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Full Record — The Bosonic Soliton Electron Program

From a kinematic ansatz for a spinning charged lump, through the Maxwell and Born–Infeld obstructions, through the Skyrme control model and the classical deformation-matrix program, to a sharp statement of what a successor model must supply. Every claim labeled. Every correction recorded.

How to Read This Document

This is a record of an extended interaction between a user and an AI assistant, working through a single question: can a purely bosonic field theory produce a localized soliton with the spin, charge, mass, and magnetic moment of the electron, without introducing fundamental fermions? The document contains established physics, several clean negative results, one verified numerical near-cancellation, one falsified hypothesis, and a precise statement of the missing mathematical structure. Failed routes are recorded alongside successful ones. Corrections are recorded next to the claims they correct. Nothing is hidden.

Labels used throughout.

Label	Meaning
Established	Measured, tested, or proven within its domain. Not the program’s invention.
Verified	Confirmed against an independent numerical or symbolic check that does not share analytic intermediates with the object being tested.
Derived	Follows from a calculation, theorem, or equation. Not put in by hand.
Computed	Numerical result obtained; not yet independently verified.
Structural	Follows from the structure of the framework. Not a derivation of a number.
Coherent	Consistent with everything measured. Not derived.
Conditional	Holds under stated assumptions that have not been verified.
Pending recheck	Previously verified; the underlying input has since changed or been called into question.
Not derived	Asserted but not shown.
Open	No calculation has settled it.
Falsified	Contradicted by data or by a computed result.

Part I — The Framing Question and the Kinematic Ansatz

The Question

The program began with a sharper version of an old question: can a bosonic field theory support a localized, finite-energy configuration that carries half-integer angular momentum and unit electric charge, and whose electromagnetic vertex matches the electron’s? The Skyrme model was the natural precedent: a purely bosonic theory whose solitons quantize fermionically. But the Skyrme model’s charge structure is baryonic, not leptonic. So the question became: what, if anything, can produce an electron-like soliton?

The First Ansatz

The starting configuration was the most basic one that can carry angular momentum in Maxwell theory:

Φ = Φ(r), A = A(r, θ) φ̂

Spherically symmetric electric potential, axisymmetric magnetic vector potential

With this ansatz, E is radial, B is poloidal, and E × B is azimuthal — a circulating field momentum. The electromagnetic angular momentum is

J_EM = ε₀ ∫ r × (E × B) d³x

For a separable profile A(r,θ) = a(r) sin θ, this reduces to

J_z = (8π/3) ε₀ ∫ r² Φ′(r) ∂_r(r a(r)) dr

The 4/3 angular integral, after the correction from the first pass

Correction 1 — The angular integral The first pass wrote the angular integral of sin³θ as 2/3. It is 4/3. The corrected prefactor is 8π/3, not 4π/3. This correction was accepted and carried forward.

Part II — Kinematic Feasibility

The Tame Radial Profile

A concrete profile was chosen to test kinematic feasibility:

Φ′(r) = E₀ (r/R) e^(−r/R), a(r) = A₀ (r/R) e^(−r/R)

This gives fields regular at the origin and exponentially localized at infinity. The radial overlap integral evaluates to

∫ r² Φ′ ∂_r(r a) dr = −(3/8) E₀ A₀ R³

J_z = −π ε₀ E₀ A₀ R³

The field energy is

U = (3π/2) ε₀ E₀² R³ + (π A₀² R)/μ₀

Both are finite. Minimizing at fixed J_z gives

U_min = √6 |J|c / R

Setting |J| = ħ/2 and U = m_ec² selects

R = (√6/2) λ_C ≈ 1.225 λ_C ≈ 4.73 × 10⁻¹³ m

Not a prediction of the electron radius — a consequence of imposing both conditions on a chosen profile

Result — Kinematic feasibility The ansatz can carry J = ħ/2 with finite energy, at a characteristic scale of order the Compton wavelength. There is no kinematic obstruction. But this is a classical angular momentum, not spin-half quantization. And the charge is zero: the smooth radial E field corresponds to a net-zero charge density.

Part III — The Charge Filter

The Three Targets

The program sharpened to three constraints:

J = ħ/2, U = m_ec², Q = −e

The toy configuration satisfies the first two by parameter choice but fails the third.

The Charged-Sphere Construction

A charged sphere was added by hand: a uniformly charged core of radius R_q and total charge −e, with the magnetic halo on a larger scale R. This passes the charge filter kinematically. But:

The charge is inserted, not derived.
Once J is fixed, U_E and U_B are not independently adjustable. The two-scale picture needs to be checked by substituting the exact A₀(R_q, R) back into U.
The Coulomb self-energy of a small core is large: U_E ≈ (3/5) e²/(4πε₀ R_q).

Correction 2 — “Passed the filter” was too strong The charged-sphere construction passes the charge filter only kinematically. It is not a self-contained field theory. The charge density is prescribed, not dynamical. Corrected status: Conditional.

Part IV — The Skyrme Model and the Spin–Charge Split

Why Skyrme

The Skyrme model is the canonical bosonic soliton with fermionic quantization:

ℒ = (F²/16) Tr(L_μL^μ) + (1/32e_S²) Tr([L_μ,L_ν][L^μ,L^ν]) + (m²F²/8) Tr(U − 1)

L_μ = U†∂_μU, U ∈ SU(2)

The B = 1 hedgehog

U₀(x) = cos f(r) + i (τ · n) sin f(r), f(0) = π, f(∞) = 0

quantizes to a spin-1/2 state via the Finkelstein–Rubinstein mechanism. That part is solid.

The Charge Problem

The gauged U(1) is isospin I₃. The Wess–Zumino–Witten term ties hypercharge to baryon number. The resulting charge formula is

Q = I₃ + B/2

For B = 1, the allowed charges are 1 (proton) and 0 (neutron). There is no Q = −1 state. The Skyrme model gives fermionic spin, but not electron-like charge.

