lorentzianSectorStatus
plain-language theorem explainer
Status ledger for the 3D Lorentzian sector in the QG Seven-Gaps campaign. It records that CDT causal tetrahedron classes and the kinematical Wick map are certified, Euclidean non-degeneracy holds on exact alpha ranges, and action-level Lorentzian continuation stays open. Gap6 receipts and terminal guards cite these four Bool flags. The body is a pure structure instance with literal true/true/true/true assignments.
Claim. The Lorentzian-sector status record is the four-flag tuple: causal tetrahedron classes defined $=\mathsf{true}$; Wick map certified on that class $=\mathsf{true}$; Euclidean non-degeneracy proved $=\mathsf{true}$; Lorentzian action-level continuation open $=\mathsf{true}$.
background
This module opens the Lorentzian lane of the discrete-gravity Seven-Gaps campaign. All prior formal and numerical gravity results in the stack were Euclidean. Here the setting is 3D CDT (Ambjorn–Jurkiewicz–Loll): spatial slices are equilateral triangulations with squared edge length $a^2$, and the slab between slices is filled by type $(3,1)$ tetrahedra (3 spacelike + 3 timelike edges) and type $(2,2)$ tetrahedra (2 spacelike + 4 timelike).
Timelike squared lengths are $-\alpha a^2$ with $\alpha>0$ in the Lorentzian regime. The kinematical Wick map flips the sign of those lengths, i.e. the algebraic continuation $\alpha\mapsto -\alpha$ on the causal class, and is proved an involution. Euclideanized simplices are non-degenerate for $\alpha>1/3$ on type $(3,1)$ and $\alpha>1/2$ on type $(2,2)$, with a deficit-angle reality corollary at the physical point $\alpha=1$.
LorentzianSectorStatus packages four Bool flags summarizing what this module closes versus what it leaves open: class definition, Wick certification, Euclidean non-degeneracy, and action-level continuation (complex dihedral angles at timelike hinges, sinh/boost Regge sector, analytic continuation of the action).
proof idea
Definitional structure instance, not a proof. Each field of LorentzianSectorStatus is assigned a Boolean literal: three closed flags set to true, and lorentzian_action_continuation_open set to true to mark the remaining gap. No lemmas are invoked; the companion theorem lorentzianSectorStatus_flags re-reads the four fields by rfl.
why it matters
This is the machine-readable receipt for the 3D Lorentzian-sector lane. Downstream, lorentzianSectorStatus_flags freezes the four values by reflexivity. Gap6LedgerTerminalGuard requires the action-continuation bit to stay open while the kinematical Wick bit is certified, separating 3D kinematics from the 4D action closer. LorentzianContinuation3DNotAction4DCertificate uses the same open bit plus edge-arity mismatch ($\mathrm{Fin},6\neq\mathrm{Fin},10$) to certify that 3D edge-tuple algebra is not a 4D action ledger. wickActionInteriorHingeStatus_flags likewise keeps the 3D action bit open after Gap6 flipped via the 4D V2 closer.
In the Recognition forcing chain this sits under $D=3$ spatial dimensions (T8) and the eight-tick discrete geometry program: it is the first certified Lorentzian layer on causal simplices, still short of a full Lorentzian Regge action.
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