How to Read This Document

This is a record of a sustained derivation. It begins from the structural condition Mκ = Λ implied by g = 2 and follows the topology and gauge routes to a sharp result. The neutral case is proved. The charged case has a principal-symbol cohomological obstruction that is reproducible from the present record; the small-e existence theorem is conditional on continuation assumptions that are asserted but not displayed. The numerical size of the window is undetermined. The broader questions — all-e continuation, the general response identity, and exceptional structures — are separate open problems.

Labels and Inspectability

Two label systems are used. The first records the logical status of a claim. The second records the evidentiary basis for it. Both matter, and they are not the same thing.

Logical status

Evidentiary basis

Summary of Results

Part I — The Sharpened Condition

From Mκ = Λ to Current–Metric Locking

The starting point is structural g = 2 for a charged rotating soliton. In collective-coordinate language, the magnetic moment κ and the moment of inertia Λ satisfy

This is equivalent to a mixed-response statement. Writing the energy as

the condition becomes

The sharpening is to ask what this requires of the moduli-space geometry. Rotation acts through rotational Killing vectors \(K_i^a\). Electromagnetism acts through a U(1) connection whose magnetic response one-form is \(\Theta_{\rm EM}^i\). The moment of inertia is \(\Lambda_{ij} = g_{ab}K_i^aK_j^b\).

Part II — The Neutral Route and Its Closure

The Metric-Independent Topological Current

The most natural candidate for a bosonic mechanism is a soliton whose electromagnetic current is a conserved topological current:

The Off-Shell Test

For the SU(2) hedgehog with unit winding, the topological density \(B^0(r)\) satisfies

But the magnetic first moment involves an extra factor of \(r^2\):

The Measure Mismatch in BPS Skyrme

The BPS Skyrme model is the most favorable bosonic soliton to test. Its Lagrangian is

On a BPS solution the two terms are pointwise equal, so the static mass is

The rotational inertia is

The topological magnetic moment is

The missing-half condition \(\mathcal K = (q/2)(\Lambda/M)\) becomes

Failed Deformation Candidates

The Profile-Preserving Deformation

The obstruction is broken by using the derivative of the topological current:

Under a rigid rotation, the convective derivative rotates the spatial covector index of \(L_i\):

After angular integration,

The Neutral Non-Redundancy Proof

Evaluate on the true adiabatic rotating solution \(\phi_\Omega = \phi_0 + \Omega Z_{\rm rot} + \Omega^2\eta_2 + \cdots\). Because \(\mathcal O_B[\phi_0]=0\) and \(D\mathcal O_B[\phi_0]=0\), the quadratic matrix element is determined entirely by the rotational zero mode.

Suppose for contradiction that \(\mathcal O_B = E_A Y^A + \partial_\mu V^\mu\). On-shell, \(E_A=0\), so \(\mathcal O_B = \partial_\mu V^\mu\). Integrating over a closed rotational period T:

The spatial boundary term vanishes by localization. The temporal term vanishes for single-valued local \(V^0\) over a closed orbit. Therefore redundancy would imply \(\int_0^T dt\int d^3x\,\mathcal O_B = 0\). But the explicit matrix element is \(TC\Omega^2 + O(\Omega^3)\), nonzero.

Part III — The Charged Case: Setup

The Operator

If U is charged under the electromagnetic U(1), then \(J_{\rm EM}^\mu\) is a Noether current, not a topological current. The natural generalization is

where the gauge-invariant baryon current must be derived from a fixed gauged action.

Part IV — Gate 1: The Mixed Hessian

The Corrected Expansion

On the static background at \(A=0\): \(\mathcal B^\mu = (\rho, 0)\), \(L_0=0\), \(L_i=l_i\), \(X_{00}^{(0)}=0\), \(X_{0i}^{(0)}=0\), \(X_{ij}^{(0)}={\rm Tr}(l_i l_j)\).

Introduce two ordinary parameters s, t:

with \(A_s^\dagger\dot A_s = s\omega\) and \(a_i = \tfrac12(B_{\rm ext}\times x)_i\). Then

At \(s=0\), the configuration is static for all t. Therefore \(\nabla_0 L_i = 0\) for all t, and hence \(\delta_B\nabla_0 L_i = 0\). The standard gauge-invariant baryon current gives \(\delta_B\mathcal B^i = 0\) on the static background. The mixed Hessian becomes

All three terms vanish. The first because \(\delta_B Y_0{}^\alpha = 0\). The second because \(\delta_B X_{i0} = {\rm Tr}(Y_{i\alpha}^{(0)}\delta_B Y_0{}^\alpha) = 0\). The third because \(\delta_B\mathcal B^i = 0\).

