boundary_threeTwo_spacelike
plain-language theorem explainer
On the closed Wick arc [0,1], the split-form cosine of the unique spacelike hinge of the (3,2) causal 4-simplex is continuous, equals −11/8 at the Lorentzian endpoint (on the arccos cut), and equals −1/4 at the Euclidean endpoint. Anyone assembling the B2 all-hinge Wick certificate or the pent-hinge path continuity cites this. The proof reduces the path to the cut-free rational (5−6z)/(6z−2) along zArc and evaluates the endpoints by direct arithmetic.
Claim. Let $\gamma_{3,4}$ be the split-form cosine path of the spacelike hinge of the $(3,2)$ causal 4-simplex (opposite vertex pair $\{3,4\}$) along the canonical upper-half-plane Wick arc $z(t)$. Then $\gamma_{3,4}$ is continuous on the closed interval $[0,1]$, $\gamma_{3,4}(0)=-(11/8)$ in $\mathbb{C}$, and $\gamma_{3,4}(1)=-(1/4)$ in $\mathbb{C}$.
background
Lane B2 of the QG Seven-Gaps campaign treats complex-first Wick continuation of every triangular hinge of the threeTwo causal 4-simplex at the physical point $a=1$, $\alpha=1$. Hinges are classified by opposite vertex pairs: one spacelike hinge (pair $(3,4)$, triangle $(0,1,2)$), six mixed hinges, and three upper-pair hinges. For the spacelike class the closed cofactors are $C_{pp}=C_{qq}=6z-2$, $C_{pq}=5-6z$, with area squared $3/16$.
Along the arc the split cosine collapses to the cut-free rational $(5-6z)/(6z-2)$. At the Lorentzian endpoint $t=0$ this value is $-(11/8)$, which lies on the arccos branch cut ($\mathrm{Im}=0$, $|\mathrm{Re}|\ge 1$): the classical Lorentzian boost angle at that hinge. Endpoint contact is allowed by the executed gate; branch regularity is stated only on the open interior. The Euclidean endpoint is the regular-4-simplex cosine $-(1/4)$.
The path is the restriction of that rational function to the canonical arc $z(t)$ of the complex-first Wick package, with denominator nonvanishing on $[0,1]$.
proof idea
Split the conjunction into three goals.
Continuity: exhibit the auxiliary map $t\mapsto(5-6z(t))/(6z(t)-2)$ as a quotient of continuous functions (constant and $z$-arc pieces) whose denominator never vanishes (denom_ne). Pass to ContinuousOn on $[0,1]$, then rewrite pointwise via the identity that the spacelike cosine path equals this rational on the whole arc.
Endpoint $t=0$: rewrite by the same identity and $z(0)$ evaluation, then norm_num yields $-(11/8)$.
Endpoint $t=1$: rewrite by the identity and $z(1)$ evaluation, then norm_num yields $-(1/4)$.
No cofactor square-root analysis is needed: class A is the rational collapse case.
why it matters
This is the class-A boundary certificate for the unique spacelike hinge of threeTwo. The B2 headline wick_continuation_threeTwo_hinges quantifies over every opposite pair and demands closed-interval continuity of every cosine path ending at the Euclidean value $-(1/4)$; the spacelike case is discharged here, with the Lorentzian cut contact disclosed rather than hidden.
Downstream, continuousOn_pentHingeCosPath_Ioc_one identifies the pent-hinge path of index 1 with this spacelike path and inherits continuity on $(0,1]$ by restriction of the closed-interval statement. The sibling product-form kill certificates (upper-pair class) sit in the same module and complete the negative half of the executed Wick-arc trace.
In the Seven-Gaps gravity finishing charter this closes the honest endpoint bookkeeping for the only hinge whose Lorentzian value sits on the arccos cut, so the all-hinge Wick package can claim closed continuity without pretending the cut is avoided at $t=0$.
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