Dimensional Constriction and Hadronic Mass — Part 11

Contents · Previous · Next

49. Status after the baryon-topology test

The following candidate sources of the missing doubling have now been explicitly rejected as automatic consequences of the present V2 structure:

• canonical coordinate/momentum quadratures;
• reciprocal variation of one interaction action;
• Dirac chirality;
• particle/antiparticle structure;
• local cubic Schrödinger self-contact;
• local scalar-Soler self-contact;
• finite-band Fubini--Study contact curvature;
• naive $K_3$ endpoint-incidence counting.

The baryon graph still supplies two independent relative Jacobi modes per spatial axis, but ordinary harmonic quantization does not turn those into a universal extra factor two multiplying the previously derived band-blind constriction scale.

Therefore the remaining factor two must come from a genuinely additional operator contribution or from a revision of which V2 quantity has been identified with the physical oscillator gap.

50. Full $3$--$4$ coupling-matrix spectrum

The remaining factor of two could in principle come from a second eigen-branch of the coupled depth-three/depth-four quadratic form. The complete coupling matrix is

$$G_{34} =\begin{pmatrix} g_{33}&g_{34}\\ g_{34}&g_{44} \end{pmatrix}.$$

Using the V2 rank-one relation

$$g_{34}^2=g_{33}g_{44},$$

its determinant vanishes exactly,

$$\det G_{34}=0.$$

Therefore its eigenvalues are

$$\boxed{ \lambda_-=0, \qquad \lambda_+=g_{33}+g_{44}. }$$

For the current couplings,

$$g_{33}=8\sqrt7, \qquad g_{44}=\frac{12}{\sqrt7},$$

so

$$\boxed{ \lambda_+ =\frac{68}{\sqrt7}. }$$

Relative to the surviving depth-three stiffness,

$$\frac{\lambda_+}{g_{33}} =1+\frac{g_{44}}{g_{33}} =\boxed{\frac{17}{14}},$$

not $2$.

Thus the full rank-one $3$--$4$ coupling matrix contains one flat branch and one positive branch, but it does not contain two equal positive eigenmodes and cannot supply the missing universal doubling.

This strengthens the no-go result: the factor two is not hidden in the eigenvalue multiplicity of the existing rank-one coupling block.

51. Virial bookkeeping and the Dirac-oscillator test

A possible rescue of the missing factor two is to interpret

$$\frac{14}{9}\mu_3g_{33}$$

as only the potential-energy half of a harmonic mode and then invoke equality of kinetic and potential energy. This does not follow from the preceding derivation.

For dimensionless canonical quadratures $[X,P]=i$, the general quadratic oscillator is

$$H =\frac{\alpha}{2}P^2 +\frac{\beta}{2}X^2.$$

Its frequency is

$$\boxed{\omega=\sqrt{\alpha\beta}.}$$

When canonical isotropy fixes

$$\alpha=\beta=\varepsilon,$$

one has

$$H =\frac{\varepsilon}{2}(P^2+X^2) =\varepsilon\left(a^\dagger a+\frac12\right),$$

so the level spacing is

$$\boxed{\Delta E=\varepsilon.}$$

The virial theorem gives equal kinetic and potential expectation values in an eigenstate, but their sum is already the expectation value of this single Hamiltonian. It does not multiply the level spacing by two.

Therefore the factor-two discrepancy cannot be repaired by adding a second copy of the canonical kinetic energy to the harmonic constriction energy.

51.1 Relativistic Dirac oscillator

The V2 master operator is Dirac type, so the normal oscillator can also be tested in the standard one-dimensional Dirac-oscillator form

$$H_D =\alpha\left(p-i\beta m\omega x\right)+\beta m.$$

Squaring gives a harmonic operator plus the spinor shift,

$$H_D^2 =p^2+m^2\omega^2x^2+m^2-m\omega\beta,$$

up to the equivalent component convention.

The positive-energy spectrum therefore has the schematic exact form

$$\boxed{ E_n^2=m^2+2m\omega n }$$

for one component, with the partner component shifted by one oscillator level.