Part V — The Source-Derived Current

The Fixed Action

Fix the action:

where \(\mathcal B_\mu\) is a background source for baryon number. The gauge-invariant baryon current is defined by

The standard descent gives

Variation on the Static Background

At \(A=0\), the background is static: \(L_0=0\), and \(\partial_0\) of any static quantity is zero.

Goldstone–Wilczek term

For the epsilon tensor to be nonzero, \(i,k,\sigma,\nu\) must be distinct. Since \(i,k,\sigma\) are spatial, \(\nu=0\). Then \(\partial_0\) of a static quantity vanishes.

Bardeen term

At \(A=0\), both \(A_\nu=0\) and \(F_{\rho\sigma}=0\), so both terms vanish.

Part VI — The Principal-Symbol Obstruction

Reduction to a Local Test

The closed-orbit argument does not carry over to the zero-frequency adiabatic setting, because the collective configuration is not an exact four-dimensional solution. The correct local test is the off-shell jet-space identity:

The best first attack is the principal time-derivative symbol.

The Quadratic Symbol of \(\mathcal O_c\)

Write \(U = U_0 e^\xi\), so \(\delta L_\mu = D_\mu\xi\). At principal order \(\delta L_\mu \approx \partial_\mu\xi\). On a static background, \(B^\mu=(\rho,0)\). For a plane-wave fluctuation \(\xi = v\,e^{-i\omega t + ikz}\),

The principal contraction is the Minkowski sum over \(\alpha\):

Therefore

The Skyrme Characteristic Matrix at \(p \parallel n\)

At a point on the hedgehog, choose the spatial 3-axis along n. The background Maurer–Cartan forms have

Let A be the sigma-model coefficient and C the Skyrme coefficient. For \(p \parallel n\), the principal quadratic form diagonalizes by internal polarization.

The Transverse Obstruction

For the radial mode, \(\omega_R^2=k^2\), so \(P_{c,\rm prin}(\omega_R,k)=0\). No obstruction.

For a transverse characteristic mode,

Therefore

At a generic point of a nontrivial hedgehog, \(\rho \neq 0\) and \(X_R = f’^2 \neq 0\), and with \(A,C > 0\),

Derived Reproducible-here

Part VII — The Divergence Loophole Removed

The Problem with the Integrated Argument

If \(\mathcal O_c = E_A Y^A + \partial_\mu V^\mu\), then on an exact periodic on-shell solution the integrated \(\mathcal O_c\) would vanish. But the zero-frequency collective configuration is not an exact four-dimensional solution. On it, \(E_A \neq 0\) at the relevant higher adiabatic order, so the EOM piece and the divergence piece are evaluated on two different configurations. They do not form a contradiction.

The Euler Operator Test

Apply the Euler–Lagrange operator to the proposed identity. Since the Euler operator annihilates local divergences identically,

redundancy requires

At quadratic order around the static background, if

then its Euler derivative is essentially the formally self-adjoint operator \(P_c\). A divergence has disappeared automatically. For an infinitesimal field redefinition with linearization R, the quadratic EOM-exact contribution has Euler operator

Now the characteristic-kernel argument becomes legitimate without any \(V^\mu\). If \(Hv=0\), then because the Hessian principal symbol is self-adjoint, \(v^\dagger H^\dagger = 0\). Hence

But the transverse Skyrme characteristic gives

Part VIII — The Gauss-Reduced Physical Pencil

Coulomb Gauge and the Gauss Constraint

Impose the Coulomb gauge condition \(\partial_i A_i = 0\) as a gauge choice. This sets the longitudinal component \(A_L = 0\). The residual gauge freedom is time-independent gauge transformations, fixed by requiring \(A_0 \to 0\) at spatial infinity.

The Gauss constraint is not \(\partial_i\dot A_i = 0\); it is

where \(\rho_{\rm matter}\) is the linearized charge density of the matter fluctuation. At fixed nonzero spatial momentum k, this gives

So \(A_0\) is a constrained field determined instantaneously by the matter fluctuation. It is not an independent dynamical degree of freedom.

The Physical Degrees of Freedom

After Coulomb gauge fixing and solving Gauss’s law, the physical dynamical fields are

with \(\xi\) a three-component matter fluctuation and \(A_T\) the two transverse photon polarizations. Total: five degrees of freedom.

The Physical Pencil

The linearized equations for the physical fields take the form

The kinetic matrix at \(e=0\)

At \(e=0\), the matter and transverse photon sectors decouple. The kinetic matrix is block diagonal:

Both are positive since \(A,C > 0\) and \(X_T, X_R \geq 0\). The last two entries are the standard Maxwell kinetic coefficients.