Correction 3 — The WZW mod-2 language An earlier pass described the SU(2) WZW term as “mod 2.” The correct statement is that the SU(2) WZW term is a mod-2 object and cannot tie an integer U(1) charge to baryon number. The mod-2 structure is relevant to the FR obstruction, not to the charge formula. The two were conflated and the language was dropped.

Result — Skyrme as proof of principle Topology can give fermions from bosons. That is established by Finkelstein–Rubinstein. But the Skyrme model does not naturally give the electron’s charge structure. The charge comes out as I₃ + B/2, which for B = 1 gives 0 or 1. The model is a proof of principle, not a candidate electron theory.

Part V — Hopfions and the Local-Current Problem

The Hopfion Alternative

The Faddeev–Niemi model has target space S², with π₃(S²) = ℤ. The Hopf invariant labels the solitons. Krusch and Speight showed that odd Hopf charge can quantize fermionically. So the spin side works.

The Obstruction

The Hopf invariant is a secondary characteristic class. There is no local, Lorentz-invariant four-current J^μ_top built polynomially from the field n and its derivatives. The usual expression H = ∫A ∧ dA requires an auxiliary one-form that is nonlocal in n. Therefore a coupling of the form A_μJ^μ_top is not well-defined for a Hopfion.

Correction 4 — “No known constructions” was too strong An earlier summary claimed “no known bosonic construction satisfies both conditions.” That was an overreach. The correct statement: the natural sigma-model choices tested (SU(2) Skyrme, SU(3) Skyrme, Faddeev–Niemi, CP¹, SU(2) × U(1)) all fail the charge condition. That is not a no-go theorem for all bosonic solitons.

Part VI — The Baseline Control Model

The Control Experiment

Rather than searching for a new target space, the program deliberately constructed the simplest theory that satisfies the two conditions by hand. Take the B = 1 Skyrme hedgehog and gauge the topological current with electromagnetic weight −e. Then:

Q = −eB = −e for B = 1

The theory is globally consistent for any coefficient k in S_top = k ∫ A_μB^μ. Large gauge transformations quantize k to an integer, but do not fix it to −1. The coefficient is an input.

The Tree-Level g-Factor

For this control model, the collective-coordinate effective action is

L_eff = −M + ½ Λ Ω² + κ q Ω·B

g_tree = 2Mκ/Λ

Using the universal massless Skyrme profile:

C_M ≈ 11.61, C_Λ ≈ 37.85, C_E ≈ 1.119

C_μ = C_E/3, g_tree^(B) = 8π C_M C_E/(3 C_Λ) ≈ 2.88

Result — Skyrmion-specific failure The control model predicts g_tree ≈ 2.88, not 2. The discrepancy is O(1), not O(α). Quantum corrections would need to produce ΔF₂(0) ≈ −0.44, followed by the tiny Schwinger remainder. That is not a Schwinger-like correction; it is a restructuring of the tree-level vertex. The control model fails the g = 2 gate before any loop calculation.

Correction 5 — The g-factor derivation An earlier pass derived g ∼ 1/R² from an incorrect scaling μ ∼ e_SΛ. The correct scaling is μ = C_μ q Ω R², and the scale dependence cancels: g_tree is a dimensionless number fixed by the profile, not a scale-dependent quantity. The obstruction disappears; the correct result is g ≈ 2.88.

Part VII — The Maxwell Obstruction

The One-Sign Sharp Bound

For any spherically symmetric, one-sign classical charge distribution with total charge e and RMS radius R_E, the electrostatic energy satisfies

U_C R_E ≥ (3√15/25) αħc ≈ 0.464758 αħc

Equality for the uniformly charged sphere

If U_C ≤ m_ec², then

R_E ≥ 0.464758 αλ_C ≈ 1.31 × 10⁻¹⁵ m

The Momentum-Space Bound

The sign-changing loophole was tested and closed. In momentum space,

U_C = (αħc/π) ∫₀^∞ dQ |G_E(Q)|²

If experiment requires |G_E(Q) − 1| ≤ ε on 0 ≤ Q ≤ Q_exp, then

U_C ≥ ((1−ε)²/π) αħc Q_exp

For a pointlike charge with Q_exp = 1/r_exp:

U_C ≥ (1/π) αħc / r_exp

At r_exp = 10⁻¹⁸ m, this gives U_C/m_ec² ≳ 900. The argument uses only ordinary Maxwell electrostatics, G_E(0) = 1, and |G_E − 1| ≤ ε on a finite interval. It does not use one-sign density, RMS radius, spherical symmetry, or a one-scale assumption.

Result — Maxwell/localized-charge failure Any classical Maxwell charge distribution with total charge e that looks pointlike out to momentum Q_exp has positive electrostatic energy at least (1/π) αħc Q_exp. For Q_exp ≫ 1/λ_C, this exceeds m_ec² by a large factor. The sign-changing loophole is closed.

Correction 6 — The numerical coefficients An earlier pass gave the one-sign constant as ≈ 0.116 k_e, off by a factor of four. The correct constant is 3√15/25 ≈ 0.464758 k_e. The factor-of-four error came from rewriting the electrostatic energy in terms of e²/(4πε₀).

Part VIII — The Born–Infeld Control and Its Exclusion

The Born–Infeld Calculation

Born–Infeld electrodynamics regularizes the point-charge self-energy. For a point charge,

U_BI = (Γ(1/4)²/(6√π)) αħc/r_BI ≈ 1.23605 αħc/r_BI

Setting U_BI = m_ec² gives

r_BI = 1.23605 αλ_C ≈ 3.48 × 10⁻¹⁵ m ≈ 1.236 r_e

The BI maximum field is b ≈ 90 E_c, where E_c is the Schwinger limit.

The Experimental Exclusion

The BI nonlinearity generates light-by-light scattering. The 2017 Ellis–Mavromatos–You analysis of the ATLAS light-by-light result requires

M_BI ≳ 100 GeV

whereas the self-energy calibration requires

M_self-energy ∼ 5.7 × 10⁻² GeV

The hierarchy is ≈ 1.8 × 10³. The two requirements pull the same parameter in opposite directions.

