Dimensional Constriction and Hadronic Mass — Part 8

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34. Projection-normalization lemma for the constriction energy

The remaining normalization question can be stated sharply:

If the $4\to3$ constriction quadratic form is projected into the same normalized depth-three scalar-contact channel that defines the ordinary $d=3$ mass operator, what dimensionful coefficient multiplies its dimensionless coupling matrix?

The answer is fixed by the existing projection theorem, provided the constriction term belongs to that same scalar channel.

34.1 The depth-three projected scalar channel has one energy coefficient

The V2 scalar self-contact projection is

$$P_{3,N} \left(\kappa_3J_3\delta_{[z]}\right) P_{3,N} = \mu_3P_N\delta_{[z]}P_N,$$

with

$$\mu_3=\kappa_3R_3B_3.$$

After the normalized spatial profile and scalar spinor contraction have been performed, $\mu_3$ is the coefficient of the scalar rest-energy endomorphism on the depth-three mode fibre.

The remaining mode dependence is dimensionless:

$$P_N\delta_{[z]}P_N =S(N+1,3)|z;N\rangle\langle z;N|.$$

Thus, before inserting a particular occupied band, the normalized depth-three scalar channel has the operator form

$$\boxed{ H_3^{\rm scalar} =\mu_3\,\widehat H_3, }$$

where $\widehat H_3$ is dimensionless.

34.2 The coupling matrix is dimensionless

The conditional fixed-point equation is

$$\frac{(M_d^\star)^2}{g_{dd}} = \frac{m_e^2}{g_{66}}.$$

Since both numerators have dimensions of energy squared, the self-couplings $g_{dd}$ are dimensionless. The rank-one relation

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

therefore also makes $g_{34}$ dimensionless.

Consequently the reduced constriction stiffness

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

is dimensionless.

34.3 Projection of the quadratic constriction block

Suppose the normal/extrinsic $4\to3$ block, after the fourth coordinate is constrained and the non-mode factors are projected onto the same normalized depth-three scalar channel, has dimensionless quadratic form

$$\widehat Q_{34} =\frac12 \begin{pmatrix}x&y\end{pmatrix} \begin{pmatrix} g_{33}&g_{34}\\ g_{34}&g_{44} \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}.$$

Because the $g$ matrix and normalized amplitudes are dimensionless, the projected physical quadratic operator must have the form

$$Q_{34}^{\rm phys} =E_{34}\widehat Q_{34}$$

for some energy coefficient $E_{34}$.

If this block is not a new independent interaction channel, but is the normal/extrinsic component of the same projected scalar contact whose tangential depth-three component is already normalized by $\mu_3$, covariance of the single projected scalar vertex requires

$$\boxed{ E_{34}=\mu_3. }$$

Introducing a different coefficient would amount to introducing an additional sector-dependent scalar normalization not present in the assumed single-vertex master action.

Therefore, under the same projected scalar vertex hypothesis,

$$\boxed{ Q_{34}^{\rm phys} =\mu_3\widehat Q_{34}. }$$

The constrained-versus-relaxed energy penalty is then

$$\Delta Q_{34}^{\rm phys} = \frac12\mu_3 \frac{g_{34}^2}{g_{44}}x^2 = \frac12\mu_3g_{33}x^2.$$

Hence the natural physical stiffness unit of the constriction block is

$$\boxed{ \mu_3g_{\rm con} =\mu_3g_{33}. }$$

34.4 No simplex-factor double counting

The factor $S(n,3)$ is the eigenvalue generated when the localized projective point contact is evaluated on a particular cumulative mode band.

The hadronic constriction term above is instead the band-independent normal stiffness of the projected scalar vertex before selecting a quark excitation level. Multiplying it by another $S(n,3)$ would therefore count the occupied-ray contact multiplicity a second time and would destroy the observed nearly universal bulk nucleon contribution per valence quark.

The appropriate energy unit for the universal normal block is thus $\mu_3$, not an individual quark mass $\mu_3S(n,3)$.

