e_231302
plain-language theorem explainer
For Fin-4 indices (2,3,1,3,0,2), the Regge midpoint mass-squared numerator equals eight times the explicit kernel integer. Gravity analysts proving the global 4D midpoint M2TT identity cite this as one cell of the 256-case kernel. The proof is a single native decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the midpoint mass-squared numerator at $(2,3,1,3,0,2)$ equals eight times the explicit kernel value at those same indices: $m_2^{\mathrm{num}}(2,3,1,3,0,2)=8\,Z(2,3,1,3,0,2)$.
background
This module is chunk 11 of a 256-cell kernel certifying that the folded midpoint numerator equals eight times an explicit integer table on every sextuple of Fin-4 indices. The setting is 4D Regge calculus at the exact midpoint configuration used in the M2TT identity analysis.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each term's contribution at those indices. The comparison table $Z(a,b,c,d,i,j)$ is an explicit pattern-matched integer function on six Fin-4 arguments (sample clauses give values such as $4$, $-2$, and so on). Both live in the kernel certificate module imported here.
The global claim is the pointwise identity $m_2^{\mathrm{num}}=8Z$ for all six indices; each chunk theorem discharges one concrete sextuple so the assembler can finish by exhaustive fin_cases.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int expressions (the fold for the numerator and the pattern match for the explicit table) and checks equality by native decision. No lemmas are invoked beyond the definitions of the two functions.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ and proves it by six nested fin_cases over Fin 4, dispatching each cell to a chunk theorem of this form. Without the full 256-cell cover, the midpoint M2TT numerator identity in 4D Regge analysis does not close. This is pure kernel bookkeeping inside the gravity analysis stack, not a forcing-chain (T0–T8) step; it underwrites the algebraic side of the exact midpoint mass-squared identity used downstream in the RS gravity development.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.