The premise already used by V2

The time variable in the V2 evolution equations is shared across sectors. The photon module already uses shared time. There is one infinite physical manifold, with one temporal direction and one nested chain of spatial submanifolds. A sector's depth counts only its spatial dimensions.

Thus a six-dimensional electron has six ordinary macroscopic spatial directions and uses the same time coordinate as three-dimensional matter and a ten-dimensional tau. Its six spatial directions are six of the tau's ten. Sharing three with an observer does not make those three different in kind, and no sector acquires a separate time dimension.

At a finite depth, a local evolution chart can be written

(t,qd),qd∈Qd,Qa⊂Qb(a<b). (t,q_d),\qquad q_d\in Q_d,\qquad Q_a\subset Q_b\quad(a<b).

The same tt appears in both charts. This notation records the framework's temporal premise; it does not construct an infinite-dimensional metric or Dirac operator. A global product with a time axis is a stronger geometric choice than merely using such local charts.

Shared coordinate time and elapsed time

One coordinate tt does not imply that every physical clock accumulates the same proper time. For example, in the finite-dimensional flat Lorentzian metric

ds2=−c2dt2+∑j=1d(dqj)2,dτ=dt1−∥q˙∥2c2, ds^2=-c^2dt^2+\sum_{j=1}^{d}(dq^j)^2, \qquad d\tau=dt\sqrt{1-\frac{\lVert\dot q\rVert^2}{c^2}},

timelike paths with different speeds have different elapsed proper times. This is a compatibility example using the usual metric definition of proper time, not a derivation of the V2 metric. See Carroll, General Relativity notes, §1, equation 1.97.

Likewise, one timelike direction alone does not forbid boosts along any particular spatial direction or prove the absence of closed timelike curves. Those questions need the actual metric, global causal assumptions, and a specification of physically accessible operations.

Finally, a block-diagonal metric need not be an exact product metric:

ds2=−c2dt2+a(t)2∑j(dqj)2 ds^2=-c^2dt^2+a(t)^2\sum_j(dq^j)^2

has no time–space cross terms, yet its spatial block depends on time. Consequently, neither exact operator separation nor ordinary observer Lorentz dynamics follows just from the common time coordinate.