Dimensional Constriction and Hadronic Mass — Part 4

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18. A closed V2-native candidate for the normal gap

Although the existing master action does not yet derive the absolute transverse gap, the committed V2 scales produce one exceptionally close candidate that deserves an explicit test.

The depth-three scale is

$$\mu_3=m_e\sqrt{\frac{g_{33}}{g_{66}}},$$

with

$$g_{33}=8\sqrt7.$$

The product

$$\mu_3g_{33}$$

has dimensions of energy and is built entirely from quantities already fixed in the V2 conditional scale chain. Numerically,

$$\mu_3g_{33}=99.519575\ {\rm MeV}.$$

For a $4\to3$ transition, the dimension-incidence ratio

$$\frac{d_{\rm in}+d_{\rm out}}{d_{\rm out}} =\frac{4+3}{3} =\frac73$$

produces the candidate pre-constriction normal quantum

$$\boxed{ \Delta E_\perp^{\rm cand} =\frac73\mu_3g_{33} =232.212342\ {\rm MeV}. }$$

Fixed-action redistribution then gives

$$\boxed{ \Delta E_3^{\rm cand} =\frac43\Delta E_\perp^{\rm cand} =\frac{28}{9}\mu_3g_{33} =309.616456\ {\rm MeV}. }$$

No proton or neutron mass enters this numerical value once $\mu_3$ and $g_{33}$ are taken from the existing V2 scale construction.

Using

$$m_u=2.16\ {\rm MeV}, \qquad m_d=4.70\ {\rm MeV},$$

the resulting nucleon predictions are

$$\boxed{ m_p^{\rm cand} =2m_u+m_d+3\Delta E_3^{\rm cand} =937.869369\ {\rm MeV}, }$$

compared with

$$m_p=938.272088\ {\rm MeV}.$$

The difference is

$$\boxed{-0.403\ {\rm MeV}},$$

or about

$$-0.043\%.$$

For the neutron,

$$\boxed{ m_n^{\rm cand} =m_u+2m_d+3\Delta E_3^{\rm cand} =940.409369\ {\rm MeV}, }$$

compared with

$$m_n=939.565421\ {\rm MeV},$$

a difference of

$$\boxed{+0.844\ {\rm MeV}},$$

or about

$$+0.090\%.$$

Thus the same V2-native candidate confinement quantum lands between the proton and neutron requirements:

$$\epsilon_p=309.750696\ {\rm MeV},$$

$$\Delta E_3^{\rm cand}=309.616456\ {\rm MeV},$$

$$\epsilon_n=309.335140\ {\rm MeV}.$$

18.1 Why the $7/3$ coefficient is not yet derived

The numerical agreement alone is not sufficient to establish

$$\Delta E_\perp=\frac73\mu_3g_{33}.$$

There are two structural observations behind the candidate coefficient:

1. the spatial transition has incoming depth four and surviving depth three, so

   $$\frac{d_{\rm in}+d_{\rm out}}{d_{\rm out}}=\frac73;$$

2. each spatial direction carries a position and momentum quadrature, so the reciprocal $4\to3$ phase-space contact involves eight incoming and six surviving real quadratures, giving the same ratio

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

The V2 cross-depth action is reciprocal: the lower-depth state samples the complete marginal of the higher-depth state, while the higher-depth state feels the lower-depth state uniformly across the additional fibre. This makes a normalization involving both incident and surviving sectors plausible.

However, the currently written action does not contain a theorem saying that the reciprocal contact energy is normalized by

$$\frac{2d_{\rm in}+2d_{\rm out}}{2d_{\rm out}}.$$

Therefore the factor $7/3$ remains a conjectured cross-sector normalization, not a derived result.

18.2 The decisive normalization calculation

The candidate becomes a derivation only if the quadratic cross-sector action, after projection onto normalized $d=4$ and $d=3$ oscillator states, yields

$$\boxed{ R_{34}^{\rm norm} =\frac{2d_4+2d_3}{2d_3} =\frac73. }$$

Equivalently, starting from the normalized reciprocal interaction

$$S_{34} =g_{34}\int dt\,I_{34},$$

one must evaluate the phase-space contraction and show that its scalar normal-mode eigenvalue is

$$\boxed{ \lambda_\perp =\frac73\mu_3g_{33}. }$$

If that calculation instead yields any of the more obvious alternatives

$$\frac43, \qquad \frac74, \qquad \sqrt{\frac43}, \qquad \sqrt{\frac73},$$

then the nucleon-scale agreement disappears by tens of percent.

