6.1. Three different arrows
A generative ordering says which operation precedes another. A record ordering says which results are available to later observers. A thermodynamic arrow concerns the behavior of entropy under specified physical conditions. These arrows may be related, but none is identical to the others by definition.
An oriented reversible rotation, for example, has a preferred sign for its generator while remaining invertible. A reduced record can lose access to correlations even though the full system evolves reversibly. An entropy increase additionally depends on the state, interactions, environmental conditions, and chosen coarse description. The familiar difficulty of connecting reversible microscopic laws with irreversible behavior therefore survives in a more precise form.
The published dual-time geometry studies coupled temporal parameters, connections, and reductions with memory [8]. These parameters serve a mathematical modeling role. They should not be identified without argument with all the nested inner levels described in the earlier books.