Continuity in \(e\)

The covariant derivative introduces the coupling \(D_\mu\xi = \partial_\mu\xi – ieA_\mu[Q,\xi]\). At fixed k, the coupling between matter and transverse photons enters at \(O(e)\). The Coulomb backreaction through \(A_0 = \rho_{\rm matter}/k^2\) enters at \(O(e^2)\). Both are bounded on the physical sector for a smooth static background. Therefore

Since \(K_{\rm phys}(0,k)\) is positive definite, its eigenvalues are bounded below by a positive constant. Small perturbations preserve invertibility:

Part IX — Small-Coupling Persistence (Conditional)

The Reduced Matrix

Define

This is continuous in \(e\) on the physical sector.

The Spectral Gap at \(e=0\)

At \(e=0\), the transverse matter cluster has eigenvalue

of multiplicity two, isolated from the radial matter branch and the transverse photon branch by

The Riesz Projector Argument

Choose a contour \(\Gamma\) surrounding \(\lambda_T\) and no other eigenvalue. Define

For sufficiently small \(e\), \(\Pi_T(e)\) exists, satisfies \(\Pi_T(e) \to \Pi_T(0)\), and has rank 2. Conditional on the asserted continuation

The Quadratic Form on the Cluster

At \(e=0\),

By continuity,

For sufficiently small \(e\), this restricted quadratic form is nondegenerate on the entire perturbed two-dimensional cluster. Conditional on the asserted continuation

Part X — The O(e) Block Identities

The Three Identities

The target is

If all three hold, then

is matter–photon off-diagonal, so for every matter vector \(v\),

The first-order correction to a transverse matter eigenvector is purely photonic:

Then \(P_0 w^{(1)} = 0\) because \(P_0\) acts only on the matter sector, and \(v^\dagger P_1 v = 0\) from the block condition. Therefore

The Structural Argument for \(K_1^{mm} = S_1^{mm} = 0\)

The quadratic action at \(e=0\) is block diagonal. At \(O(e)\), the only source of coupling is the covariant derivative \(D_\mu U = \partial_\mu U – ie\,a_\mu[Q,U]\). Every explicit \(\partial_e\) at \(e=0\) introduces exactly one gauge fluctuation \(a_\mu\), provided:

1. The background matter field has no \(O(e)\) displacement.
2. The background gauge potential starts at \(O(e)\).

Both hold: the Coulomb field \(A_0^{(0)}\) is \(O(e)\), and the profile shift is \(O(e^2)\). Therefore \(H_1\) is block off-diagonal with no matter–matter component.

Gauss reduction. The constrained variable satisfies \(a_0 = e\,G\,\xi + \cdots\). Substituting back produces \(e^2\,\xi G\xi\), not \(e\,\xi^2\). So the Gauss reduction preserves \(H_1^{mm} = 0\).

The \(P_1^{mm}\) Check

The operator is \(\mathcal O_c = \mathcal B^\mu \mathcal B^\nu\,{\rm Tr}(\nabla_\mu L_\alpha \nabla_\nu L^\alpha)\). The expansion of \(L_\mu\) and \(\nabla_\mu L_\alpha\) yields only \(e\,a\xi\) terms at \(O(e)\), never \(e\,\xi^2\). Therefore

Conclusion

Therefore \(\delta\lambda_T^{(1)} = 0\), \(\delta\gamma^{(1)} = 0\), and the leading deterioration is \(O(e^2)\).

The Second-Order Effective Operator

Part XI — The Window-Size Specification (Draft)

The Frozen Input

The CAS-ready specification for the window-size calculation is:

together with one fixed explicit Goldstone–Wilczek current representative:

and with the charge matrix fixed as \(Q = {\rm diag}(2/3, -1/3) = \tau_3/2 + \mathbf 1/6\).

Background Maurer–Cartan forms in the frozen local frame with \(n = e_3\):

One verified building block from a direct recomputation, useful as a cross-check for the CAS:

The Three Stages

1. Unreduced local Hessian. Compute \(H_{0,1,2}\) and \(P_{0,1,2}^{\rm explicit}\) on the fixed background, retaining \(A_1, A_1’\) and any necessary background jets. Keep \(U_2\) entirely out.
2. Gauss reduction. Construct the physical \(K, S, P\) pencil from the unreduced system.
3. Profile response. Add \(D_{U_2}H_0\) and \(D_{U_2}P_0\) only when solving the physical \(O(e^2)\) window problem.

The Definitions

followed by Gauss reduction of the pair \((H, P)\).

