diffMaskB
plain-language theorem explainer
For each oriented triangle slot in a Kuhn 4-simplex, the second difference mask is the bitwise XOR of the middle and top vertex masks of that triangle. Combinatorial gravity and Regge-calculus workers cite it when classifying hinge orbits by popcount pairs and when building slot area covariances. The body is a two-line unpack of the ordered triple of vertex masks followed by XOR.
Claim. Given a Kuhn simplex index $s\in\{0,\ldots,23\}$ and a triangle slot $t\in\{0,\ldots,9\}$ with ordered vertex masks $(m_0,m_1,m_2)$, the second difference mask is $b:=m_2\oplus m_1\in\mathbb{N}$.
background
This module classifies triangle hinges in one unit 4-cube Kuhn triangulation up to lattice translation and triangulation-preserving symmetry. It is a combinatorial prerequisite for assembling the flat Hessian of the 4D Regge action from per-orbit star kernels; it does not evaluate those kernels or complete the Hessian.
Each of the 24 Kuhn simplices carries vertex masks (bitsets on four coordinate axes). A triangle slot is an ordered index triple inside a simplex; the upstream helper returns the three corresponding masks $(m_0,m_1,m_2)$ as a monotone chain. Difference masks are the successive XORs $a=m_1\oplus m_0$ and $b=m_2\oplus m_1$. Realizable pairs are disjoint and nonzero, and the hinge type is the popcount pair $(|a|,|b|)$ in ${(1,1),(1,2),(2,1),(1,3),(3,1),(2,2)}$.
The sibling first-difference mask is defined analogously as $a=m_1\oplus m_0$. Together they feed orbit typing, complement symmetry, and later Bloch-fold area tables.
proof idea
Definitional unpack only. Bind the ordered triple of vertex masks for simplex $s$ and slot $t$, then return the natural-number XOR of the third component with the second. No lemmas, tactics, or case splits.
why it matters
Difference masks are the translation-invariant coordinates of a triangle hinge. Downstream Bloch-fold code builds the slot area covariance as a 15-class table supported exactly on the two difference masks (each weight $1/4$), proves that support claim, and forms the integer $\times 4$ version. Transport permutations of slots are read off the pair $(a,b)$, and phased class-dot vanishing theorems for types $(1,2)$ and $(2,1)$ take hypotheses that the second mask equals 2 or 1 respectively.
In the module deliverable, every index-triple triangle is a monotone mask chain with disjoint nonzero differences; cell counts and $S_4$ (plus complement) orbit classification are stated in those popcount pairs. This definition is the second half of that pair and is therefore on the critical path from Kuhn combinatorics to flat-kernel assembly. It does not itself close gap-action recovery or Einstein-Hilbert convergence of the RS action.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.