Result — Standard Born–Infeld is excluded Standard BI QED, calibrated so that the electron’s BI self-energy accounts for its observed mass, is excluded by light-by-light scattering data. This does not establish a theorem excluding every possible nonlinear ℒ(X,Y). A more complicated NLED could introduce additional scales separating the strong-field self-energy from the four-photon operator. Whether that can be done consistently is a separate question.

Correction 7 — The inequality direction An earlier pass wrote “excludes a BI scale ≳ 100 GeV.” The inequality was backwards. The correct statement: light-by-light scattering excludes M_BI ≲ 100 GeV, equivalently requires M_BI ≳ 100 GeV.

Part IX — The Deformation-Matrix Program

The Question After the Controls

The two classical control theories — ordinary Maxwell and standard Born–Infeld — are both excluded. But that does not eliminate the entire space of classical bosonic theories. The program pivoted to a sharper question: whether the symmetries of the tested Skyrme family can protect g = 2.

The Collective-Coordinate Formulation

For one rotational mode,

L_coll = −M + ½ Λ Ω² + q κ Ω·B + ⋯

J = ΛΩ, μ = qκΩ, g = 2Mκ/Λ

Structural g = 2 means

Λ − Mκ = 0

for all symmetry-preserving deformations.

The Diagnostic

For a deformation with direct and response contributions,

D_a = D_a,dir + D_a,resp, C_a = −D_a,resp/D_a,dir

The ratio C_a is invariant under rescaling of the specified tangent direction. It is not invariant under a general change of basis mixing directions. That is sufficient for comparing the physically specified operators.

Correction 8 — “Coordinate-independent” was too strong C_a is normalization-invariant along a specified tangent direction, not invariant under arbitrary coordinates on the whole tangent space. The precise statement was adopted.

The Two Questions

Question 1: dC(v) = 0 for every allowed v?

Question 2: Are allowed tangent directions generically nearly tangent to C = 0?

One admissible v₆ with dC(v₆) ≠ 0 answers Question 1 in the negative. Question 2 requires a normalization/metric on deformation space.

Part X — The Three Deformations: L₄, L₆, L₈

The Operators

Operator	Form	Direct normalization
L₄	A² + B, with A = Tr(L_μL^μ), B = Tr(L_μL_ν)Tr(L^μL^ν)	128π/3
L₆	B_μB^μ	16π/3
L₈	(B_μB^μ) Tr(L_νL^ν)	16π/3

The Results

D_a,dir = I_a/C_M − (c_aπ/3) J_a/C_Λ, D_a,resp = R_κ^(a) − R_Λ^(a)

C_a = −D_a,resp/D_a,dir

Direction	I_a	J_a	D_a,dir	D_a,resp	D_a,total	C_a
L₄	98.38693	4.43084	−7.10	+4.28	−2.82	0.60
L₆	3.78938	1.65948	−0.408	+0.434	+0.026	1.06
L₈	28.59602	11.61584	−2.641	+5.229	+2.588	1.98

Result — The three-direction classical test

Protection: No evidence that the shared symmetries fix g. The L₆ deformation preserves all shared symmetries while giving D₆ ≠ 0.
Generic cancellation: No. L₄ gives C₄ ≈ 0.60.
B^μ sufficiency: No. L₈ gives C₈ ≈ 1.98, despite sharing the same 16π/3 direct normalization and the same B^μ construction as L₆.
Relaxation opposition: D_dir < 0 and D_resp > 0 in all three directions. Observed, not yet explained.

The sextic near-cancellation is the point where two competing effects happen to nearly balance, not evidence for a generic B^μ-based protection mechanism.

Correction 9 — The near-origin source coefficient An earlier pass gave the near-origin coefficient of −½S̃₄ as 0.94x³. That omitted the prefactor 16a₀⁵/(3(2a₀²+1)) ≈ 19.19. The correct coefficient is ≈ 18.04x³. The sign statement survived; the magnitude was corrected.

Correction 10 — The Hessian normalization An earlier pass inferred that the sextic response terms should be halved because of a factor-of-two convention mismatch. A fresh numerical solve with the corrected operator confirmed that the earlier R_κ ≈ 1.90 was already correctly normalized. The inference 1.90 → 0.95 was wrong.

The Virial Convention

For the Derrick scaling \(F_\lambda(r) = F(r/\lambda)\), increasing λ makes the soliton larger, and

E₂(λ) = λE₂, E₄(λ) = λ⁻¹E₄, E₆(λ) = λ⁻³E₆

E₂ − E₄ − 3E₆ = 0

The inverse convention reverses all powers. The convention must be fixed before using the virial mode to explain response signs. The observed sign pattern is not yet a theorem: positivity of H⁻¹ does not determine the sign of the mixed bilinear form ⟨G_g, H⁻¹S_a⟩.

Part XI — The Quantum Gate

Setup

After FR quantization, the B = 1 hedgehog has collective coordinates A(t) ∈ SU(2). The quantized states are |I, I₃, J, J₃⟩ with I = J. The electromagnetic current operator is

J^μ_EM(x) = q B^μ(x)

Charge and Magnetic Moment

Q = q ∫ B⁰ d³x = q B = −e

μⁱ = (q/3Λ) ⟨r²⟩_B Jⁱ

Tree-Level Result

g_tree = 2M⟨r²⟩_B/(3Λ)

F₂⁽⁰⁾(0) = Mκ/Λ − 1

For the baryon-current Skyrmion control,

g_tree ≈ 2.88, F₂⁽⁰⁾(0) ≈ 0.44

Correction 11 — F₂(0) = 0 does not mean κ = 0 Setting κ = 0 would give g_tree = 0 and F₂⁽⁰⁾(0) = −1, even farther from the electron. The desired condition is F₂⁽⁰⁾(0) = 0, which means Mκ = Λ, i.e., Dirac g = 2. The tree-level magnetic moment is nonzero.

