Spin-lifted null dynamics and a parity correction on Lorentzian tori

Definitions and mathematical statement

Problem 16 asks to relate $h(g)$ to null-geodesic dynamics, without defining $h$. The paired aid interprets it as the complex dimension of the full smooth Dirac kernel on a Lorentzian spin manifold. Write $h(g,s)=\dim_{\mathbb C}\ker D_{g,s}$ to retain the spin structure explicitly; chirality, completions or boundary conditions would define different quantities.

Its exact test case is the unit-period torus $(\mathbb R/\mathbb Z)^2$ with \[ g=-\lambda_t^2dt^2+\lambda_x^2dx^2,\qquad \lambda_t,\lambda_x>0. \] Let $e_t,e_x\in\{0,1\}$ prescribe spin periodicities $(-1)^{e_t}$, $(-1)^{e_x}$. The two null directions have slopes $dx/dt=\pm\lambda_t/\lambda_x$. If $\lambda_t/\lambda_x=m/n$ with coprime positive integers $m,n$, the aid reports \[ h(g,s)=\begin{cases} \infty,&ne_t+me_x\equiv0\pmod2,\\ 0,&ne_t+me_x\equiv1\pmod2. \end{cases} \] Here infinity means infinite-dimensional, not a specified cardinality. A primitive closed null geodesic has winding $(n,\pm m)$; its spin holonomy is $ (-1)^{ne_t\pm me_x}$, with the same parity for both signs. Thus the condition is precisely trivial spin holonomy along the primitive null orbits.

For irrational $\lambda_t/\lambda_x$, the reported full-kernel answer is \[ h(g,s)=2\quad\text{if }(e_t,e_x)=(0,0),\qquad h(g,s)=0\quad\text{otherwise}. \] The two-dimensional value counts the constant full complex spinors. In the Fourier convention of these periodicities, null frequencies obey \[ n(p+e_t/2)=\pm m(q+e_x/2),\qquad p,q\in\mathbb Z, \] in the rational case. The aid uses the parity condition to correct or qualify rational-slope entries in a table following Witt's Fourier equation, not reproduced in the supplied record.

The same metric and ordinary null dynamics therefore can have different Dirac-kernel dimensions for different spin structures, including zero versus infinity. Any proposed determination from dynamics must retain spin-lifted transport, rather than only null orbit geometry.

The aid adds two broader qualifications. Microlocally, Dirac polarizations propagate along null geodesics by spin parallel transport; this concerns characteristic singularity/polarization transport and is not itself a count of the global smooth kernel. On a globally hyperbolic spin spacetime of dimension at least two, unrestricted smooth Dirac solutions already form an infinite-dimensional space through free Cauchy data. Thus the meaning of $h$, global causal assumptions and the solution space must be fixed before a finite-dimensional dynamical relation is sought. The torus result is a special-case obstruction and correction, not a general formula for every Lorentzian metric.

Short English statement

Under the supplied full-Dirac-kernel interpretation, ordinary null dynamics does not determine $h$: changing spin structure on a fixed flat Lorentzian torus can change the kernel from zero to infinite-dimensional. The rational case needs an exact winding-parity condition. Microlocal transport and globally hyperbolic solution spaces supply further context, not a general counting formula.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[373], Problem 16. Title: "Spin-lifted null dynamics and a parity correction on Lorentzian tori."
• AIM workshop Conformal structure in geometry, analysis, and physics, supplied reference https://aimath.org/WWN/confstruct/confstruct.pdf. Original remarks and literature fields are empty; the extracted question leaves $h(g)$ undefined. This draft does not access a PDF.
• Supplied research attempt 1: flat-torus Fourier/parity criterion, a qualification of Witt's rational-slope table, spin transport context and free-Cauchy-data observation. No full Witt citation or authoritative references for the general propagation/Cauchy statements are supplied.
• Registry dated 14 August 2026: "partially_solved," "PARTIAL-PROGRESS," statement status "exact"; attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.