The $7/3$ normalization is therefore highly falsifiable rather than adjustable.

19. Current strongest hadronic-mass statement

The present calculation has reached a useful fork.

The following relation is derived conditional on assumed frequency-rescaling constriction:

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

The following absolute gap is a V2-native conjecture:

$$\boxed{ \hbar\omega_4 \stackrel{?}{=} \frac73\mu_3g_{33} =232.212342\ {\rm MeV}. }$$

Together they predict

$$\boxed{ \hbar\omega_3 \stackrel{?}{=} \frac{28}{9}\mu_3g_{33} =309.616456\ {\rm MeV}. }$$

That one number reproduces the dominant proton and neutron mass excesses to better than one part in a thousand.

The remaining proof obligation is now extremely narrow: derive or reject the reciprocal $7/3$ normalization directly from the normalized $P_3D_\infty P_4$ cross-sector action.

20. Canonical-normalization test of the $7/3$ candidate

The proposed factor

$$\frac73 =\frac{2d_4+2d_3}{2d_3} =\frac{8+6}{6}$$

must be checked against the normalization of an actual reciprocal bilinear rather than accepted from dimension counting alone.

Model the local phase-space carriers by normalized vectors

$$z_4\in\mathbb C^8, \qquad z_3\in\mathbb C^6,$$

and let

$$P:\mathbb C^8\to\mathbb C^6$$

be the orthogonal projection onto the six shared quadratures. A canonical normalized cross term has the form

$$I_{34}=\langle z_3,Pz_4\rangle+\text{c.c.}$$

or the corresponding Hermitian quadratic block.

For isotropic normalized states, the natural dimension factors generated by this construction are

$$\frac{6}{8}=\frac34$$

for shared norm fraction,

$$\sqrt{\frac68}=\sqrt{\frac34}$$

for amplitude overlap, and

$$\frac{2\cdot6}{8+6}=\frac67$$

for a symmetrically normalized reciprocal shared weight.

Its inverse is

$$\frac76,$$

not

$$\frac73.$$

The only direct way to obtain

$$\frac73$$

from the raw quadrature counts is

$$\frac{8+6}{6},$$

which counts all incident plus surviving quadratures and divides only by the surviving count. That is not the coefficient produced by the canonical normalized bilinear itself.

Therefore

$$\boxed{ \text{canonical reciprocal normalization does not derive the }7/3\text{ factor.} }$$

The excellent nucleon prediction

$$\Delta E_3^{\rm cand} =\frac{28}{9}\mu_3g_{33} =309.616456\ {\rm MeV}$$

must consequently remain a numerical conjecture unless a different, explicitly derived cross-sector energy functional produces the asymmetric normalization

$$\frac{8+6}{6}.$$

This is an important negative result: the simplest normalized phase-space overlap does not support promoting the candidate formula to a theorem.

21. Revised first-principles status

The hadronic calculation now has a clean hierarchy of confidence.

Conditional frequency-rescaling input

The separate frequency-rescaling hypothesis is

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

Empirical target

The proton and neutron require

$$\boxed{ \hbar\omega_3\approx309.5\ {\rm MeV}, \qquad \hbar\omega_4\approx232.1\ {\rm MeV}. }$$

Rejected simple derivations

The target is not supplied by

$$\frac{3}{2R}$$

as the lowest round-$S^3$ Dirac mode,

$$\frac{\sqrt3}{R}$$

as the first scalar momentum scale, or by the canonical normalized reciprocal phase-space contraction.

Remaining open derivation

The absolute normal gap must therefore come from genuine dynamics in the normal/extrinsic block:

$$\boxed{ \mathcal R_{34} \supset H_\perp }$$

with

$$\boxed{ \operatorname{gap}(H_\perp)\approx232\ {\rm MeV}. }$$

The local algebraic scalar self-contact presently written in V2 fixes sector rest-mass contact eigenvalues. V2 also admits an isotropic harmonic coordinate mean field, but the written derivation does not fix its physical stiffness or show that the self-contact coefficient determines the transverse oscillator frequency. A first-principles hadronic-mass derivation therefore requires an additional closure relation tying the normal harmonic curvature or localization width to the normalized master interaction.

The close formula

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

should be retained as a target pattern to explain, not used as evidence until its coefficient is derived.

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