Result — Control model fails at tree level The quantized calculation reproduces the classical result: g_tree ≈ 2.88, F₂⁽⁰⁾(0) ≈ 0.44. The failure is not a loop effect; it is a mismatch in the leading electromagnetic vertex. Quantum corrections would need to produce ΔF₂ ≈ −0.44, followed by the tiny Schwinger remainder. That is not a Schwinger-like correction; it is a restructuring of the tree-level vertex.

Part XII — The Three-Strength Hierarchy and the Successor Search

The Hierarchy

numerical coincidence < on-shell identity < symmetry/structural identity

A numerical coincidence — Mκ ≈ Λ for one particular solution — is weak; the deformation analysis showed exactly why. An on-shell identity derived using the profile equations is much stronger. Strongest is a Ward identity, first-order/BPS relation, moduli-space geometry, or other structural principle that forces the equality and controls its quantum corrections.

Stability under quantization. An on-shell identity is not automatically stable under quantization. The Skyrme virial theorem E₂ − E₄ − 3E₆ = 0 is an on-shell identity, and it does not survive loop corrections without counterterms. A BPS relation is stronger than a numerical coincidence but still not a Ward identity. Only a symmetry, first-order structure, or moduli-space geometry that constrains the effective action directly controls quantum corrections without additional input.

The Successor Gates

Gate 1 Topology and quantization. B = 1, J = ½ via FR or equivalent. Gate 2 Charge. F₁⁽⁰⁾(0) = 1. Gate 3 Structural Dirac moment. Mκ − Λ = O(αΛ), stable under all unsuppressed symmetry-allowed corrections. Gate 4 Pointlike tree-level vertex. F₁⁽⁰⁾(q²) ≃ 1 and F₂⁽⁰⁾(q²) ≃ 0 over the experimentally relevant momentum range, not just at q² = 0. Gate 5 Radiative gate. F₂(0) = α/2π + ⋯, with the QED-like coefficient emerging from the quantum theory rather than being inserted.

The Symbolic Search Order

Candidate field theory ↓ Spinorial soliton quantization ↓ L_coll[A_μ] ↓ {Λ, κ} ↓ Identity/protection test ↓ F_1,2⁽⁰⁾(q²)

Only candidates surviving that chain deserve a numerical profile calculation.

The Litmus Test

The interesting structural question is whether a single functional governs both rotational inertia and electromagnetic response. If one could derive

Λ = ∫ d³x W[Φ], Mκ = ∫ d³x W[Φ]

on the soliton equations — not merely equal numerical integrals for one profile — then Mκ = Λ would follow automatically. A candidate should be rejected early if symmetry permits an independent local operator that changes Ω² without producing the corresponding change in (q/M) Ω·B, or vice versa.

Rejection criterion. If the two quantities depend on manifestly independent profile functionals, the tuning problem exposed by the deformation analysis returns. Do not solve the model numerically first. Derive the collective-coordinate kinetic metric and the electromagnetic moment functional symbolically and ask whether an identity can possibly relate them.

Part XIII — Current State and Open Items

What Is Established

The Skyrme model gives fermionic spin but not electron-like charge. Q = I₃ + B/2, which for B = 1 gives 0 or 1.
Hopfions can quantize fermionically, but the Hopf invariant has no local Lorentz-invariant four-current.
Ordinary Maxwell with a pointlike classical charge gives excessive positive self-energy. The momentum-space bound U_C ≥ (1/π) αħc Q_exp survives arbitrary sign-changing densities.
Standard Born–Infeld, calibrated to the electron mass, is excluded by light-by-light scattering by a factor ≈ 1.8 × 10³.
The symmetries shared by the two- and four-derivative Skyrme theory and its sextic extension do not fix g.
The sextic near-cancellation does not recur along the quartic or octic directions. The B^μ-dynamics hypothesis is falsified.
The control model with gauged baryon current gives F₂⁽⁰⁾(0) ≈ 0.44 at tree level. It fails the Dirac vertex gate before any loop calculation.

What Is Verified or Computed

Result	Value	Status
Kinematic J = ħ/2 with finite energy	Yes, at R ≈ 1.225 λ_C	Derived
Maxwell one-sign bound	U_CR_E ≥ 0.464758 αħc	Derived
Maxwell momentum-space bound	U_C ≥ (1/π) αħc Q_exp	Derived
Born–Infeld self-energy	U_BI = 1.23605 αħc/r_BI	Derived
Born–Infeld calibration	r_BI = 1.236 r_e ≈ 3.48 fm	Derived
Light-by-light exclusion	M_BI ≳ 100 GeV	Established
Skyrmion g_tree^(B)	≈ 2.88, F₂⁽⁰⁾(0) ≈ 0.44	Computed
Quartic D₄^(unit), C₄	≈ −2.82, ≈ 0.60	Computed
Sextic D₆, C₆	≈ +0.026, ≈ 1.06	Computed
Octic D₈, C₈	≈ +2.588, ≈ 1.98	Computed
B^μ-dynamics hypothesis	Falsified by C₈ ≉ 1	Falsified

What Is Open

Item	Status	Blocking
L₄ convergence table	Open	D₄ ≈ −2.82 is strongly indicated, not publication-grade frozen
Systematic relaxation opposition	Open	D_dir < 0, D_resp > 0 observed in all three directions. Not yet explained.
Successor model with Mκ = Λ	Open	Requires symbolic search, not numerical. Rejection criterion: independent profile functionals.
Pointlike tree-level vertex	Open	F₁⁽⁰⁾(q²) ≃ 1, F₂⁽⁰⁾(q²) ≃ 0 over the experimental range.
Radiative gate	Open	F₂(0) = α/2π + ⋯, with the QED-like coefficient emerging, not inserted.
Electron mass, size, magnetic moment	Open	The classical program does not reach them. The quantum soliton program is the real gate.

The Compact Statement

FR topology can supply fermionic spin, but no tested bosonic structure supplies a protected Dirac electromagnetic vertex.

