Dimensional Constriction and Hadronic Mass — Part 15

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64. Action and frequency are distinct normalization questions

The fixed-radius oscillator construction describes one band. It does not supply a conservation law for a change in depth. Even if total action were conserved, Section 2 shows that this would imply $3I_{\rm per}^{(3)}=4I_{\rm per}^{(4)}$, not a frequency ratio.

Consequently the trial quantity

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

is not a derived constriction quantum, nor an established half of a nucleon contribution. It combines three independent, unclosed identifications:

1. a scalar energy normalization $\mu_3g_{33}$;
2. a chosen reciprocal normalization $7/6$;
3. a frequency multiplier $4/3$.

The static rank-one Schur identity alone does not make any product of these factors a spectral pole. The companion trial value $(7/6)\mu_3g_{33}=116.106171\ {\rm MeV}$ is likewise not an independently established physical energy. Sections 66--76 examine those remaining normalization and spectral questions.

An action-transfer law and a stationary Hamiltonian must be specified separately. No transfer of action alone fixes the missing frequency.

65. Symplectic-reduction test: canonical projection does not generically produce the $4/3$ action transfer

The assumed frequency-rescaling hypothesis can be tested at the level of classical phase-space geometry.

Start with four oscillator directions with canonical coordinates

$$(q_i,p_i), \qquad i=1,2,3,4,$$

and symplectic form

$$\Omega_4 =\sum_{i=1}^4dq_i\wedge dp_i.$$

Let the corresponding action variables be

$$I_i =\frac{1}{2\pi}\oint p_i\,dq_i.$$

For an isotropic uncoupled oscillator,

$$H_4 =\omega_4\sum_{i=1}^4I_i.$$

Now perform the simplest canonical reduction to three surviving directions by restricting to the constraint surface

$$q_4=0, \qquad p_4=0.$$

The pullback of the symplectic form is

$$\iota^*\Omega_4 =\sum_{i=1}^3dq_i\wedge dp_i =\Omega_3.$$

Thus the reduced system is perfectly symplectic on the surviving phase space.

But its action is

$$I_{\rm red} =I_1+I_2+I_3,$$

not

$$I_1+I_2+I_3+I_4.$$

The eliminated normal action is simply absent unless some additional dynamics transfers it before or during reduction.

Therefore canonical/symplectic reduction by itself yields

$$\boxed{ I_{\rm total}^{(3)} =I_{\rm total}^{(4)}-I_4, }$$

not

$$I_{\rm total}^{(3)}=I_{\rm total}^{(4)}.$$

For an initially isotropic state with

$$I_1=I_2=I_3=I_4=I,$$

the direct reduced Hamiltonian is

$$H_3 =3\omega_4 I,$$

so the surviving oscillator frequency remains

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

under simple deletion/restriction.

The separate frequency-rescaling hypothesis is

$$\omega_3=\frac43\omega_4$$

and does not follow merely from an action-transfer process satisfying

$$I_4 \longrightarrow \Delta I_1+\Delta I_2+\Delta I_3,$$

with

$$\Delta I_1+\Delta I_2+\Delta I_3=I_4.$$

Equal redistribution would give

$$\Delta I_i=\frac13I_4,$$

but neither symplecticity nor dimensional reduction forces that sharing rule.

65.1 Adiabatic invariance does not rescue the transfer automatically

For a slowly varying Hamiltonian with a fixed number of independent oscillator modes, each action variable is an adiabatic invariant to leading order:

$$I_i\simeq\text{const}.$$

That statement does not imply that when one degree of freedom disappears its action is reassigned to the others. In fact, ordinary adiabatic invariance favours preservation of the surviving $I_i$ rather than the specific transfer

$$I_4\to(I_1,I_2,I_3).$$

A change in the number of degrees of freedom typically involves a singular limit, constraint activation, mode coupling, or separatrix-type dynamics, at which ordinary adiabatic-invariant arguments require separate analysis.

Hence

$$\boxed{ \text{neither canonical symplectic reduction nor generic adiabatic invariance derives the }4/3\text{ factor.} }$$

65.2 Action transfer would still not determine the frequency

To derive action transfer, V2 would need explicit coupled projection dynamics. Schematically one needs a Hamiltonian

$$H(q_i,p_i;\lambda)$$

with a constriction parameter $\lambda$, together with mode-coupling terms such that evolution from the four-dimensional regime to the three-dimensional regime satisfies

$$\lim_{\lambda\to\lambda_{3D}} \left(I_1+I_2+I_3\right) = I_1^{\rm in}+I_2^{\rm in}+I_3^{\rm in}+I_4^{\rm in}.$$

Only after such a transfer theorem is established does isotropy imply

$$3I_{\rm out}=4I_{\rm in}$$

for the per-direction action ratio. This does not imply $\omega_3/\omega_4=4/3$. The Hamiltonian and energy exchange must additionally determine the frequency; conserving total energy as well as total action in this isotropic model instead gives equal frequencies.

The present V2 action contains no such explicit constriction Hamiltonian or transfer map.

Therefore the current status is stronger than merely "unproved":

$$\boxed{ \text{the }4/3\text{ factor is not the generic result of symplectic }4\to3\text{ reduction.} }$$

It requires additional mode-transfer dynamics.

65.3 Consequence for the hadronic scale

Dropping the unsupported frequency multiplier leaves the numerical trial $(7/6)\mu_3g_{33}\approx116.106171$ MeV, not an established physical scale. The scalar-unit and reciprocal-normalization identifications also remain unclosed.

A complete calculation must derive the interacting spectral operator, including its normalization and response to constriction. Neither total-action transfer nor proximity to a constituent residual fixes that operator. There is no established half-contribution that the calculation is obliged to double.

66. Reciprocal-normalization audit: $7/6$ is canonical, but not forced by the current cross-depth action

The remaining reciprocal factor was audited against the independent V2 cross-depth modules rather than against its repeated use in this hadronic note.

The V2 cross-depth theorem establishes three relevant facts:

1. the higher-depth density enters through its complete marginal over the extra fibre;
2. the lower-depth density is cylindrically extended over that fibre;
3. functional differentiation of a symmetric interaction action produces reciprocal response in both sectors.

Schematically,

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

with

$$\frac{\delta S_{ab}}{\delta\rho_a} \propto\bar\rho_{b\to a}, \qquad \frac{\delta S_{ab}}{\delta\rho_b} \propto\rho_a.$$

These statements establish full contact and exchange symmetry, but they do not specify a unique averaging functional over the dimensions of the two phase-space carriers.

For a $4\to3$ phase-space comparison, the real quadrature counts are

$$N_4=8, \qquad N_3=6.$$

If one chooses the symmetric arithmetic normalization of the shared carrier, the shared fraction is

$$C_{34} =\frac{2N_3}{N_4+N_3} =\frac{12}{14} =\frac67,$$

and its inverse is

$$\boxed{ C_{34}^{-1}=\frac76. }$$

This is mathematically natural: it treats the two reciprocal sectors symmetrically and reduces to unity when their dimensions are equal.

However, the written V2 action does not contain an independent theorem requiring this particular arithmetic normalization. Full marginalization fixes the total normalized contact weight, and self-adjointness/exchange symmetry constrain the cross-sector blocks, but neither condition uniquely selects

$$\frac{2N_3}{N_4+N_3}$$

over other dimensionless symmetric normalizations that also equal one at $N_4=N_3$.

Therefore the current status of the factor is

$$\boxed{ \frac76\text{ is the canonical symmetric reciprocal normalization used in this construction, not yet a uniquely derived coefficient of the V2 master action.} }$$

This is stronger caution than the earlier statement that $7/6$ is merely preferable to $7/3$. The audit shows that current V2 decisively rejects the naive $7/3$ double count, but it does not yet prove that $7/6$ is the only allowed normalized reciprocal coefficient.

66.1 Consequence for the primitive hadronic scale

The previously isolated pre-redistribution value

$$\varepsilon_{\rm pre} =\frac76\mu_3g_{33} \approx116.106171\ {\rm MeV}$$

must therefore also be classified as conditional on a normalization convention/theorem not yet derived from the master cross-depth action.

The hierarchy of certainty is now:

1. Algebraic rank-one result $$\boxed{g_{\rm con}=g_{33}}$$ follows directly from $$g_{34}^2=g_{33}g_{44}.$$

2. Physical scalar energy unit $$\boxed{E_{\rm stiffness}=\mu_3g_{33}}$$ is conditional on the same-projected-scalar-vertex normalization.

3. Reciprocal phase-space normalization $$\boxed{C_{34}^{-1}=7/6}$$ is a natural symmetric canonical choice, but is not uniquely forced by the written cross-depth action.

4. Action redistribution $$\boxed{\omega_3/\omega_4=4/3}$$ requires additional mode-transfer dynamics and is not the generic result of symplectic reduction.

Thus even the earlier $116.1$ MeV pre-redistribution scale is not yet a first-principles V2 prediction. The most primitive dimensionful quantity supported by the current scalar-block construction is

$$\boxed{ \mu_3g_{33} \approx99.519575\ {\rm MeV}, }$$

conditional only on the same-scalar-vertex identification and on the already adopted V2 coupling/scale relations.

66.2 Revised proof target

Before any hadronic mass closure can be called derived, the projected cross-depth action must itself determine the bilinear normalization. One needs an operator-level calculation of

$$P_3D_\infty P_4$$

with its actual Hilbert-space measures and adjoint structure, producing a definite scalar coefficient rather than inserting a dimension-counting average afterward.

A successful calculation may reproduce $7/6$, produce another coefficient, or show that no standalone dimension factor should appear at all.

Until then,

$$\boxed{ \mu_3g_{33} }$$

is the last comparatively primitive hadronic scale in the present chain, while both $7/6$ and $4/3$ remain additional projection assumptions of different strengths.

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