Three Precursor E-Spheres.
One Proton Wave.

1 · The physical picture

2 · The exact three-precursor control

AFor the candidate formation channel

the signed sum is exactly

If three free muonic-scale precursors share the proton rest energy equally, each has

Three equal translational momenta separated by $120^\circ$ in one plane sum to zero:

The reciprocal Doppler control is then

3 · Why three is the minimum—not an assumption hidden as a theorem

One extra positive precursor gives

If the precursors are equal-energy free muonic modes,

C$N=3$ is the MDL-leading search because it is the smallest composite with the required net sign, half-integer closure and a natural three-sector response. The frozen solver should nevertheless include $N=5$ and $N=7$ as blind competitors. A unique stable $N=3$ minimum would select three rather than preload it.

4 · Spin: one exact parity result and one branch the wave must select

BIf every precursor is a lifted spherical-rotation mode that changes sign under $2\pi$, any odd locked collection obeys

This is a cross-lock between two ledgers, not an identification of electric charge with spin hand. Reversing charge reverses the phase/sign relation to the background sea; reversing spin exchanges the dominance of the two spherical rotation hands. Nature permits either charge with either spin orientation.

The two positive roles may couple through $S_{++}=0$ to $J=\tfrac12$, or through $S_{++}=1$ to $J=\tfrac12$ or $J=\tfrac32$. The proton solution must select $J^P=\tfrac12^+$; the aligned $J=\tfrac32$ branch supplies a natural excitation question, not an automatic derivation of the observed $\Delta$ resonance.

5 · $C_3$ is a basis for the fused wave

The signed ancestry $(1,1,-1)$ is not itself a pure eigenvector of cyclic permutation. Use the exact $C_3$ basis

AThe chiral vectors are neutral under the simple sum:

Therefore the fused proton may contain a charged symmetric component plus neutral chiral circulation components. These are not three separately locatable particles. They are three coordinated roles of one displacement field, just as normal modes can delocalise the motion of an entire elastic object.

The solution may realise the signed ancestry through: a charged $v_0$ component with neutral $v_\pm$ components; a combined spatial rotation and internal phase operation; or lower instantaneous signed symmetry whose cycle-averaged exterior monopole is spherical. The three-dimensional action must choose.

6 · Odd charge, even persistence

Let each precursor sign be $s_i=\pm1$. For the one-extra-positive family,

For the minimal sign count in neutral hydrogen, proton $(+,+,-)$ plus electron $(-)$ gives

Reversing every sign for antihydrogen leaves the even count unchanged. This supplies a clean WSM reason why electric neutrality need not remove gravity and why matter and antimatter can share the same gravitational sign. It is compatible with ALPHA-g’s observation of antihydrogen behaving consistently with attraction toward Earth; it does not yet calculate the acceleration.

7 · Charge topology and baryon topology must be different

A smooth direction field on an enclosing sphere can carry integer degree when its target is $S^2$:

CThis is a promising real-wave route to an indivisible signed exterior charge: the curve/orientation pattern cannot unwind without passing through a zero or changing the boundary map.

But electric winding cannot explain proton longevity. Under the precursor bookkeeping, the reverse channel

is open by rest-energy arithmetic, with roughly $621.3$ MeV available. Yet the proton is extraordinarily stable. The finished proton must therefore acquire a closure possessed neither by three separated leptonic e-spheres nor by electric charge alone.

8 · Size: four radii, not one hard shell

A standing wave has no classical particle edge. Keep these quantities distinct:

The geometric clue $4\bar\lambda_\mu/9\approx0.830$ fm lies close to the measured proton charge-radius scale near $0.84$ fm. Treat it as a Tier-C skeleton target: the physical radius is the slope of a calculated response function, and nonlinear deformation supplies the necessary correction. Neither “precision derivation” nor automatic rejection is warranted before that response is solved.

9 · Magnetism is organised motion, not the electron anomaly enlarged

The proton magnetic moment must be obtained from the action’s conserved current for the entire fused displacement pattern:

The measured proton $g$ factor is a composite-geometry observable. It is not the Dirac baseline plus the electron’s self-relay anomaly multiplied by a convenient ratio. A common-angular-speed point-lobe model may be useful as a no-go control, but it cannot be promoted to the proton ontology.

