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---
## What it means to rotate a localized structure
ASD constructs spin from the collective orientation and quantization of closed spatial configurations. It supplies collective rotational energy and a route to half-integer representations under the Finkelstein–Rubinstein condition. The value $j=1/2$ is the lowest allowed representation of that sequence; identifying the physical ground state still depends on the complete configuration, inertia and quantization conditions. Internal $2\pi$ phase closure does not replace this argument.
Matter has angular momentum and orientation response as well as translation. ASD studies spin through collective rotation of an entire closed configuration: how core and surrounding fields rotate, how rotational kinetic energy enters the action, and which representations quantization permits. The construction neither assumes a rigid classical electron or proton nor assigns spin from a mechanical rotation speed.
Four objects must be separated. Spatial rotation changes orientation relative to exterior coordinates. Internal rotation changes field organization. A configuration-space path describes continuous evolution of all fields. A quantum state represents quantized collective freedoms. Internal phase closure can relate to them, but is not the same angle under four names.
## Collective coordinates and rotational energy
Let $Q_0(\mathbf x)$ be an actual stable representative, $R(t)\in SO(3)$ a spatial rotation, and $A(t)\in SU(2)$ its double-cover variable. Start with a spatially rotated family and insert it into the complete action:
$Q_0$ is an orientation field and $A$ a collective rotation variable, not interchangeable objects. $\Omega_a$ is collective angular velocity with inverse-time units, rather than displacement curl. $\Lambda_{ab}$ is an inertia tensor in mass-times-length-squared units obtained by integrating the complete kinetic energy.
The manuscript then defines $J_a=\partial L_{\rm rot}/\partial\Omega_a=\Lambda_{ab}\Omega_b$. The quadratic rotational energy is $\tfrac12\Lambda_{ab}\Omega_a\Omega_b$; reducing inertia to a scalar requires suitable symmetry. Independent internal rotations need separate coordinates with duplicate descriptions removed.
Rotations that leave the configuration unchanged form its stabilizer. They cannot be counted again as independent physical modes in inertia or state space. The actual fields, symmetry and retained components determine the collective rotation space.
## A double cover and a configuration-space path
$SU(2)$ double covers $SO(3)$: both $A$ and $-A$ represent the same spatial rotation $R$. A continuous $2\pi$ rotation can lift from $A$ to $-A$, and a second rotation returns to $A$. This supplies the mathematical origin of a $4\pi$ restoration picture.
Particle topology, however, concerns the complete allowed configuration space. One must check whether the $2\pi$ rotation loop is noncontractible with amplitude and accompanying fields retained, and how configuration symmetries affect it. An $SU(2)$ target alone does not establish the required loop. Amplitude zeros or other permitted deformations may alter the conclusion.
The [Finkelstein–Rubinstein route](https://doi.org/10.1063/1.1664510) uses such nontrivial paths to organize quantization. Chapter 4 adopts it as a candidate ASD route to half-integer rotational representations, rather than claiming every particle spin from a diagram.
## The quantization choice and allowed representations
Once the required nontrivial $\mathbb Z_2$ loop is established, the manuscript chooses
$\psi(A)$ is a quantum state on collective coordinates, not orientation $Q$ or the spatial matter-wave envelope. The minus sign is an explicit quantization choice allowing half-integer $j$; the lowest allowed value can be $1/2$. The actual lowest-energy representation depends on inertia, constraints and further collective coupling.
A $2\pi$ rotation changes the state’s sign and $4\pi$ restores it. An isolated overall sign does not change a state’s probability, while relative phases between paths can participate in interference. Double-cover geometry, configuration loops and quantization choices play distinct roles in this argument.
## Exchange and magnetic response
Exchanging two identical structures gives another path in the two-structure configuration space. Fermionic exchange requires its relation to rotational paths and a consistent quantization. The single-structure $4\pi$ picture does not replace this calculation, whose inputs still include full fields and boundaries.
A magnetic moment also needs the charged rotating configuration’s current and external-field coupling. The same [charge identification](/en/theory/matter/charge) and [electromagnetic response](/en/theory/electromagnetism/magnetic-field) must connect angular momentum with magnetic response. Neither $g\approx2$ nor the anomalous moment is an automatic spin-label output, and a borrowed Dirac result is not an ASD derivation.
The manuscript supplies collective rotational energy and a conditional half-integer route. [OP-MAT-004](/en/open-problems/matter/spin-statistics) continues with the actual configuration space, inertia, exchange path and magnetic response. Internal $2\pi$ or $6\pi$ particle labels must be matched to each of these objects.