Correlated length rates

For local flow time $t=\tau -{\tau }_a$, write

Chosen units make the rate parameter dimensionless. It is not identified with a Lorentz rapidity between face frames.

For particle slice advance $s=s_a+v_{s}t$, $v_s\ge 0$, incoming active finite lengths have zero rates. In the outgoing epoch,

and $c_i,t_i,o_i$ are frozen.

For example, on $e_i=[B_i,A_i]$,

Hence

and its rate is the first row of eq:particle-rates. On $u_i=[P,B_i]$, both endpoints have time $2s$, so $\ell (u_i)=Bs$. The shrinking and growing rates are correlated by the same vertices and parameter; they are not independently assigned.

For $\sigma ={\sigma }_a+v_{A}t$, $v_A\ge 0$, the four-face central rates are

on their respective epochs. The arm rates are

Cap-side rates are $2v_A/3,2v_A/5$, corner-spoke rates are $-v_A\sqrt{34}/15$, and outer rectangle sides are frozen. Additional ambient rates are the declared laws of contract:A.