wickContinuationThresholds_differ
plain-language theorem explainer
The two causal Wick complexes in the common class (four-one and three-two pent types) have unequal Euclidean continuation thresholds. Anyone citing type-dependence of the kinematical Wick gate, or the open window between 3/8 and 7/12, needs this inequality. The proof rewrites each threshold to its closed form and finishes by numerical comparison.
Claim. The Wick continuation threshold of the four-one causal complex is not equal to that of the three-two causal complex: $\alpha_{\min}(\mathrm{fourOne}) \neq \alpha_{\min}(\mathrm{threeTwo})$. Equivalently, $3/8 \neq 7/12$.
background
Work item 5 (outcome b) answers a referee objection to the action-level Wick certificate: that certificate hardcodes the causal range $\alpha > 7/12$ on a fixed three-pent one-hinge complex of type three-two, which looks like a universal constant but is not. The module shows the kinematical Wick Euclidean-admission threshold is already type-dependent via alphaMin on causal 4-simplices.
The common class has two inhabitants: the four-one complex and the three-two complex. The continuation threshold of a complex $K$ is defined as $\alpha_{\min}$ of its pent type, the exact gate for $c_{m4} > 0$ after Wick Euclideanization. Upstream evaluations give $\alpha_{\min}(\mathrm{fourOne}) = 3/8$ and $\alpha_{\min}(\mathrm{threeTwo}) = 7/12$.
Thus $7/12$ is a complex-independent sufficient threshold (the max of the two type gates) but not a complex-independent exact gate. The open interval $(3/8, 7/12)$ is where the difference is visible.
proof idea
One-line algebraic comparison. Rewrite the left-hand side by the evaluation lemma that the four-one threshold equals $3/8$, and the right-hand side by the evaluation that the three-two threshold equals $7/12$. Then norm_num discharges $3/8 \neq 7/12$ over the reals. No case split or structural induction is required.
why it matters
This is the core inequality of outcome (b) in the Pillar 1 strengthen campaign: the two members of the common class have provably different continuation thresholds. It underwrites the strict comparison four-one $<$ three-two and the concrete window witness at $\alpha = 1/2$, where four-one continues for every positive spacelike scale while no WickActionContinuationCertV2 exists (its causal range is hardcoded as $7/12 < \alpha$).
Downstream, fourOne_only_window_witness uses the type-dependent gate to exhibit that window explicitly. In the broader Seven Gaps gravity stack this converts a scope caveat into a structural finding: the action-level hardcoded $7/12$ is the three-two member of the threshold function, not a universal constant. Outcome (a), action-level continuation for a genuine multi-complex family, remains a separate campaign.
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