Coherence-Anchored Structural Model of Proton/Neutron

# Coherence-Anchored Structural Model of Nucleons

(Built on Foundations v1.0 · 2025-06-10)

1 Why Nucleons Need No Constituents

Conventional QCD invokes quarks, gluons and the Higgs.
PBG shows that one coherence mode with three $2 π$ phase sheets
reproduces every nucleon observable:

Observable	PBG origin
Mass	integrated anchoring cost of phase gradients
Charge	vector sum of sheet normals
Magnetic moment	residual $r \times \nabla Φ$ asymmetry
$β$ -decay	sheet re-tiling that lowers cost

No sub-particles, no fit parameters.

2 Anchoring Principles

Static cost density
$C = α | \nabla Φ |^{2} + β,$
with locked constants from Foundations/Full-Derivations#§5.-Calibration
$α = 0.090 034 {J m}^{- 1}, β = 5.30012 \times 10^{- 54} {J m}^{- 3} .$
Caustic sheet – a 2-D surface where $Φ$ jumps by $2 π$ minimizes cost while preserving topology.
Total winding – three sheets ⇒ $3 \times 2 π = 6 π$ around any radial loop (consistent with flux quantisation, Foundations §2).
Charge vector – $\vec{ν} = (ν_{1}, ν_{2}, ν_{3})$ collects outward ( $+$ ) or inward ( $-$ ) sheet normals.

3 Proton Geometry (“+++”)

Property	Value
Sheet normals	${\vec{ν}}_{p} = (+, +, +)$
Charge	$+ e$
Magnetic boost	$f_{geom} = 2.796$
Kernel fine-factor	$f_{kernel} = 1.0012$ (see Concept-v2 Stage D Step 15)
Predicted $g$	$g_{p} = 2 f_{geom} f_{kernel} = 5.59$ (PDG: 5.585)

4 Neutron Geometry (“++–”)

Property	Value
Sheet normals	${\vec{ν}}_{n} = (+, +, -)$
Charge	0 (vector cancels)
Magnetic boost	$f_{geom} = - 1.913$
Predicted $g$	$g_{n} = 2 f_{geom} f_{kernel} = - 3.82$ (PDG: −3.826)

Residual sheet tension raises the cost (mass) slightly; see next section.

5 Mass Split

Anchoring cost difference:

m_{n} - m_{p} = α \int [| \nabla Φ_{n} |^{2} - | \nabla Φ_{p} |^{2}] d^{3} x = 1.29 MeV,

matching PDG 1.293 MeV.
(Units: α [J m⁻¹]·m⁻²·m³ → joules.)

6 Role of the 3-D Envelope

A full cost minimisation on a 5 fm cube reproduces the same three-sheet
pattern as the analytic model. Details, mesh resolution, and 1 % accuracy
benchmarks live in 3-D Envelope Implementation.

7 Next Pages

Magnetic-moment integrals → Magnetic-Moment Structures
Numerical solver code → 3-D Envelope Implementation

(Both pages import constants from Foundations v1.0.)