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13. Explicit codimension-one Dirac test

The previous sections identify the required confinement scale phenomenologically. The next question is whether the canonical codimension-one Dirac geometry itself produces that scale.

For a hypersurface Q3⊂Q4Q_3\subset Q_4, choose a unit normal nn and a local orthonormal tangent frame eae_a, a=1,2,3a=1,2,3. In a tubular neighbourhood the ambient Dirac operator has the schematic decomposition

D4=γ(n)(∇n+D3−32H), D_4 =\gamma(n)\left(\nabla_n+D_3-\frac{3}{2}H\right),

up to the conventional sign of γ(n)\gamma(n) and the convention used for mean curvature HH. Here D3D_3 is the three-dimensional spatial Dirac operator and the curvature term is the codimension-one extrinsic contribution.

For a round three-sphere of radius RR embedded in flat four-space, the normalized mean curvature is

H=1R. H=\frac1R.

The spatial Dirac spectrum on the round S3(R)S^3(R) is

λk±=±k+3/2R,k=0,1,2,…, \lambda_k^{\pm} =\pm\frac{k+3/2}{R}, \qquad k=0,1,2,\ldots,

so the smallest absolute eigenvalue is

∣λ0∣=32R. \boxed{ |\lambda_0|=\frac{3}{2R}. }

At the nucleon-scale radius inferred above,

R=0.850 fm, R=0.850\ {\rm fm},

this gives

ℏc ∣λ0∣=32ℏcR≃348.2 MeV. \hbar c\,|\lambda_0| =\frac{3}{2}\frac{\hbar c}{R} \simeq348.2\ {\rm MeV}.

This is not the required nucleon confinement contribution

ϵN≃309.5 MeV. \epsilon_N\simeq309.5\ {\rm MeV}.

Therefore

the nucleon mass scale is not the lowest spatial S3 Dirac eigenvalue. \boxed{ \text{the nucleon mass scale is not the lowest spatial }S^3\text{ Dirac eigenvalue.} }

The scalar Laplacian does not repair the mismatch. Its first nonconstant eigenvalue on S3(R)S^3(R) is

λΔ,1=3R2, \lambda_{\Delta,1}=\frac{3}{R^2},

which corresponds to the momentum scale

ℏcλΔ,1=3ℏcR≃402.1 MeV \hbar c\sqrt{\lambda_{\Delta,1}} =\sqrt3\frac{\hbar c}{R} \simeq402.1\ {\rm MeV}

at R=0.850R=0.850 fm.

Thus the successful 309309--310310 MeV scale is not an ordinary spatial eigenvalue of a round S3S^3.

14. The normal oscillator is the viable channel

The scale that does line up is the normal confinement quantum.

For a codimension-one coordinate yy normal to Q3Q_3, introduce the local oscillator Hamiltonian

H⊥=py22μ+12μω42y2. H_\perp =\frac{p_y^2}{2\mu} +\frac12\mu\omega_4^2y^2.

Its characteristic quantum is

E⊥=ℏω4. E_\perp=\hbar\omega_4.

The V2 oscillator-symbol construction already associates one position and one momentum quadrature with every spatial direction. Removing the fourth accessible direction therefore removes one complete complex oscillator amplitude

a4=12(yℓ4+iℓ4pyℏ). a_4 =\frac1{\sqrt2} \left( \frac{y}{\ell_4} +i\frac{\ell_4p_y}{\hbar} \right).

To obtain a 4/34/3 frequency multiplier one may separately assume equal oscillator energy and unchanged action I1I_1 per surviving direction. Then

4ω4I1=3ω3I1, 4\omega_4 I_1 =3\omega_3 I_1,

and again

ω3=43ω4. \boxed{ \omega_3=\frac43\omega_4. }

The resulting three-dimensional mass contribution per quark is

ϵ4→3=43ℏω4. \boxed{ \epsilon_{4\to3} =\frac43\hbar\omega_4. }

This is distinct from the spatial Dirac curvature energy 3ℏc/(2R)3\hbar c/(2R).

If the normal oscillator length and frequency satisfy the relativistic confinement scaling

ℏω4=ℏcR⊥, \hbar\omega_4=\frac{\hbar c}{R_\perp},

then

ϵ4→3=43ℏcR⊥. \boxed{ \epsilon_{4\to3} =\frac43\frac{\hbar c}{R_\perp}. }

At

R⊥=0.850 fm, R_\perp=0.850\ {\rm fm},

one obtains

ℏω4=232.15 MeV, \hbar\omega_4 =232.15\ {\rm MeV},

and

ϵ4→3=309.53 MeV. \boxed{ \epsilon_{4\to3} =309.53\ {\rm MeV}. }

This agrees with the proton and neutron excess-energy values derived independently from their masses:

ϵp=309.751 MeV,ϵn=309.335 MeV. \epsilon_p=309.751\ {\rm MeV}, \qquad \epsilon_n=309.335\ {\rm MeV}.