The important point is structural: the spinor term shifts the two components relative to one another; it does not create two equal additive copies of the oscillator energy.

In the nonrelativistic limit,

$$E_n-m\simeq n\omega,$$

so the physical spacing tends to the ordinary harmonic spacing, not $2\omega$.

For finite $m$,

$$E_{n+1}-E_n =\sqrt{m^2+2m\omega(n+1)} -\sqrt{m^2+2m\omega n},$$

which is not a universal factor-two multiple of $\omega$ and in fact decreases with $n$.

Hence

$$\boxed{ \text{Dirac-oscillator spinor structure does not supply the missing universal factor }2. }$$

The relativistic normal operator therefore leaves the same hard discrepancy as the canonical harmonic analysis.

52. Cross-depth normalization audit and current ledger

The existing V2 cross-depth modules were searched for an independent normalization theorem that could replace the canonical reciprocal factor

$$\frac76$$

by

$$\frac73.$$

No such theorem is present in the current V2 files.

The cross-depth framework does establish:

• asymmetric higher-to-lower marginalization;
• reciprocal response of the two participating sectors;
• invariant/singlet polarization selection;
• common geometric measures for self-contact and cross-depth contact;
• rank-one depth-$3$--depth-$4$ coupling through $$g_{34}^2=g_{33}g_{44}.$$

But none of the written cross-depth actions supplies an extra factor of two in the normalized scalar oscillator quantum.

The canonical phase-space carrier has eight real quadratures before $4\to3$ projection and six shared quadratures after projection. A symmetric normalized reciprocal overlap therefore gives

$$\boxed{ C_{34}^{-1} =\frac{8+6}{2\cdot6} =\frac76. }$$

The apparently attractive ratio

$$\frac{8+6}{6}=\frac73$$

counts the same shared carrier twice and is not the normalization of the canonical symmetric bilinear.

The independent reciprocal zero-point trace reaches the same conclusion when converted to a per-surviving-direction oscillator energy:

$$\frac{E_{0,4}+E_{0,3}}{3} =\frac76\hbar\omega.$$

Thus the current V2-supported chain remains

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

$$\varepsilon_{34}^{\rm scalar} =\mu_3g_{33},$$

$$\varepsilon_{34}^{\rm recip} =\frac76\mu_3g_{33},$$

and under the assumed frequency-rescaling $4\to3$ redistribution,

$$\boxed{ \varepsilon_{\rm constr} =\frac43\frac76\mu_3g_{33} =\frac{14}{9}\mu_3g_{33} =154.808228\ {\rm MeV}. }$$

The empirical light-nucleon bulk term remains almost exactly

$$2\varepsilon_{\rm constr} =309.616456\ {\rm MeV},$$

but the second factor of two has now failed the following independent tests as an automatic consequence of the present theory:

1. raw phase-space dimension counting;
2. reciprocal zero-point normalization;
3. coordinate/momentum quadrature counting;
4. virial kinetic-plus-potential bookkeeping;
5. Gaussian self-contact curvature once exact profiles are imposed;
6. exact cubic Schrödinger self-localization;
7. exact scalar-Soler localization;
8. Fubini--Study coherent-contact curvature;
9. reciprocal Euler--Lagrange variation;
10. Dirac chirality;
11. particle/antiparticle doubling;
12. normal-connection counting without a fixed norm;
13. $K_3$ endpoint-incidence counting;
14. $K_3$ collective-mode multiplicity;
15. the spectrum of the rank-one $3$--$4$ coupling matrix;
16. parity/even-level selection in the presently written V2 modules;
17. ordinary and Dirac-oscillator quantization;
18. reinterpretation as a mass-squared eigenvalue.

The last point is especially firm because the V2 self-contact construction explicitly defines the projected interaction as a Lorentz-scalar rest-energy endomorphism entering the Dirac equation linearly,

$$\left(i\gamma^a\nabla_a-m(n,d)\right)\psi_{N,d}=0,$$

with

$$m(n,d)=\mu_dS(n,d).$$

Therefore the missing doubling cannot be recovered by taking a square root of two equal quadratic mass contributions without changing the present mass theorem.

