certV2_above_threeTwo_threshold
plain-language theorem explainer
Any action-level Wick continuation certificate forces the CDT ratio strictly above the three-two kinematical threshold 7/12. Gravity and CDT workers cite it when separating certificate scope from geometry. The proof is a one-line field projection after identifying the three-two threshold with alphaMin.
Claim. If $\alpha\in\mathbb{R}$ admits a version-2 action-level Wick continuation certificate (interior-hinge form), then $\alpha$ lies strictly above the Wick continuation threshold of the three-pent one-hinge complex, i.e. $\mathrm{thresh}(3{+}2)<\alpha$.
background
In causal dynamical triangulations, 4-simplices come in two causal types: four-one and three-two. After Wick rotation, Euclidean admission is gated by a type-dependent lower bound on the CDT ratio $\alpha$: $\alpha_{\min}(4{+}1)=3/8$ and $\alpha_{\min}(3{+}2)=7/12$. Those bounds are exact iff-gates for the post-Wick volume factor $c_{m4}>0$.
The module treats the action-level certificate WickActionContinuationCertV2 as specialized to a collapsed three-two one-hinge Möbius path. Its causalRange field hardcodes $\alpha>7/12$. The kinematical threshold function on complexes therefore coincides with $\alpha_{\min}$ on each type, so the three-two threshold is exactly $7/12$.
Local setting (Pillar 1 strengthen campaign): convert the referee objection that $7/12$ looks like a universal constant into a structural fact that thresholds are type-dependent, while $7/12$ remains a complex-independent sufficient bound as the max of the two type thresholds.
proof idea
One-line term proof. Rewrite the three-two continuation threshold via the identification lemma that equates it with $\alpha_{\min}$ of the three-two type (hence $7/12$). The goal becomes the certificate's own causalRange inequality, which is projected directly from the hypothesis.
why it matters
Feeds the action-level window theorems: no_certV2_in_fourOne_only_window (any $\alpha$ below the three-two threshold cannot carry a CertV2) and the concrete witness at $\alpha=1/2$, where four-one continues for every positive spacelike scale yet no certificate exists. Also sits under the broader claim that no single real is an exact gate for every causal type.
In the Recognition gravity stack this is the precise sense in which $7/12$ is a scope boundary of the certificate rather than of the geometry. It closes the honesty gap flagged in the module: the hardcoded constant is defensible as a sufficient condition and indefensible as a universal exact threshold. Outcome (a), genuine multi-complex action-level continuation, remains a separate campaign; this result only pins CertV2 above the three-two member of the threshold function.
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