Calculation: Proton Structure & Nuclear Physics

Step 1 — The j₀ Waveform

The zeroth-order spherical Bessel function is:

This is the fundamental spherical mode — the lowest-energy standing wave in a three-dimensional spherical cavity. It has no angular nodes (l = 0), only a radial structure that falls smoothly from a maximum at the center to zero at the first node.

Step 2 — Confinement to One Period

The proton waveform occupies one full period of j₀, from the center (r = 0) to the first zero at r = πR_c/π = R_c. Here R_c is the cavity radius — the physical boundary of the standing wave. The waveform is:

At r = 0: ψ = 1 (maximum). At r = R_c: ψ = 0 (node). This is the proton.

Step 3 — RMS Radius of j₀

The measured “proton radius” is the root-mean-square charge radius, which is the RMS of the probability distribution |ψ|² weighted by r². For j₀ confined to [0, R_c]:

Evaluating term by term:

Step 1 — Input Values

Step 2 — Compton Wavelength

The proton’s reduced Compton wavelength is:

Step 3 — Proton Radius = 4 Compton Wavelengths

The virial factor d + 1 = 4 gives the confinement scale. The proton radius is four Compton wavelengths:

Step 4 — Cavity Radius

Inverting the j₀ RMS relation from Section 1:

This is the physical boundary of the proton standing wave — the actual extent of the proton waveform. Everything within R_c is the proton; beyond it, the wave amplitude is zero (in the idealized j₀ model) or evanescent (in reality).

Step 1 — The QCD Scale

From the strong coupling chain (see Strong Coupling & QCD):

This is derived from the lattice constants with zero free parameters.

Step 2 — The Three Factors

The proton mass involves three physical effects, each with a clean wave-mechanical origin:

Step 3 — The Gibbs Factor in Detail

The Gibbs factor arises from the sine integral at the confinement boundary:

This is the Gibbs overshoot correction — the j₀ standing wave has a finite Fourier series that converges to Si(π)/π at the cavity boundary. Physically, confinement means the wave is truncated, and the Gibbs factor accounts for this truncation.

Step 4 — Compute the Proton Mass

Combining all three factors:

The product 0.532 × Si(π)/π = 0.532 × 0.5895 = 0.3136 provides a sub-leading correction that refines the mass gap derivation. The dominant relation m_p = 4Λ_QCD already captures the answer.

Step 1 — Pion Decay Constant f_π

The decay constant measures how strongly the pion couples to the weak axial current. In GWT, f_π is set by the antibonding geometry of the d=3 lattice: 2d−1 = 5 spatial antibonding modes × 2 (particle/antiparticle) gives a denominator of 10:

The factor 2(2d−1) = 10 counts the antibonding modes available to the pion in 3D: the kink condensate has 2d−1 = 5 spatial modes (one per lattice axis minus parity), doubled by particle/antiparticle symmetry.

Step 2 — Chiral Condensate

The vacuum condensate density counts virtual q¯q pairs. In GWT, this is determined by the number of coupling channels in the Wyler domain D_IV(d+2), normalized by the d-cube unit cell volume:

where Λ_QCD = m_p/4 = 234.6 MeV, d(d+2) = (d+1)²−1 = 15 counts the coupling channels, and 2^d = 8 is the d-cube vertex count (volume normalization). The same D_IV(5) geometry that gives α = 1/137.042.

Step 3 — Pion Mass via GMOR Relation

The Gell-Mann–Oakes–Renner relation is exact in QCD:

Using GWT-derived quark masses: m_u = m(13,31) = 2.21 MeV, m_d = m(5,30) = 4.78 MeV:

Step 4 — Pseudoscalar VP Correction (possible correction)

The GMOR result is the bare pion mass. The physical pion, a quark-antiquark pseudoscalar, may lose mass through vacuum polarization — the same mechanism that dresses all fermion masses. Each of d = 3 spatial axes contributes one π^−α attenuation factor:

This follows the VP sign rule: fermionic content → mass decreases, with d = 3 spatial loop axes. Equivalently, two fermion constituents each receive π^−dα/2, and the squared result is π^−dα. This correction is physically motivated and pattern-consistent with the tau (π^−α, 1 axis) and Z (π^−α/4, 4 axes) corrections, but is not yet formally derived from the lattice Lagrangian.