This is more specific than “bosonic electrons are impossible” and more honest than “a Skyrmion could be an electron.” It names the missing mathematical structure: a principle relating the rotational kinetic metric to the electromagnetic moment functional in the same collective EFT, strong enough to survive quantization.

The next useful material is a symbolic search for a candidate field theory whose collective-coordinate kinetic metric and electromagnetic moment functional are governed by the same profile functional. Until that exists, there is nothing further to infer numerically without weakening the independence of the test.

Appendix — Conventions, Corrections, and Labels

Notation

Symbol	Meaning
U₀	Hedgehog background field, SU(2) valued
f(r)	Classical profile
n	Unit radial vector x/r
τ	Pauli matrices
B^μ	Topological baryon current
J^μ_EM	Electromagnetic current; in the control model, qB^μ
M, Λ, κ	Mass, moment of inertia, magnetic coupling coefficient
D_a	d ln g / dη_a
C_a	−D_a,resp/D_a,dir
F, e_S, m	Pion decay constant, Skyrme parameter, pion mass
μ² = m²/(e_S²F²)	Dimensionless mass parameter
x = e_SF r	Dimensionless radius

Key Equations

Electromagnetic angular momentum: J_EM = ε₀ ∫ r × (E × B) d³x

Reduced form: J_z = (8π/3) ε₀ ∫ r² Φ′ ∂_r(r a) dr

Skyrme Lagrangian: ℒ = (F²/16) Tr(L_μL^μ) + (1/32e_S²) Tr([L_μ,L_ν][L^μ,L^ν]) + (m²F²/8) Tr(U−1)

Hedgehog profile equation (dimensionless):

(x² + 2 sin²F) F″ + 2x F′ + sin 2F (F′² − 1 − sin²F/x²) − μ² x² sin F = 0

Skyrme charge formula: Q = I₃ + B/2

Collective-coordinate action: L_coll = −M + ½ Λ Ω² + q κ Ω·B + ⋯

g = 2Mκ/Λ, structural g = 2 ⟺ Λ − Mκ = 0

F₂⁽⁰⁾(0) = Mκ/Λ − 1

Maxwell one-sign bound: U_C R_E ≥ (3√15/25) αħc ≈ 0.464758 αħc

Maxwell momentum-space bound: U_C ≥ ((1−ε)²/π) αħc Q_exp

Born–Infeld self-energy: U_BI = (Γ(1/4)²/(6√π)) αħc/r_BI ≈ 1.23605 αħc/r_BI

Deformation direct contributions: D_a,dir = I_a/C_M − (c_aπ/3) J_a/C_Λ

c₄ = 128, c₆ = c₈ = 16

Cancellation ratio: C_a = −(R_κ^(a) − R_Λ^(a)) / D_a,dir

Source checks: −½S̃₄ ∼ 18.04 x³ (x→0), −½S̃₆ ∼ −36a⁵/x¹⁴ (x→∞), −½S̃₈ ∼ 3645.28 x³ (x→0), −½S̃₈ ∼ 324a⁷/x²⁰ (x→∞)

Corrections Recorded

#	Correction
1	Angular integral of sin³θ is 4/3, not 2/3. Prefactor 8π/3.
2	Charged-sphere construction passes the charge filter only kinematically, not as a self-contained field theory.
3	The SU(2) WZW term is a mod-2 object. It is not what ties integer charge to baryon number.
4	“No known bosonic constructions” was too strong. Only the natural sigma-model choices tested are excluded.
5	The g ∼ 1/R² obstruction was an artifact of an incorrect scaling. g_tree is a dimensionless profile number.
6	The one-sign constant is 0.464758 k_e, not 0.116 k_e. Factor-of-four error.
7	Light-by-light excludes M_BI ≲ 100 GeV, not ≳ 100 GeV. Inequality was backwards.
8	C_a is normalization-invariant along a specified direction, not coordinate-independent on the whole tangent space.
9	Near-origin coefficient of −½S̃₄ is 18.04x³, not 0.94x³. Prefactor 19.19 was omitted.
10	The sextic response terms were already correctly normalized. The inferred halving was wrong.
11	F₂⁽⁰⁾(0) = 0 does not mean κ = 0. It means Mκ = Λ, i.e., Dirac g = 2.

Status Label Legend

Label	Meaning
Verified	Confirmed against an independent numerical or symbolic check that does not share analytic intermediates with the object being tested
Established	Measured, tested, or proven within its domain
Derived	Follows from a calculation, theorem, or equation
Computed	Numerical result obtained; not yet independently verified
Coherent	Consistent with everything measured; not derived
Conditional	Holds under assumptions not yet verified
Pending recheck	Previously verified; the underlying input has since changed or been called into question
Open	Not yet computed or not yet checked
Not derived	Asserted but not shown
Falsified	Contradicted by data or by a computed result

Full Record — The Non-Protection Program

From the general question of whether a bosonic soliton can have a structurally Dirac electromagnetic vertex, through the deformation-matrix idea elevated to a theorem, to a candidate non-protection conjecture with an identifiable missing lemma. Every claim labeled. Corrections recorded.

How to Read This Document

This is the record of the final phase of the bosonic soliton electron program: the attempt to formulate a general no-go theorem about structural protection of the Dirac condition \(g=2\) for bosonic solitons. It follows the earlier records:

The classical control program (Maxwell, Born–Infeld, Skyrme deformation matrix).
The target-space filter and CP¹ local-current theorem.
The instanton branch, which reached a conditional endpoint at the flat-BPST mode-classification statement \(\mathcal{T}_3^{(IV)}\).

This document captures the pivot from model-by-model investigation to a general theorem.

Labels used throughout.

Label	Meaning
Established	Measured, tested, or proven within its domain.
Verified	Confirmed against an independent check that does not share analytic intermediates.
Derived	Follows from a calculation, theorem, or equation.
Computed	Numerical result obtained; not yet independently verified.
Structural	Follows from the structure of the framework. Not a derivation of a number.
Coherent	Consistent with everything measured. Not derived.
Conjecture	Proposed but not proven.
Conditional	Holds under stated assumptions not yet verified.
Open	No calculation has settled it.
Falsified	Contradicted by data or by a computed result.

