wickContinuationThreshold_threeTwo
plain-language theorem explainer
The kinematical Wick Euclidean-admission threshold on the three-pent one-hinge complex equals exactly 7/12. Anyone citing the action-level certificate's hardcoded causal range, or comparing typewise gates in the fourOne/threeTwo class, uses this identification. The proof is a one-line simplification through the threshold definition and the already-proved alphaMin value on threeTwo.
Claim. The Wick continuation threshold of the three-pent one-hinge causal complex equals $7/12$ as a real number: $\mathrm{thresh}(C_{3+2}) = 7/12$.
background
Work item 5 (outcome b) answers a referee objection: the action-level Wick certificate hardcodes the causal range $\alpha > 7/12$ on one fixed three-pent one-hinge complex, which looks like a universal constant but is not. The module shows that the kinematical Wick Euclidean-admission gate for causal 4-simplices is already type-dependent via alphaMin on causal pent types.
Two exact gates are in play: $\mathrm{alphaMin}(\mathrm{fourOne}) = 3/8$ and $\mathrm{alphaMin}(\mathrm{threeTwo}) = 7/12$. These are iff conditions for the post-Wick Cayley-Menger quantity $c_{m4} > 0$. The Wick continuation threshold of a complex is defined to be that type's alphaMin, so evaluating it on the threeTwo complex is pure arithmetic identification.
Upstream, alphaMin_threeTwo is the reflexivity fact $\mathrm{alphaMin}(\mathrm{threeTwo}) = 7/12$. The present theorem simply transports that equality through the threshold wrapper on the named threeTwo complex inhabitant.
proof idea
One-line simp only through three definitions: the threshold function (which projects a complex to its type's alphaMin), the threeTwo complex inhabitant, and the upstream theorem alphaMin_threeTwo that pins that minimum at $7/12$. No arithmetic or case split remains after unfolding.
why it matters
This is the threeTwo half of outcome (b): thresholds are complex-dependent, and the certificate's hardcoded $7/12$ is exactly the threeTwo gate, not a universal exact threshold. Downstream, hardcodedConstant_eq_threeTwo_threshold is its symmetric form; wickContinuationThresholds_differ and wickContinuationThreshold_fourOne_lt_threeTwo compare it to the fourOne value $3/8$; universal_sufficient_threshold_eq_max elevates $\max(3/8,7/12)=7/12$ to a complex-independent sufficient bound; and certV2_above_threeTwo_threshold shows every action-level CertV2 lives strictly above this gate via its causalRange field.
In the Seven Gaps gravity campaign this converts a scope caveat into a structural finding: $7/12$ is defensible as a sufficient condition and indefensible as a typewise exact gate. The open window $(3/8,7/12)$ is where fourOne continues kinematically while no CertV2 exists. Outcome (a), genuine multi-complex action-level continuation, remains a separate campaign.
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