The modular statement being compared

For a von Neumann algebra $\mathcal{M}$ and a cyclic separating vector $\Omega$, Tomita–Takesaki theory supplies modular unitaries ${{\Delta }_{\mathcal{M}}}^{it}$. A half-sided modular inclusion is an inclusion $\mathcal{N}\subseteq \mathcal{M}$ with a specified standard vector and a one-sided modular invariance condition. Under its precise hypotheses, this structure yields an associated one-parameter group and positivity properties Wiesbrock,ArakiZsido.

Wedge modular geometry has established QFT examples and hypotheses BisognanoWichmann,BGL. The Family 282 record reports a further use of these structures in its scalar reconstruction. That use is not verified here.

A modular group or positive null-cut generator is not yet a channel from a declared logical code to a declared accessible output. It does not select an environment, receiver, or deterministic quantum-return map.