Part I — The General Question

The Question

Does there exist a purely bosonic field theory in \(3+1\) dimensions with: 1. A stable, finite-energy soliton with \(\pi_3\) topology (or other topology supporting fermionic quantization). 2. A compact \(U(1)\) global symmetry that can be gauged to electromagnetism. 3. The soliton’s electromagnetic vertex satisfying \(F_1(0) = 1\) and \(F_2(0) = 0\) (Dirac \(g=2\)) as a structural consequence, not a tuning.

The Structural Condition

The condition \(g=2\) is equivalent to \[ M\kappa = \Lambda \] in the collective-coordinate effective action \[ L = -M + \tfrac12\Lambda\Omega^2 + q\kappa\,\boldsymbol\Omega\cdot\mathbf{B} + \cdots \] This is a relation between three different responses of the same soliton: – \(M\): the mass (response to time translation). – \(\Lambda\): the moment of inertia (response to rotation). – \(\kappa\): the magnetic coupling (response to an external magnetic field).

Known Mechanisms

Mechanism	Requires	Status
BPS states in \(N=2\) SUSY	Fermions, central charge	Not purely bosonic
Kerr–Newman black holes	Gravity, no-hair theorem	Not a soliton in flat space
Particle on Kähler manifold, \(B = \omega_K\)	Moduli space Kähler, current = Kähler form	Attempted in instanton branch; fails

Part II — What the Earlier Program Established

The Classical Controls

Ordinary Maxwell with a pointlike classical charge gives excessive positive self-energy. The momentum-space bound \(U_C \ge \frac{1}{\pi}\alpha\hbar c\,Q_{\rm exp}\) survives arbitrary sign-changing densities.
Standard Born–Infeld, calibrated to the electron mass, is excluded by light-by-light scattering by a factor \(\approx 1.8 \times 10^3\).
The Skyrme control model with gauged baryon current gives \(g_{\rm tree} \simeq 2.88\), failing the \(g=2\) gate at tree level.
The symmetries of the tested Skyrme family do not fix \(g\). The sextic deformation \(B_\mu B^\mu\) preserves those symmetries while giving \(D_6 \neq 0\).
The sextic near-cancellation does not recur along the independent quartic or octic directions. The \(B^\mu\)-dynamics hypothesis is falsified.

The Target-Space Filter

Structural table \[ \begin{array}{c|cc} & \text{local topological current} & \text{promising Kähler geometry} \\ \hline S^3 \text{ degree} & \checkmark & \times \\ S^2 \text{ Hopf} & \times & \checkmark \end{array} \] The search produced opposite halves. The successor problem is a target with both.

The Instanton Branch

The hyper-Kähler identity \(dK^\flat = 2\mathcal{N}\omega_3\) on the centered one-instanton moduli space was proved.
The rotational metric at logarithmic order: \(a_L = a_{L,\rm dir} > 0\), from the \(13/5\) tensor calculation.
The isoscalar logarithmic coefficient: \(a_B = 0\), from the small-\(Z\) structure of the HSS derivation.
The isovector propagator contribution: \(a_{V,\rm propagator} = 0\), by the exact completeness sum.
The isovector source contribution \(a_{V,\rm source}\) was reduced to a single flat-BPST mode-classification statement: \[ \mathcal{T}_3^{(IV)} = \operatorname{span}\{Z_{\rm rot}^1, Z_{\rm rot}^2, Z_{\rm rot}^3\} \quad ? \]

What the instanton branch showed. Even the most promising known geometry does not automatically produce a Dirac vertex. The hyper-Kähler identity holds on the moduli space, but the physical electromagnetic current does not select the distinguished one-form beyond the leading scale-covariant term. The obstruction is dynamical, not topological.

Part III — The Deformation-Matrix Formulation

The Structural Protection Question

The meaningful theorem is about structural protection: \[ \boxed{\text{When can } C[\Phi] \equiv \Lambda[\Phi] – M[\Phi]\kappa[\Phi] \text{ vanish identically on an allowed theory space?}} \] Trying to prove \(M\kappa \neq \Lambda\) for every bosonic soliton is too strong. Accidental equality can occur. The meaningful theorem is that \(C=0\) is not protected by the stated symmetries and topology.

The Functional

Define \[ DC: \delta\mathcal{L} \longmapsto \delta\Lambda – \kappa\,\delta M – M\,\delta\kappa. \] Structural protection means that \(DC\) vanishes on the entire space \(\mathscr{D}\) of admissible local deformations: \[ \boxed{DC|_{\mathscr{D}} = 0.} \] Each variation has two parts: \[ \delta M = \delta M_{\rm dir} + \delta M_{\rm resp}, \qquad \delta\Lambda = \delta\Lambda_{\rm dir} + \delta\Lambda_{\rm resp}, \qquad \delta\kappa = \delta\kappa_{\rm dir} + \delta\kappa_{\rm resp}. \] The direct parts are operator insertions on the undeformed soliton. The response parts come from the profile deformation solving the linearized equations.

The Envelope Theorem for the Mass

Because the undeformed static soliton extremizes the energy, \[ \delta M_{\rm resp} = 0 \] at first order, with collective coordinates and topological sector fixed. But neither \(\Lambda\) nor \(\kappa\) is an extremized functional, so generally \[ \delta\Lambda_{\rm resp} \neq 0, \qquad \delta\kappa_{\rm resp} \neq 0. \] This is the same envelope-theorem distinction encountered in the instanton calculation.

Part IV — The Failed Concrete Example and Its Corrections

The Proposed Example

The initial concrete example was a gauged complex scalar with three deformations: potential, kinetic, and current. The claim was that these give rank 3, hence non-protection.