Step 1 — Pion Compton Wavelength

Step 2 — Neutral Radius

The nuclear force range is the proton cavity radius plus the evanescent zone where pion-mediated attraction operates:

Step 1 — Nuclear Radius Constant r₀

The nuclear radius scales as R = r₀ × A^1/3, where r₀ is set by the ratio of the first zero of j₁(x) to π:

The j₁ zero enters because the nuclear density profile is set by the first excited spherical mode — the mode that determines how proton standing waves pack together.

Accuracy: 0.3%

Step 2 — Nuclear Potential Depth V₀

The depth of the nuclear potential well follows from the uncertainty relation applied to a cavity of radius r₀. The factor of 3 in the denominator is the spatial dimension:

Accuracy: 1.2%

Step 3 — Fermi Momentum k_F

The Fermi momentum is the highest occupied wave number inside the nuclear volume. In a cavity of size r₀, exactly half a wavelength fits:

Accuracy: 2%

Step 4 — Nuclear Saturation Density ρ₀

Nuclear matter fills a Fermi sphere of radius k_F with two spin states per mode (yin-yang duality):

Accuracy: 6%

Step 5 — Fermi Energy T_F

The kinetic energy of the highest-occupied nuclear mode:

Step 1 — The 5/6 Factor

For a Fermi gas in d = 3 dimensions, the average kinetic energy is (3/5)T_F. The ratio of average to maximum is:

This dimensional factor accounts for averaging over the Fermi distribution.

Step 2 — Compute a_V

Step 1 — Condensate Factor

The condensate density is proportional to the number of coupling channels in D_IV(d+2), normalized by the d-cube unit cell volume:

d(d+2) = (d+1)²−1 = 15 counts the coupling channels from the Wyler domain. 2^d = 8 is the d-cube vertex count (volume normalization).

Step 2 — Compute Condensate

Energy Density Scaling

On a log-log plot, all confined wave systems obey two parallel lines separated by 8/α:

This 8/α factor is exact — it is the ratio of QCD to EM coupling strengths, multiplied by 8 from the geometric difference between spherical and orbital confinement. See the interactive energy density plot.

Step 1 — The Analytic Formula

The charge distribution is |j₀|² ∝ sin²(πr/R_c) / r². Its Fourier transform gives:

where:

Step 2 — Key Properties

The only input is R_c = 1.581 fm, which is itself derived from Λ_QCD and the j₀ RMS factor 0.532.

Step 3 — Comparison with the Standard Dipole

The conventional dipole form factor used in most textbooks is:

This is an empirical fit with one free parameter (0.71 GeV²). The GWT j₀ form factor has zero free parameters and naturally explains the experimentally observed deviations from the dipole at high Q². The soft boundary of the j₀ wave produces a form factor that falls off more gradually than the dipole, matching the data.

Step 1 — The (d−1)/d Pattern

In d = 3 spatial dimensions, a wave confined to a spherical cavity distributes its energy as 1/3 longitudinal and 2/3 transverse. The magnetic moment of the neutron (the flipped-phase partner of the proton) carries the transverse fraction with opposite sign:

Step 2 — Numerical Comparison

Step 1 — Mode Filling in 3D

Each standing-wave mode is labeled by quantum numbers (n, l), where n is the radial node count and l is the angular momentum. Each (n, l) level holds 2(2l + 1) states, where the factor of 2 is the yin-yang (spin) degeneracy:

Step 2 — Shell Closures

Filling these modes in order of increasing energy produces shell closures at cumulative totals:

Step 3 — Spin-Orbit Splitting

The first three magic numbers (2, 8, 20) emerge from pure mode filling. The remaining four (28, 50, 82, 126) require the spin-orbit interaction, which splits each l > 0 level into j = l + ½ and j = l − ½ sub-levels. In GWT, this is not an ad hoc addition:

This coupling is a consequence of wave mechanics in three dimensions. The same spin-orbit interaction that produces the fine structure of atomic spectra produces the magic numbers of nuclear physics. It is not a new force — it is geometry.

Step 1 — Nuclear Energy Scale

The nuclear seesaw energy — the pion recoil energy, analogous to ionization energy E_H = m_eα²/2 in atomic physics:

Step 2 — Nuclear Bohr Radius

The pion Compton wavelength — the natural length scale of nuclear forces, analogous to the Bohr radius a₀:

Step 3 — Phase and Binding

The deuteron charge radius R_d = 2.1421 fm sits just below the first node at π/2 = 1.571 in nuclear Bohr units (2.30 fm). This is why the deuteron is barely bound — it lives near the node of the standing wave: