Dimensional Constriction and Hadronic Mass — Part 1

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Current Reading and Scope — 1 November 2026

A collection of internal common-frequency portions is not an identification of observed quarks as constituents of every particle. Composite mass requires complete rest energy, relative motion, support and reciprocal interaction. The compact cluster law supplies a new conditional mechanical comparison; its count offset and source normalization are premises, and it does not yet predict a nucleon mass.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. Question

The quark sectors in V2 are assigned different spatial depths:

$$Q_4 \quad \text{for up-type quarks},$$

and

$$Q_3 \quad \text{for down-type quarks}.$$

The cross-depth geometry is asymmetric. A four-dimensional mode projected into a three-dimensional hadronic sector loses one independent spatial oscillator direction. The three-dimensional state does not gain one.

This raises a specific dynamical question:

Can the large difference between light current-quark masses and observed hadron masses be interpreted as the energy cost of constraining a four-dimensional oscillator into three dimensions?

The calculation below separates what follows mathematically from dimensional constriction from what remains an empirical or dynamical input.

2. Oscillator action and the separate frequency-rescaling hypothesis

For an isotropic oscillator with $d$ directions,

$$E_d=\omega_d I_{\rm total}^{(d)} =d\,\omega_d I_{\rm per}^{(d)}.$$

Conservation of total action under a hypothetical $4\to3$ transfer gives

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

This fixes the action per surviving direction, not the frequency. If both total action and total oscillator energy are conserved, the displayed Hamiltonian instead gives $\omega_3=\omega_4$.

The equation $4\omega_4 I_1=3\omega_3 I_1$ assumes equal energy and the same per-direction action on both sides. It yields $\omega_3=4\omega_4/3$, but changes total action from $4I_1$ to $3I_1$. It is not a consequence of total-action conservation.

The numerical explorations below use

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

as a separate frequency-rescaling hypothesis. No current V2 dynamics derives it. Transferring eliminated action into surviving directions, by itself, does not establish this hypothesis.

3. Mass-like energy from the constriction

For a relativistic bound mode, oscillator or confinement energy contributes to the rest energy observed in the lower-dimensional sector. Define the effective confinement contribution per valence quark by

$$\epsilon_h = \frac{M_h-\sum_i m_{q_i}}{N_q},$$

where

• $M_h$ is the observed hadron mass;
• $m_{q_i}$ are the current-quark masses used by V2;
• $N_q$ is the number of valence quarks in the minimal quark assignment.

If the observed three-dimensional confinement quantum is the constricted oscillator quantum,

$$\epsilon_h=\hbar\omega_3,$$

then the corresponding pre-constriction four-dimensional quantum is

$$\boxed{ E_{4,h} =\hbar\omega_4 =\frac34\epsilon_h. }$$

If the four-dimensional oscillator has characteristic spatial scale $R_h$, dimensional analysis gives

$$E_{4,h}\sim\frac{\hbar c}{R_h},$$

so the effective constriction radius inferred from a hadron mass is

$$\boxed{ R_h =\frac{4\hbar c}{3\epsilon_h}. }$$

Equivalently,

$$\boxed{ M_h =\sum_i m_{q_i} +N_q\frac43\frac{\hbar c}{R_h}. }$$

This is the simplest quantitative hadronic-mass law produced by the $4\to3$ constricted-oscillator hypothesis.

4. Nucleon test

Using the V2 light-quark masses

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

the proton current-quark sum is

$$2m_u+m_d=9.02\ {\rm MeV}.$$

With

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

the excess mass is

$$\Delta M_p =929.252088\ {\rm MeV}.$$

Per valence quark,

$$\boxed{ \epsilon_p =309.750696\ {\rm MeV}. }$$

The inferred pre-constriction oscillator quantum is

$$E_{4,p} =\frac34\epsilon_p =232.313022\ {\rm MeV},$$

and therefore

$$\boxed{ R_p^{\rm constr} =\frac{\hbar c}{E_{4,p}} =0.8494\ {\rm fm}. }$$

The neutron gives almost the same result. With

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

and

$$m_u+2m_d=11.56\ {\rm MeV},$$

one finds

$$\boxed{ \epsilon_n =309.335140\ {\rm MeV}, }$$

$$E_{4,n}=232.001355\ {\rm MeV},$$

and

$$\boxed{ R_n^{\rm constr} =0.8505\ {\rm fm}. }$$

Thus proton and neutron independently select essentially the same constriction scale:

$$\boxed{ R_N^{\rm constr}\simeq0.850\ {\rm fm}, }$$

with a three-dimensional confinement quantum

$$\boxed{ \hbar\omega_3\simeq309.5\ {\rm MeV}. }$$

The near equality of the proton and neutron values is nontrivial because their current-quark sums differ.

5. Relation to the measured nucleon size

The proton charge radius is approximately

$$r_p\simeq0.84\ {\rm fm}.$$

The mass-derived constriction radius

$$R_p^{\rm constr}\simeq0.849\ {\rm fm}$$

is therefore within roughly one percent of the observed proton charge-radius scale.

Conversely, inserting a radius near $0.84$ fm into

$$\epsilon=\frac43\frac{\hbar c}{R}$$

gives a confinement quantum near

$$313\ {\rm MeV},$$

close to the $309$--$310$ MeV required by the proton and neutron masses.

This is the strongest numerical result of the dimensional-constriction test.

It does not by itself prove the mechanism because the relation

$$E_4=\hbar c/R$$

is a characteristic confinement estimate rather than a derived eigenvalue of the present V2 master operator. What is striking is that the independent mass excess and the physical nucleon size select nearly the same scale once the exact $4/3$ constriction factor is imposed.

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