CausalSimplex4DStatus
plain-language theorem explainer
Status record for the 4D Lorentzian causal-simplex module: four boolean flags tracking whether the CDT 4-simplex classes are defined, the exact Cayley-Menger thresholds certified, Lorentzian cm4 negativity proved, and the action-level (Regge boost/sinh) continuation still open. Downstream status instances fill the flags. Pure structure definition with no proof content.
Claim. A four-field boolean status record for the 4D Lorentzian lift: (i) whether the causal 4-simplex classes are defined; (ii) whether the exact Cayley-Menger thresholds $\alpha > 3/8$ (type $(4,1)$) and $\alpha > 7/12$ (type $(3,2)$) are certified; (iii) whether Lorentzian tuples satisfy strict $cm_4 < 0$; (iv) whether the action-level continuation (complex dihedral angles at timelike hinges) remains open.
background
This module is Phase 3a of the QG Seven-Gaps Lorentzian-sector lane: the 4D lift of the kernel-checked 3D causal-simplex machinery. Spatial slices are equilateral tetrahedra of squared edge length $a^2$. Between slices one fills with two CDT 4-simplex types: $(4,1)$ (six spacelike + four timelike edges) and $(3,2)$ (four spacelike + six timelike). Timelike squared lengths are $-\alpha a^2$ with $\alpha > 0$; Wick rotation is the algebraic continuation $\alpha \mapsto -\alpha$.
The 4-simplex Cayley-Menger determinant $cm_4$ is the bordered $6\times 6$ determinant from the dimension-parametric geometry library. Euclidean non-degeneracy requires $cm_4 > 0$ above exact thresholds ($\alpha > 3/8$ for $(4,1)$, $\alpha > 7/12$ for $(3,2)$), with degeneracy exactly at threshold; Lorentzian tuples are proved to have strict $cm_4 < 0$.
Sibling status patterns in the campaign (discrete Lichnerowicz operator convergence, RS-native units) use the same boolean-flag schema to separate proved kinematical layers from deliberately open dynamical continuations.
proof idea
No proof: this is a structure declaration. Four Bool fields name the module's deliverable checklist. The concrete assignment lives in the downstream value causalSimplex4DStatus, which sets the first three flags true and leaves the action-level flag true to mark wave 3b as open.
why it matters
Gives the module a machine-readable claim surface for the 4D kinematical Wick rotation. Downstream, the single status instance records that causal classes, exact $cm_4$ thresholds, and Lorentzian negativity are closed, while the boost/sinh sector of the 4d Regge action (complex dihedral angles at timelike triangular hinges) stays open by design.
In the Seven-Gaps campaign this seals Phase 3a kinematics before action-level continuation. It sits beside other gravity-lane status objects (discrete Lichnerowicz TT-axis convergence proved, curved backgrounds open) so auditors can see what is certified versus deferred. No direct T0-T8 forcing step; the link is the discrete Lorentzian geometry needed for continuum gravity recovery.
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