52.1 Conditional trial value, not a derived half-contribution

The numerical combination

$$\varepsilon_{\rm trial}=\frac{14}{9}\mu_3g_{33} =154.808228\ {\rm MeV}$$

uses the scalar-unit identification, reciprocal normalization, and $4/3$ frequency-rescaling ansatz. It has not been obtained as a pole of a derived hadronic generator. Its doubled value, $309.616456$ MeV, is a comparison with an allocated constituent residual, not a separately measured spectral line.

The preceding tests reject particular doubling prescriptions under their stated assumptions. They do not establish this trial value first and leave only a missing factor of two, nor exclude all alternative master-operator constructions. Sections 64--76 analyze the more basic normalization and spectral questions.

53. Codimension-one extrinsic Dirac term: real second operator, but no V2 closure yet

The projected Dirac geometry does contain a genuinely independent lower-order term that is not the same scalar contact oscillator counted twice.

For a codimension-one hypersurface $Q_3\subset Q_4$, the ambient Dirac operator has the standard schematic decomposition

$$D_4 =\gamma(n)\left(\nabla_n+D_3-\frac{3}{2}H\right),$$

up to sign and mean-curvature conventions.

Thus the projection naturally produces an algebraic extrinsic contribution of magnitude

$$\boxed{ E_H=\frac{3}{2}\hbar c\,|H|. }$$

This is structurally different from the scalar self-contact/rest-energy endomorphism and therefore could, in principle, provide a legitimate second contribution rather than a double count.

The missing light-nucleon contribution is

$$\varepsilon_{\rm constr} =154.808228\ {\rm MeV}.$$

Requiring the extrinsic Dirac term to equal this gives

$$\frac{3}{2}\hbar c\,|H_*| =154.808228\ {\rm MeV},$$

hence

$$\boxed{ |H_*| =0.52301761\ {\rm fm}^{-1}. }$$

If represented by a round hypersphere with normalized mean curvature $H=1/R$, this corresponds to

$$\boxed{ R_*=1.9119815\ {\rm fm}. }$$

The important point is not the numerical radius itself, since a measured three-dimensional hadronic radius cannot be identified with a pre-projection four-dimensional confinement radius. The decisive issue is whether V2 already fixes the effective mean curvature $H$ from its existing couplings, projectors, contact scale, or normal geometry.

The present V2 files do not provide such an equation.

The projected-geometry theorem states only that normal-connection and normal-Clifford data survive as lower-order algebraic terms,

$$\mathcal D_d =i\gamma_d^a(\partial_a+\omega_a+A_a^\perp)+\Phi_d,$$

while the CKM module places second-fundamental-form and related extrinsic terms inside the remainder $\mathcal R_{34}$. Their existence is therefore supported, but their norm is not fixed.

For codimension one there is also no nontrivial multi-direction normal rotation that automatically supplies an additional universal scalar magnitude. The normal bundle has only one real normal direction, so an independent non-Abelian normal-connection multiplicity cannot be invoked here as a factor of two.

Therefore

$$\boxed{ \text{extrinsic curvature is a legitimate second operator channel, but its coefficient is not derived by current V2.} }$$

This is materially different from the previously rejected doubling mechanisms: the mean-curvature term is not bookkeeping duplication. It is a real candidate closure term. But obtaining

$$E_H=\varepsilon_{\rm constr}$$

would presently be an additional geometric condition unless the master embedding equations can be shown to imply

$$\boxed{ |H|=\frac{2}{3\hbar c}\,\varepsilon_{\rm constr} =\frac{28}{27\hbar c}\,\mu_3g_{33}. }$$

The next proof target is therefore sharply defined: derive or falsify this mean-curvature relation from the existing embedding/projector geometry rather than imposing it from the nucleon mass.

Contents · Previous · Next