Correction 1 — A single complex scalar does not furnish the \(\pi_3\) soliton The ordinary \(U(1)\) vacuum manifold of a single complex scalar has \(\pi_3(S^1) = 0\). Gauged scalar theories can have non-topological solitons, but that is a different class. A gauged Skyrme-type field is the legitimate test bed for the topology required.

Correction 2 — The current deformation is not gauge invariant The term \(\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}K_\rho A_\sigma\) is not generically gauge invariant because of the explicit \(A_\sigma\). Under \(A_\sigma \to A_\sigma + \partial_\sigma\alpha\), its variation is not automatically a harmless boundary term merely from \(\partial_\rho K^\rho = 0\). It cannot serve as the independent magnetic-only deformation.

Correction 3 — The response terms are first order in \(\epsilon\) The response terms cannot be discarded as higher order. Only \(\delta M_{\rm resp} = 0\) by stationarity. \(\delta\Lambda_{\rm resp}\) and \(\delta\kappa_{\rm resp}\) are first-order in the deformation.

Result — The example does not establish rank 3 With the corrected field content and the corrected gauge-invariant operator basis, the concrete example does not establish rank 3. The Skyrme deformation matrix provides a legitimate concrete example, but a general construction for an arbitrary bosonic soliton theory requires care.

Part V — Rank Is Not Necessary

The Key Observation

Rank 3 is a much stronger statement than required. At a point satisfying \[ C = \Lambda – M\kappa = 0, \] structural protection means that \(DC\) vanishes on the entire admissible deformation space. To disprove protection, we need only one admissible deformation for which \[ \boxed{DC[\mathcal{O}] \neq 0.} \]

The Geometry

Let \(\mathscr{T}\) be a class of local Lorentz-invariant bosonic theories with fixed field representations, topology, and compact \(U(1)\). Let \[ \mathcal{S}: \mathscr{T} \to \mathbb{R}^3, \qquad \mathcal{S}(\mathcal{L}) = (M, \Lambda, \kappa). \] Define \(C(\mathcal{L}) = \Lambda(\mathcal{L}) – M(\mathcal{L})\kappa(\mathcal{L})\). Ordinary symmetry and topology protect Dirac \(g=2\) only if \[ \boxed{DC_{\mathcal{L}}[\mathcal{O}] = 0 \quad \forall \mathcal{O} \in T_{\mathcal{L}}\mathscr{T}} \] whenever \(C(\mathcal{L}) = 0\). In other words, the symmetry-allowed theory space must be tangent everywhere to the hypersurface \(C=0\). If there exists even one allowed operator with \(DC[\mathcal{O}] \neq 0\), then \(C=0\) has positive codimension in theory space: it is a tuning, not a consequence of those symmetries.

What Rank 3 Would Give

Rank 3 would show that \(M, \Lambda, \kappa\) are locally independent observables. Rank 2 could still destroy \(g=2\); what matters is whether the image of the derivative happens to lie inside the two-dimensional tangent plane \[ \delta\Lambda – \kappa\,\delta M – M\,\delta\kappa = 0. \] So rank 3 is a stronger statement, not a necessary one.

Part VI — The Candidate Non-Protection Conjecture

The Conjecture

Bosonic Dirac-vertex non-protection conjecture.

Consider a local Lorentz-invariant bosonic theory in \(3+1\) dimensions possessing a stable finite-energy topological soliton, a compact \(U(1)\) symmetry with conserved current \(J^\mu\), and collective rotational coordinates. Suppose no additional Ward identity, first-order identity, or geometric identity relates the \(U(1)\) current to the stress tensor or the collective-coordinate metric. Then \(C = \Lambda – M\kappa = 0\), if realized, is not stable under all symmetry- and topology-preserving local deformations. Equivalently, \[ C = 0 \quad\Longrightarrow\quad \exists\,\mathcal{O} \in T_{\mathcal{L}}\mathscr{T}: \quad DC[\mathcal{O}] \neq 0. \]

What This Would and Would Not Establish

Establishes	Does not establish
That \(g=2\) cannot be a consequence merely of the ordinary ingredients: local Lorentz invariance, gauge invariance, topology, and a conserved \(U(1)\).	That no bosonic model can have \(g=2\).
That a model exhibiting structural \(g=2\) must possess an additional identity tying \(J^\mu\) to \(T^{\mu\nu}\) or to the moduli-space metric.	That \(M\kappa = \Lambda\) is impossible for a bosonic soliton.

The Missing Lemma

The conjecture has an identifiable missing lemma: \[ \boxed{ \begin{array}{c} \text{Construct one admissible transverse local deformation}\\ \text{for a sufficiently general class of bosonic soliton theories,}\\ \text{without assuming functional independence of } M, \Lambda, \kappa \text{ in advance.} \end{array}} \] The statement “\(M, \Lambda, \kappa\) are independent functionals” must remain a motivation, not an assumption. If we assume their independence, we have essentially assumed the desired conclusion. The theorem must derive transversality from the allowed local operator space.

The Converse Direction

A counterexample to the conjecture would be extremely informative. Suppose a purely bosonic model satisfies \[ DC[\mathcal{O}] = 0 \] for every allowed local deformation despite having no apparent SUSY or gravitational mechanism. Then rather than merely falsifying the no-go program, it would signal a hidden identity. One could work backward from \[ \delta\Lambda – \kappa\,\delta M – M\,\delta\kappa = 0 \quad \forall \mathcal{O} \] to identify the structure enforcing it.

Part VII — The Two Open Problems

The Separation

\[ \boxed{ \begin{aligned} &\textbf{Existence:} && \text{Does any purely bosonic soliton possess structurally protected } M\kappa = \Lambda? \\[2mm] &\textbf{Non-protection:} && \text{Can ordinary bosonic symmetry and topology alone be proved insufficient?} \end{aligned}} \] The Skyrme deformation matrix gives evidence for the second. The instanton/HK investigation tested one particularly strong candidate mechanism for the first. Neither establishes the universal statement.

