency of the scalar self-contact profile.

The exact scalar self-contact channel itself is now ruled out as the source of the missing factor.

41. Fubini--Study coherent-contact curvature test

The next V2-native possibility is that the missing normal curvature comes from the projective coherent-contact kernel itself rather than from a coordinate-space self-density profile.

For normalized degree-$N$ coherent states in

$$\operatorname{Sym}^N(\mathbb C^{d+1}),$$

the squared overlap of two rays separated by Fubini--Study angle $\theta$ is

$$\boxed{ |\langle z;N|u;N\rangle|^2 =(\cos^2\theta)^N. }$$

Near the occupied ray,

$$(\cos^2\theta)^N =1-N\theta^2 +\frac{N(3N-1)}{6}\theta^4 +O(\theta^6).$$

Therefore an attractive coherent-contact well of depth $A$,

$$V_N(\theta) =-A(\cos^2\theta)^N,$$

has local expansion

$$V_N(\theta) =-A+AN\theta^2+O(\theta^4),$$

and exact second derivative

$$\boxed{ V_N''(0)=2AN. }$$

Thus the projective coherent-contact curvature is proportional to the cumulative oscillator degree $N=n-1$.

This immediately rules it out as the universal source of the bulk hadronic confinement quantum. For the committed quark modes the relevant degrees include

$$N_d=0, \qquad N_s=3, \qquad N_u=2, \qquad N_c=19, \qquad N_t=71,$$

with the bottom pair at $N=15,16$.

A restoring frequency derived directly from the coherent-state overlap curvature would therefore vary strongly with particle band. In particular,

$$\boxed{N_d=0\Longrightarrow V_0''(0)=0,}$$

so the lightest depth-three mode would receive no projective restoring curvature at all.

That is incompatible with the approximately common bulk nucleon contribution required for both proton and neutron.

Hence

$$\boxed{ \text{the finite-band Fubini--Study coherent-contact profile is not the universal normal harmonic well.} }$$

This is distinct from the successful simplex mass theorem: the diagonal value of the reproducing kernel correctly supplies the mass multiplier $S(n,d)$, but its off-diagonal ray curvature is band dependent and therefore cannot supply the observed nearly band-independent hadronic bulk scale.

42. Surviving mathematical core after profile tests

The following pieces have survived direct calculation:

$$\boxed{ \frac{g_{34}^2}{g_{44}}=g_{33} }$$

from rank-one coupling and constrained-versus-relaxed quadratic energy,

$$\boxed{ C_{34}=\frac76 }$$

from canonical reciprocal normalization, and

$$\boxed{ \omega_3=\frac43\omega_4 }$$

from assumed frequency-rescaling dimensional constriction.

Three natural candidates for the missing normal profile have now been tested:

1. Gaussian self-consistent density: gives the desired curvature factor $2$, but only by assuming the Gaussian oscillator profile.
2. Exact local cubic Schrödinger/Dirac self-contact: gives at most $\sqrt2$, so it fails quantitatively.
3. Projective coherent-contact kernel: gives curvature proportional to $N$, so it fails universality.

Therefore the remaining normal dynamics must be a genuinely separate coordinate-space harmonic or finite-width kernel whose curvature is band blind after projection.

The target is now extremely precise. A successful V2 normal kernel $K_\perp$ must satisfy all three conditions:

$$\boxed{ \text{(i) band-blind curvature}, }$$

$$\boxed{ \text{(ii) physical energy normalization }\mu_3\frac{g_{34}^2}{g_{44}}, }$$

and

$$\boxed{ \text{(iii) curvature/depth relation yielding the required normal gap.} }$$

If no such kernel follows from the master geometry, the $309.6$ MeV agreement remains an unexplained numerical coincidence rather than a derived V2 result.

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