Part XII — The Computational Boundary

Why the Calculation Stops Here

The explicit \(P_{2,\rm explicit}^{mm}\) matrix requires evaluating

at \(O(e^2)\), with the fixed GW current and the frozen \(L, R\) expansion. This requires:

• explicit \(\mathfrak{su}(2)\) index contraction of the background \(l_i^a\) with the fluctuation \(\xi^a\),
• expansion of \(L_\mu + R_\mu\) in the local frame with \(k \parallel e_3\),
• the \(e^2\) GW current evaluated on the Coulomb background \(A_1(r)\), including \(A_1′(r)\),
• the derivative structure \(D_\mu\xi\), \(\partial_\mu\xi\), and their commutators.

Doing this by hand in text would produce errors that cannot be detected — sign errors in commutators, misplaced \(U_0\) conjugations, index mismatches, confusion between \(A_1\), \(A_1’\), and the constrained \(a_0\). Any one would propagate into the matrix and produce a wrong result that looks plausible.

What Can Be Checked Once the Matrix Exists

1. Hermiticity. The matrix should be Hermitian.
2. Reduction at \(e=0\). It should reduce to \(P_0^{mm}\).
3. Static limit. The \(\xi = 0\) limit should reproduce the static background.
4. Residual symmetry. The matrix should transform covariantly under the residual symmetry of the full background including EM spurions, not simply radial = transverse.
5. Invariant cross-check. The pole value \(l_\theta|_{\rm pole} = i(\sin f\cos f\,\tau_1 + \sin^2 f\,\tau_2)\) and the invariant \(-\tfrac12{\rm Tr}(l_\theta^2) = \sin^2 f\) should be reproduced by the background algebra used in the CAS input.
6. Chirality cross-check. The Cartesian combination \(l_x – i l_y = \tfrac{i\sin f}{r}e^{if}\tau_-\) should be reproduced by the CAS background.
7. Dominant term. Identify whether the Schur term or the direct \(N_2\) term is larger, and what the resulting scaling of \(\epsilon\) is.

Part XIII — Corrections Along the Way

Part XIV — Addendum: Pole Phase-Lock Structure

Purpose

This addendum records a background-algebra result that has been independently reconstructed in this record: the single-chirality, \(e^{if}\)-phase structure of the transverse Maurer–Cartan combination at the pole. It also clarifies the attribution of Correction 23 and flags an open structural subtlety about the transverse fluctuation at the pole.

Section 1 — Clarification on Correction 23

Correction 23’s substantive claim is correct with respect to the transcript text as received: the transcript’s quoted pole value of \(l_\theta\) differs from the direct calculation. Correction 23 already gives the direct-calculation value \(i(\sin f\cos f\,\tau_1 + \sin^2 f\,\tau_2)\), which is the correct form. It attributes the \(\tfrac12\sin^2 f\) form to the transcript. If the underlying session produced \(\sin f\cos f\), then the discrepancy is inside the transcript’s transcription, and the correct action is to reword Correction 23’s attribution rather than overturn it. The invariant check confirms the direct-calculation form and the failure of the \(\tfrac12\sin^2 f\) form.

Section 2 — Direct value of \(l_\theta\)

Under \(U_0 = \cos f + i(\tau\cdot n)\sin f\), at the north pole \(\theta = 0\), \(\varphi = 0\):

\(l_\theta|_{\rm pole} = i(\sin f\cos f\,\tau_1 + \sin^2 f\,\tau_2).\)

The \(\tau_1\) coefficient is \(\sin f\cos f = \tfrac12\sin 2f\). It is independent of the left- or right-acting Maurer–Cartan convention.

Derived Reproducible-here

Section 3 — Invariant check

The invariant \(-\tfrac12{\rm Tr}(l_\theta^2) = \sin^2 f\) holds for the form above in the Pauli convention \(\mathrm{Tr}(\tau_a\tau_b) = 2\delta_{ab}\):

\(l_\theta^2 = -\sin^2 f\cos^2 f\,\mathbf 1 – \sin^4 f\,\mathbf 1 = -\sin^2 f\,\mathbf 1.\)

The alternative \(\tfrac12\sin^2 f\) form gives \(l_\theta^2 = -\tfrac14\sin^4 f\,\mathbf 1 – \sin^4 f\,\mathbf 1 = -\tfrac54\sin^4 f\,\mathbf 1\), hence \(-\tfrac12{\rm Tr}(l_\theta^2) = \tfrac54\sin^4 f\), which fails the invariant except when \(\sin^2 f = 4/5\). The invariant cleanly distinguishes the two forms and confirms the Section 2 value.