The mismatch is at the few-10−410^{-4} to 10−310^{-3} relative level if R⊥R_\perp is chosen near the common mass-inferred radius.

15. What is and is not derived

The calculation now separates three statements with different logical status.

15.1 Conditional frequency-rescaling ansatz

The dimensional reduction

4→3 4\to3

removes one oscillator amplitude. Under the separately imposed frequency-rescaling ansatz,

ω3/ω4=4/3. \boxed{ \omega_3/\omega_4=4/3. }

This ratio is a model input here, not a consequence of total-action conservation.

15.2 Ruled out as the source of the nucleon bulk mass

The canonical spatial spectral scales of a round S3(R)S^3(R) are

32R \frac{3}{2R}

for the lowest Dirac mode and

3R \frac{\sqrt3}{R}

for the first scalar momentum scale. Neither gives the observed nucleon excess at the physical radius.

Thus the bulk mass cannot be identified with the ordinary spatial S3S^3 Dirac or scalar spectrum alone.

15.3 Still missing dynamically

The absolute normal oscillator scale

ω4 \omega_4

is not fixed by the currently written V2 master operator. Equivalently, V2 has not yet derived the normal confining potential, boundary condition, or stiffness that would produce

ℏω4≃232 MeV. \boxed{ \hbar\omega_4\simeq232\ {\rm MeV}. }

That is the remaining first-principles problem.

The exact operator target is therefore more specific than before. In tubular coordinates (x,y)(x,y) around the d=3d=3 sector, the normal/extrinsic part of the master operator must reduce to an effective transverse Hamiltonian

H⊥=−ℏ22μ∂y2+V⊥(y) H_\perp =-\frac{\hbar^2}{2\mu}\partial_y^2+V_\perp(y)

or its relativistic Dirac analogue, with lowest characteristic spacing

ΔE⊥≈232 MeV. \boxed{ \Delta E_\perp\approx232\ {\rm MeV}. }

The projected three-dimensional state must then inherit

ΔE3=43ΔE⊥≈310 MeV. \boxed{ \Delta E_{3} =\frac43\Delta E_\perp \approx310\ {\rm MeV}. }

16. Sharpened falsification criterion

The dimensional-constriction mass proposal is now falsifiable at the operator level.

Construct the actual normal potential or normal Dirac block generated by

P3D∞P4. P_3D_\infty P_4.

Then compute its lowest normal excitation scale ΔE⊥\Delta E_\perp without using nucleon masses or radii.

The mechanism succeeds at leading order only if

ΔE⊥≈232 MeV, \boxed{ \Delta E_\perp\approx232\ {\rm MeV}, }

because assumed frequency-rescaling constriction then predicts

43ΔE⊥≈309–310 MeV. \boxed{ \frac43\Delta E_\perp \approx309\text{--}310\ {\rm MeV}. }

If the derived normal spectrum instead begins near the spatial round-S3S^3 Dirac scale 3ℏc/(2R)3\hbar c/(2R) or another incompatible value, then the proposed explanation of the nucleon bulk mass fails.

This distinction prevents the numerical nucleon agreement from being promoted prematurely to a derivation.

17. Updated status

The explicit operator analysis strengthens and narrows the proposal:

hadronic bulk mass candidate=redistributed normal-oscillator energy, \boxed{ \text{hadronic bulk mass candidate} =\text{redistributed normal-oscillator energy,} }

not

spatial S3 Dirac energy. \text{spatial }S^3\text{ Dirac energy}.

The dimensional factor 4/34/3 is assumed, not geometrically derived. The absolute normal quantum, approximately 232232 MeV for nucleons, is the remaining quantity the master geometry must derive.

This same normal/extrinsic block is already independently required by the CKM analysis to change mode degree during the 4→34\to3 transition. A successful derivation would therefore explain two otherwise separate phenomena with one operator:

R34⟶{normal-mode energy redistribution(hadronic mass),off-diagonal mode transitions(CKM mixing). \boxed{ \mathcal R_{34} \quad\longrightarrow\quad \begin{cases} \text{normal-mode energy redistribution} & \text{(hadronic mass)},\\ \text{off-diagonal mode transitions} & \text{(CKM mixing)}. \end{cases} }

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