34.5 Consequence

Combining the projection lemma with the previous Schur-complement result gives

$$\boxed{ E_{\rm stiffness} =\mu_3\frac{g_{34}^2}{g_{44}} =\mu_3g_{33}. }$$

Canonical reciprocal normalization then gives the central well depth

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

Gaussian density curvature gives

$$\boxed{ \hbar\omega_{\rm common} =2A_{34} =\frac73\mu_3g_{33}. }$$

Fixed-action $4\to3$ redistribution finally gives

$$\boxed{ \hbar\omega_3 =\frac43\hbar\omega_{\rm common} =\frac{28}{9}\mu_3g_{33} =309.616456\ {\rm MeV}. }$$

35. Exact status of the closure

The normalization problem is therefore closed conditional on one identifiable V2 statement:

$$\boxed{ \mathcal R_{34}^{\rm normal} \text{ is a component of the same normalized scalar master vertex that defines }\mu_3. }$$

Under that statement, no additional dimensional constant is allowed in the projected normal quadratic block, and the $\mu_3$ prefactor follows from the existing sector normalization.

If instead $\mathcal R_{34}^{\rm normal}$ is an independent derivative, tensor, or nonlocal vertex with its own projected coefficient, then a new energy normalization is allowed and the hadronic formula remains conditional.

So the remaining question is no longer numerical or algebraic. It is operator-theoretic:

$$\boxed{ \text{Is the constriction block contained in the scalar self-contact vertex, or is it an independent master-action vertex?} }$$

36. Scalar normal-singlet route: the confinement block need not be in $\mathcal R_{34}$

The operator-classification issue can be resolved without reclassifying the remainder $\mathcal R_{34}$.

The V2 coupling geometry already states that when a lower-depth operator is a scalar under the normal group $Spin(q)$, integration over the normal fibre retains the $Spin(q)$-singlet part of the higher-dimensional current.

For a $4\to3$ reduction the normal fibre is one-dimensional, so a scalar lower-depth contact has a surviving normal-singlet channel under fibre integration.

The leading cross-sector decomposition is

$$P_dD_\infty P_e = \mathcal I_{de}^{\rm pol} \otimes \mathcal C_{de}^{\rm mode} + \mathcal R_{de},$$

where $\mathcal C_{de}^{\rm mode}$ is the pushed-forward coherent contact and $\mathcal R_{de}$ contains extrinsic, derivative, and nonlocal corrections.

Therefore the order-zero scalar constriction energy does not have to be assigned to $\mathcal R_{34}$. It can arise from the scalar-singlet component of the leading pushed-forward coherent contact:

$$\boxed{ \mathcal C_{34}^{\rm scalar} \subset \mathcal I_{34}^{\rm singlet} \otimes \mathcal C_{34}^{\rm mode}. }$$

This matters because the leading coherent-contact channel is precisely the one for which the scalar self-contact projection supplies the normalized energy coefficient $\mu_3$.

The operator picture can therefore be separated consistently:

$$\boxed{ \text{bulk hadronic constriction energy} \longleftarrow \mathcal C_{34}^{\rm scalar} }$$

while

$$\boxed{ \text{generation-changing CKM structure} \longleftarrow \mathcal R_{34} \text{ and/or nontrivial mode contact}. }$$

This removes the apparent conflict between the hadronic normalization argument and the explicit definition of $\mathcal R_{34}$ as an extrinsic/derivative/nonlocal remainder.

37. Conditional closure of the bulk nucleon scale

Under the following V2-native assumptions:

1. the $4\to3$ scalar constriction is the normal-singlet component of the leading coherent-contact channel;
2. that scalar channel inherits the same normalized depth-three contact-energy unit $\mu_3$;
3. the coupling block obeys the rank-one relation $g_{34}^2=g_{33}g_{44}$;
4. the reciprocal normalized phase-space stiffness contributes the inverse symmetric factor $7/6$;
5. the localized normal mean field is self-consistent and Gaussian near its minimum;
6. total isotropic oscillator action is conserved during the $4\to3$ constriction;

the chain is

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

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

$$\hbar\omega_{\rm common} =2A_{34} =\frac73\mu_3g_{33},$$

and

$$\boxed{ \hbar\omega_3 =\frac43\hbar\omega_{\rm common} =\frac{28}{9}\mu_3g_{33}. }$$

With the current V2 values,

$$\boxed{ \hbar\omega_3 =309.616456\ {\rm MeV}. }$$

The remaining gap to a theorem is concentrated in two assumptions rather than an unexplained numerical coefficient:

• prove that the scalar normal-singlet pushed-forward contact inherits exactly the $\mu_3$ normalization;
• derive the self-consistent normal profile, or replace the Gaussian approximation with the exact profile and compute its curvature factor.

The $g_{33}$ factor, the $7/6$ reciprocal normalization, and the $4/3$ redistribution no longer need to be inserted ad hoc once these operator identifications are adopted.

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