wickContinuationThreshold_fourOne
plain-language theorem explainer
The kinematical Wick Euclidean-admission threshold on the four-one causal 4-simplex equals exactly 3/8. Anyone comparing type-dependent continuation gates in the SevenGaps Wick family cites this evaluation. The proof is a one-line simp unfolding the threshold definition through the four-one complex tag onto the already-proved alphaMin value.
Claim. The Wick continuation threshold of the four-one causal Wick complex equals $3/8$: if $K$ is the complex whose pent type is four-one, then $\alpha_{\min}(K) = 3/8$.
background
This module answers a referee objection to the action-level Wick certificate: the hardcoded causal range $\alpha > 7/12$ is not a complex-independent constant. It is the three-two member of a type-dependent threshold function already present at the kinematical level.
A CausalWickComplex packages a causal pent type (four-one or three-two). The threshold function on that class is defined by wickContinuationThreshold K := alphaMin K.ty, the exact gate for non-degeneracy of the Wick-rotated 4-volume factor $c_{m4} > 0$. Upstream, alphaMin_fourOne records alphaMin fourOne = 3/8 by reflexivity on the definition of alphaMin.
The companion three-two value is $7/12$. The open interval $(3/8, 7/12)$ is therefore the window where four-one continues and three-two does not.
proof idea
One-line wrapper. Unfold wickContinuationThreshold to alphaMin of the complex's type, unfold fourOneComplex to the four-one tag, and discharge by alphaMin_fourOne (itself rfl). No arithmetic beyond the definitional equality $3/8$.
why it matters
This evaluation is the four-one half of outcome (b) in the Pillar 1 strengthen campaign: continuation thresholds are provably type-dependent. Downstream, wickContinuationThresholds_differ and wickContinuationThreshold_fourOne_lt_threeTwo rewrite both sides and compare $3/8$ with $7/12$ by norm_num. The gap witness wickThreshold_gap_witness and the concrete window witness at $\alpha = 1/2$ (fourOne_only_window_witness) both need the four-one side equal to $3/8$ so that $1/2$ sits strictly above it.
Framework role: it converts the CertV2 hardcoded $7/12$ from an apparent universal into the maximum of two exact gates, defending it only as a sufficient (not exact) complex-independent bound. Action-level multi-complex continuation remains a separate campaign; this lemma stays at the kinematical $c_{m4}$ gate.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.