tStar
plain-language theorem explainer
Defines the exact interior arc parameter $t_\star = 1 - \arccos(1/3)/\pi \approx 0.608$ at which the product of diagonal Cayley–Menger cofactors hits the complex square-root branch cut. Anyone citing the C11 Wick hinge-data crossing or the forced split-denominator form uses this constant. It is a closed-form real definition, not a proved existence claim.
Claim. Let $t_\star := 1 - \arccos(1/3)/\pi \in \mathbb{R}$. Numerically $t_\star \approx 0.6081734480$. This is the canonical interior parameter on the upper-half-plane Wick arc at which the product-form cofactor argument crosses the negative real axis.
background
Module C11 formalizes complex-first 4D Wick continuation of 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 canonical 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 3D dihedral denominator is a single square root of a product of diagonal cofactors. On the 4D arc that product can leave the slit plane, so the formalization uses a split square-root denominator. The numeric gate (RESULTS.txt §3) isolates one interior “canonical offender” parameter where the product equals $-32$ exactly and therefore sits on the branch cut.
$t_\star$ is that parameter, written in closed form from the cosine identity that forces $\operatorname{Re} z = 1/3$ on the unit-circle arc.
proof idea
Pure definition: unfold to $1 - \arccos(1/3)/\pi$. No tactics or lemmas. Downstream lemmas recover the geometric content by algebra: $\pi(1-t_\star)=\arccos(1/3)$ and $\cos(\pi(1-t_\star))=1/3$, which pin $z(t_\star)$ and the cofactor product identity.
why it matters
This constant is the named crossing locus for the C11 gate. Downstream theorems arg_tStar, cos_arg_tStar, and product_form_crossing_value show that at $t_\star$ one has $z=1/3+i\sin(\arccos(1/3))$ and $(6z-2)^2=-32$ exactly. product_form_crossing packages the FAIL event: $t_\star\in(0,1)$, the product of diagonal cofactors equals $-32$, and $-32\notin\mathrm{slitPlane}$, killing any literal single-sqrt transcription of the 3D formula and forcing the split denominator.
The module is hinge-data continuation only; the FullTheoryLedger gap on genuine action-level 4D continuation (C12) stays open. $t_\star$ is the exact algebraic witness that the branch cut is hit in the open arc interior, not merely at an endpoint.
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