arg_tStar

Module C11 formalizes the complex-first 4D Lorentzian Wick continuation of Regge hinge data (complex Cayley–Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path on the timelike squared edge is the upper-half-plane arc $z(t) = \alpha a^2 \exp(i\pi(1-t))$ for $t\in[0,1]$, with Lorentzian endpoint at $t=0$ and Euclidean endpoint at $t=1$.

The scalar $t_\star$ is the exact interior crossing parameter of the product form on this hinge: $t_\star = 1 - \arccos(1/3)/\pi \approx 0.6081734480$ (RESULTS.txt §3). It marks where the arc argument hits the geometric angle whose cosine is $1/3$, the canonical offender in the split-sqrt denominator analysis.

This lemma simply recovers the phase $\pi(1-t_\star)$ from that definition, so downstream cosine evaluations can invoke $\cos\circ\arccos$ on the closed interval $[-1,1]$.

Unfold $t_\star$. The left-hand side becomes $\pi\bigl(1 - (1 - \arccos(1/3)/\pi)\bigr)$. A ring identity collapses the nested subtraction to $\arccos(1/3)/\pi$. Commute the product and cancel the nonzero factor $\pi$ via div_mul_cancel₀ and Real.pi_ne_zero. Pure algebraic cancellation; no analysis beyond the definition.

Feeds cos_arg_tStar, which states $\cos(\pi(1-t_\star)) = 1/3$ and is the exact crossing value used for the arc point $z = 1/3 + i\sin(\arccos(1/3))$ and the product-form denominator argument on the open arc interior.

In the QG Seven-Gaps C11 lane this pins the branch certificate for the PATH-SELECTED boundary continuation of hinge data. It does not close the FullTheoryLedger gap wick_action_continuation_4d (action-level continuation remains open under C12). Within Recognition Science gravity work it is a local exact identity on the Wick phase, not a forcing-chain (T0–T8) step.