3-D Envelope of Proton/Neutron

3-D Envelope Implementation

(Proton & Neutron numeric validation – rebuilt 2025-06-10)

Purpose – Show that a full 3-D cost-minimisation reproduces the
three-sheet analytic model for nucleons and matches all observables
to ≈ 1 %.
Cost functional, constants $α, β, γ$ are imported from
[[Foundations/Full-Derivations#§1.-Core-Action-with-Matter-Coupling]].

1 Domain & Discretisation

Item	Choice
Simulation box	Cube $[- 5, 5]$ fm³
Boundary condition	Absorbing faces ( $Φ \to 0$ )
Grid spacing	$Δ x = 0.05$ fm (200³ nodes)
Unknowns per node	Density $ρ$ and 8 phase components $θ^{a}$

Comment – the 8 real $θ^{a}$ fields encode the local phase vector
in an $S U (3)$ basis; caustic sheets appear as steep (≈ 0.05 fm) but
continuous ramps in one or more $θ^{a}$ directions.

2 Initial Seed

Superpose three radial $2 π$ ramps (choose +++ or ++– normals) to
guarantee total winding $6 π$ (see
[[Coherence-Anchored Structural Model]]).
Envelope $ρ$ : Gaussian with width $σ = 0.68$ fm reproduces the
empirical charge radius $r_{E}$ of the proton.

3 Minimisation Loop

Cost evaluation — adopt static part of anchoring action $C = \int ρ^{2} [α Tr (D_{i} U^{†} D_{i} U) + β] d^{3} x,$ where $U = e^{i θ^{a} λ^{a}}$ and
$D_{i} U = \partial_{i} U - i (\partial_{i} Φ) U$
(constants $α, β$ from Foundations §5).
Gradient-flow step on $ρ$ and all $θ^{a}$ .
Update coherence field by solving $(α \nabla^{2} - β) Φ = - \nabla \cdot J,$ with $J = ρ^{2} \nabla Φ$ (see Foundations §1).
Convergence check — stop when $Δ C / C < 10^{- 6}$ .

Any finite-element or FFT-based lattice solver can implement the loop.

4 Diagnostics

Check	Criterion
Colour neutrality	$\int ρ^{2} θ^{a} d^{3} x = 0$
Total anchoring cost	$\approx 938$ MeV (proton)
Loop holonomy	$6 π$ phase change across each sheet
Observables	$r_{E}, r_{M}, g$ within 1 % of analytic targets

5 Expected Numeric Outputs

Quantity	Proton	Neutron
Charge radius $r_{E}$	0.83 fm	0.83 fm
Magnetic radius $r_{M}$	0.83 fm	0.86 fm
$g$ -factor	5.59	–3.82
Core density $ρ (0)$	$1.6 \times 10^{44}$ m⁻³	—

Core density converts to an effective $α_{s} ≃ 0.12$ —consistent
with PDG values at $Q^{2} \sim 1$ GeV².

Second/third-order kernel — import $k^{2}, k^{3}$ corrections from
Concept-v2 Stage D Step 15.
Edge sharpness — add saturation term
$λ ρ^{4}$ to tune confinement radius.
Spin-½ test — rotate envelope by $2 π$ ; cost is unchanged but
the wavefunction flips sign, returning only after $4 π$ .
Hyperons — move one sheet to a radial node; re-minimise to predict
$Λ, Σ, Δ$ masses.

Back to nucleon overview → [[Coherence-Anchored Structural Model]]
Magnetic‐moment derivation → [[Magnetic-Moment Structures]]