Derived Reproducible-here

Section 4 — Circular combination and phase-lock

Direct Cartesian calculation at the pole, with \(\partial_x n = e_1/r\), \(\partial_y n = e_2/r\), gives

\(l_x = \tfrac{i}{r}(\sin f\cos f\,\tau_1 + \sin^2 f\,\tau_2),\)

\(l_y = \tfrac{i}{r}(-\sin^2 f\,\tau_1 + \sin f\cos f\,\tau_2).\)

Consequently,

The single-chirality structure and the \(e^{if}\) phase are reproducible here. The factor \(2\) appears only when using \(T_-\). The overall \(i/r\) factor is not part of the transcript’s bare coefficient; whether the transcript’s exact statement \(2\sin f\,e^{if}\) referred to the coefficient after stripping the conventional \(i/r\) factor cannot be determined without the underlying calculation.

Derived Reproducible-here for the \(e^{if}\tau_-\) structure. The exact transcript string remains Asserted-not-inspectable.

Section 5 — Open item: the transverse fluctuation at the pole

A transverse fluctuation of the soliton at the pole is like a p-orbital vanishing at its origin: its value vanishes, its angular derivative does not. The well-defined object at the pole is the derivative of the transverse components in the \(x, y\) directions, not their value. This was identified during the session and flagged, not resolved. Any phase-locked ansatz for the transverse fluctuation must be built with this distinction explicit from the start.

Open

Section 6 — What the addendum does not establish

• The “twice the rate” precession claim. The \(e^{if}\) phase in the Maurer–Cartan form implies a covariant derivative with phase rate \(f’\), not \(2f’\), unless the comparison is to a different quantity. The claim requires its own derivation.
• The exact transcript string \(2\sin f\,e^{if}\). The mathematical form \(l_x – i l_y = \tfrac{i\sin f}{r} e^{if}\tau_-\) is reproducible here. Whether the transcript stripped the \(i/r\) factor and how it named the coefficient cannot be determined.
• The remaining transcript-reported items — \(l_\varphi\), \(\nabla_\mu l_\alpha\), any \(e^2\) coefficient — remain asserted-not-inspectable.

Closing State

The derivation produced the following frozen status.

What Remains Open

1. The coupled-branch continuation. Weighted-space setup, Fredholm invertibility, and Riesz-projector smoothness. Not displayed. The small-e theorem is conditional on them.
2. The finite-k subprincipal transverse operator \(P_{\rm sub}^{(T)}\). Whether, after summing every contribution at the relevant symbol order, \(P_{\rm sub}^{(T)} = p_T I_2\) or contains a surviving \(p_A J_3\) with \(p_A \neq 0\). This is a question about the full local quadratic operator beyond its principal symbol; it does not affect the principal-symbol obstruction.
3. The circular-polarization cancellation. Follows from the above.
4. \(P_{2,\rm explicit}^{mm}\) and \(D_{U_2}P_0^{mm}\). Requires a CAS, with the frozen input above.
5. The numerical size of the window. Requires the above.
6. The transverse fluctuation at the pole. A transverse fluctuation vanishes at the pole like a p-orbital at its origin; the well-defined object is its derivative in the \(x, y\) directions, not its value. The phase-locked ansatz must be rebuilt with this distinction explicit.
7. The “twice the rate” precession claim. The \(e^{if}\) phase implies a covariant derivative with phase rate \(f’\), not \(2f’\), unless the comparison is to a different quantity. Requires its own derivation.
8. All-e continuation. The result holds for sufficiently small nonzero e, not for all e up to the physical coupling.
9. The general response identity. The classification of local identities capable of enforcing \(\mathscr R_\Theta = (q/M)\mathscr R_g\) modulo EOM, total derivatives, and gauge identities is open.
10. Exceptional structures. Whether any purely bosonic structure — supersymmetry, hidden symmetry, constrained operator algebra, geometric locking — can enforce current–metric locking is untouched.

The Honest Summary

The derivation produced a proved neutral lemma, a reproducible principal-symbol cohomological obstruction for the charged case, and a small-e charged theorem whose e = 0 anchor is reproducible but whose continuation portion is not. The pole value of the transverse Maurer–Cartan form has been independently verified, as has the invariant \(-\tfrac12{\rm Tr}(l_\theta^2) = \sin^2 f\) and the pole phase-lock structure \(l_x – i l_y = \tfrac{i\sin f}{r} e^{if}\tau_-\). The numerical window and the exceptional-structure question remain open. The remaining transcript-reported items are excluded pending inspectable output.

The problem is no longer “find a clever soliton profile that gives g = 2.” It is: does there exist a microscopic identity whose local response content forces the electromagnetic magnetic-response functional and the rotational kinetic functional to be the same functional up to q/M?