Spin as a Transformation Class of Rotating Wave Modes

Core results

• Photon literalness. For circularly polarized light the rotating-wave language is physically pertinent: the transverse field components rotate, the angular momentum is mechanically measurable (Beth 1936), and the Riemann–Silberstein combinations $\mathbf{F}_\pm = \mathbf{E} \pm ic\mathbf{B}$ separate the helicity sectors after complexification and restriction to positive-frequency components.
• Transformation-law criterion. A complex scalar wave rotates in its internal plane yet carries spin zero: rotation of components is not sufficient. Spin is fixed by $\Psi(\mathbf{x}) \mapsto D^{(j)}(R)\,\Psi(R^{-1}\mathbf{x})$ — the internal rotation counts only insofar as its plane is geometrically coupled to physical space.
• Weights versus representations. The axial winding intuition is rigorous for the weight $m$ (the $U(1)$ character $e^{-im\alpha}$), but the values of $j$ arise from the highest-weight theory of $SU(2)$, not from the $\mathbb{Z}_2$ topology of $SO(3)$ alone.
• Interferometric spinor sign. The $4\pi$ periodicity of neutron interference (Rauch et al., Werner et al., 1975) shows that the transformation class of the state under rotations is physical, while the rotation phase is unobservable in isolation.
• Univalence as an observable-algebra rule. The Wick–Wightman–Wigner rule does not forbid writing boson–fermion vector sums; it states that no admissible observable measures their relative phase, making such superpositions operationally equivalent to mixtures.
• Superluminal objection dissolved. The historical objection constrains rigid bodies at the classical electron radius, not internal rotations of wave components; the Belinfante–Ohanian energy-flow reading on the Compton scale removes the need for a surface velocity, its literal hydrodynamic interpretation requiring a separate causal analysis.
• Ontological constraint. Any rotational ontology of spin must involve a relational, extended, or topologically attached wave configuration (belt trick, orientation-entanglement, skyrmions under Finkelstein–Rubinstein quantization), never a point particle endowed with a mere internal axis.

Implications for the Cosmochrony programme

• Established. The admissibility construction selects $V_\rho \simeq \mathbb{C}^2$, with $\operatorname{Sym}^2(V_\rho) = V_1$ and $\wedge^2(V_\rho) = V_0$.
• Exact algebraic consequence. The central element acts on $V_\rho^{\otimes n}$ as $(-1)^n$; conditional on the Lorentz-spin identification of $V_\rho$, this tensor grading becomes the physical univalence grading.
• Correction. In Q7, $e_0$ is the spin-weight $m = 0$ within the $j = 1$ module, not a $j = 0$ state; the genuine singlet is $\wedge^2(V_\rho)$.
• Diagnostic. Lorentzian spin and the weak doublet must act on distinct tensor factors.
• Open. The representational problem — constructing the full $\operatorname{Spin}(3)$ action with $\rho_{\mathrm{rot}}(-I) = (-1)^n$ — which a one-parameter evolution cannot solve; and the stronger dynamical problem — whether the cascade canonically traces an axial one-parameter orbit, with axis and angle relation both derived.

The soldering audit (v1.1)

• Real-form classification (proved). The quaternionic structure of the admissible doublet induces a circle $W_\phi = e^{i\phi}W_0$ of distinct compact real forms of $\operatorname{Sym}^2(V_\rho)$, permuted by the center of $U(2)$; each is $SU(2)$-invariant with induced $SO(3)$ action of kernel exactly $\{\pm I\}$; neither the unitary structure nor the invariant form selects the phase.
• Veronese obstruction (proved, unconditional). Nonzero pure tensors $w w^{\mathsf T}$ lie on the null cone of the invariant form, while every $W_\phi$ is positive definite for its rescaled form: the O28 outer products cannot select the real form — the first internal selector candidate is excluded exactly.
• Indeterminacy of the spinorial soldering (conditional on the corpus inventory). The derived data determine the complex carrier and its univalence but neither the spatial realisation, nor a physical rotation action, nor its scale. The missing soldering is typed as three independently missing data: a phase-sensitive real-form selector, an independent rotation action $\rho_{\mathrm{sp}}$ (structurally blocking), and a Casimir-independent normalisation.

Relation to the Cosmochrony programme

This note belongs to the fermionic matter sub-programme. Upstream, it takes from O29 / Q7 the admissible module $V_\rho \simeq \mathbb{C}^2$; downstream, it types the open front of Q14 and its companions by separating the rotational representation problem from its possible dynamical realization by the projected cascade.

References

Jérôme Beau. Spin as a Transformation Class of Rotating Wave Modes. Working paper, 2026. 10.5281/zenodo.21380026