10 · Form factors are pictures made by scattering real waves

The charge radius is only the first low-$Q^2$ coefficient. A real proton solution must produce the full probe response: $G_E(Q^2)$, $G_M(Q^2)$, higher moments, polarizabilities and their time-like continuation. At short wavelength, a probe resolves directional roles and correlations within the one wave; response peaks need not be permanent fractional particles.

Fractional quark charges, colour factors, running, deep-inelastic scaling and jets remain empirical structures that WSM must reproduce in its scattering operator. “Three roles” is a promising geometrical seed; it is not a substitute for those quantitative results.

11 · Formation and finished proton: the causal wave sequence

1. Two positive and one negative muonic-scale e-sphere arrive with balanced total momentum and the required total energy.
2. Their real incoming and outgoing plane waves already occupy the same Space; overlap changes directional energy density rather than causing hard-body collisions.
3. By the One Law, changed $E_d$ changes propagation speed $c'$, so the three wave eggs continually rewrite one another’s travel times, phases and shapes.
4. A successful capture is a new global reclosure. The old independent centres no longer persist.
5. The fused pattern radiates the difference between the precursor history and the final recurrent history through changed curves on outgoing plane waves.
6. The completed proton is maintained by the full all-direction sea and answers later probes as one object with three coordinated roles.

12 · Neutron: a bifurcation, not a fourth stored lepton

A literal four-independent-spinor construction would acquire $(−1)^4=+1$ under $2\pi$, which conflicts with the neutron’s half-integer spin unless non-factorisable topology changes the counting. Therefore the compact notation

should mean a collective phase/boundary bifurcation of a $B=1$ spinorial core, not a proton with a free electron-like lobe stored inside. Beta decay is then a global rearrangement

whose outgoing modes are formed by the changing whole wave. The electron need not have existed as an independently closed electron inside the neutron before decay. The lifetime cannot be inferred from “deep binding”; it requires the transition overlap and available phase space.

13 · The relative energy must subtract the living sea correctly

A compact excitation is measured relative to the background wave state $Z_{\rm sea}$. If the background is not an extremum of the restricted variable set, subtract both its value and tangent:

This prevents background work from being miscounted as proton mass. The density must include coherent cross terms, gradients, phase currents and orientation stress; $\sum_i|\phi_i|^2=1$ is not an energy law for a nonlinear fused wave.

A stationary proton is not “no out-wave.” It requires balanced real in- and out-wave action with zero mean outgoing four-momentum, while every instant continues to contain waves crossing the centre and leaving.

14 · Minimal three-dimensional solver

The computational object is a relative-periodic three-dimensional solution, not a radial hedgehog with three labels pasted onto it:

where $\mathcal R_{2\pi/3}$ rotates the spatial roles and $\mathcal U$ is any required phase/orientation operation. The same frozen constitutive law and action used for the electron must be used here.

Input

• one displacement/coherence action and its electron calibration;
• total signed winding $Q=+1$ and candidate baryon sector $B=1$;
• zero total momentum, lifted $J^P=\tfrac12^+$, finite relative action;
• balanced real all-direction boundary waves;
• formation families $N=3,5,7$ supplied as initial histories, not fitted final shapes.

Blind output

• existence, stability and uniqueness of the compact mode;
• $m_p$, spatial scales and energy–stress distribution;
• signed exterior response and integer charge;
• $\mu_p$, $G_E$, $G_M$, axial current and stress form factors;
• excitation spectrum and aligned-spin branch;
• absence or suppression of the three-free-muon breakup channel;
• high-$Q^2$ response, running and jet-like final-state structure.

15 · Quantitative tribunal

16 · Corrections bank

17 · What WSM may compress

This is not trivial. From one substance and one propagation law, WSM attempts to connect formation, charge, spin, composite geometry, gravity, antimatter and scattering without declaring each a separate substance. The simplicity is earned only if the same solved wave returns every measured read. Until then, the construction is a disciplined and unusually visual research programme: real Space moving, overlapping, reclosing and remembering its own history.