The Escape Routes

The known mechanisms that enforce \(g=2\) all require either fermions, gravity, or a specific geometric relation between the current and the moduli-space metric that the physical current does not automatically satisfy: 1. Supersymmetry: the supercurrent relates \(T^{\mu\nu}\) to \(J^\mu\), and the BPS bound fixes \(g=2\). Requires fermions. 2. Conformal symmetry: \(T^{\mu\nu}\) and a dimension-3 conserved current sit in the same multiplet. Broken by the soliton mass. 3. A Kähler moduli space with the current as Kähler form: the instanton branch suggested the physical current does not automatically select this structure. A model exhibiting structural \(g=2\) would therefore have to violate one of the theorem’s hypotheses by possessing an extra identity tying \(J^\mu\) to \(T^{\mu\nu}\) or to the moduli-space metric.

The Reversed Search

Instead of asking which bosonic soliton happens to give \(g=2\)?, ask: \[ \boxed{ \text{What identity between } J^\mu \text{ and } T^{\mu\nu} \text{ would make } M\kappa = \Lambda \text{ deformation-invariant?} } \] Then either show that no purely bosonic local theory with the desired topology can possess that identity, giving the no-go theorem, or discover the missing structural mechanism.

Part VIII — Current State

What Is Established

The classical controls (Maxwell, Born–Infeld, Skyrme deformation matrix) — all eliminated.
The target-space filter and CP¹ local-current theorem — closed.
The hyper-Kähler identity \(dK^\flat = 2\mathcal{N}\omega_3\) — proved.
Instanton branch: \(a_L = a_{L,\rm dir} > 0\), \(a_B = 0\), \(a_{V,\rm propagator} = 0\) — derived.
Instanton branch: \(a_{V,\rm source}\) reduced to \(\mathcal{T}_3^{(IV)}\) — conditional.

What Is Conjectured

The bosonic Dirac-vertex non-protection conjecture: without an additional Ward identity, first-order identity, or geometric identity relating \(J^\mu\) to \(T^{\mu\nu}\) or the moduli-space metric, \(C = \Lambda – M\kappa = 0\) is not stable under symmetry- and topology-preserving local deformations.
The missing lemma: construct one admissible transverse local deformation for a sufficiently general class of bosonic soliton theories.

What Is Open

Item	Status	Blocking
Transverse deformation for general bosonic soliton theory	Open	The missing lemma of the conjecture
Flat-BPST mode classification \(\mathcal{T}_3^{(IV)}\)	Open	The \(l=1\), isovector-vector radial system; conditional endpoint of the instanton branch
Existence of a structural \(g=2\) mechanism	Open	Requires an additional identity beyond ordinary bosonic symmetry
Classification of identities that would force \(DC = 0\)	Open	The converse of the non-protection conjecture

The Compact Statement

The program has produced a precise conjecture with an identifiable missing lemma, and two well-posed open problems: prove non-protection, or find a structural escape.

The strongest defensible formulation at this point is the candidate non-protection conjecture, with its missing lemma now identifiable. Neither the universal existence of a structural \(g=2\) mechanism nor the universal insufficiency of ordinary bosonic symmetry is established. What has actually emerged is a clean scientific position rather than a claim that bosonic solitons cannot give \(g=2\).

Appendix — Conventions, Corrections, and Labels

Notation

Symbol	Meaning
\(M\)	Soliton mass (response to time translation)
\(\Lambda\)	Moment of inertia (response to rotation)
\(\kappa\)	Magnetic coupling coefficient (response to external magnetic field)
\(C\)	\(\Lambda – M\kappa\)
\(DC\)	The linear functional \(\delta\Lambda – \kappa\,\delta M – M\,\delta\kappa\)
\(\mathscr{T}\)	A class of local Lorentz-invariant bosonic theories
\(\mathscr{D}\)	The space of admissible local deformations
\(T_{\mathcal{L}}\mathscr{T}\)	The tangent space of \(\mathscr{T}\) at \(\mathcal{L}\)
\(J^\mu\)	Conserved \(U(1)\) current
\(T^{\mu\nu}\)	Stress tensor
\(\mathcal{T}_3^{(IV)}\)	Marginal \(\xi^{-3}\) solution space in the isovector-vector channel, modulo gauge

Key Equations

Dirac condition: \(M\kappa = \Lambda\)

Collective-coordinate EFT: \(L = -M + \tfrac12\Lambda\Omega^2 + q\kappa\,\boldsymbol\Omega\cdot\mathbf{B} + \cdots\)

Protection functional: \(DC[\mathcal{O}] = \delta\Lambda – \kappa\,\delta M – M\,\delta\kappa\)

Protection criterion: \(DC|_{\mathscr{D}} = 0\)

Non-protection: \(\exists\,\mathcal{O} \in T_{\mathcal{L}}\mathscr{T}: DC[\mathcal{O}] \neq 0\)

Corrections Recorded

#	Correction
1	A single complex scalar does not furnish the \(\pi_3\) soliton; its vacuum manifold has \(\pi_3(S^1) = 0\). A gauged Skyrme-type field is the legitimate test bed.
2	The term \(\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}K_\rho A_\sigma\) is not generically gauge invariant and cannot serve as the independent magnetic-only deformation.
3	The response terms are first order in \(\epsilon\); only \(\delta M_{\rm resp} = 0\) by stationarity. \(\delta\Lambda_{\rm resp}\) and \(\delta\kappa_{\rm resp}\) are first-order.
4	Rank 3 is not necessary. One transverse deformation suffices to disprove structural protection.
5	“\(M, \Lambda, \kappa\) are independent functionals” must remain a motivation, not an assumption in the theorem. The theorem must derive transversality from the allowed local operator space.

Status Label Legend

Label	Meaning
Verified	Confirmed against an independent check that does not share analytic intermediates
Established	Measured, tested, or proven within its domain
Derived	Follows from a calculation, theorem, or equation
Computed	Numerical result obtained; not yet independently verified
Conjecture	Proposed but not proven
Conditional	Holds under stated assumptions not yet verified
Open	Not yet computed or not yet checked
Falsified	Contradicted by data or by a computed result