e_022020
plain-language theorem explainer
Pointwise check that the Regge midpoint mass-squared numerator equals eight times the explicit integer kernel at multi-index (0,2,2,0,2,0). Gravity analysts cite it as one cell of the 256-case kernel that assembles the global identity. The proof is a single native decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,0,2,0)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of a 256-cell kernel certifying the algebraic identity $m_2^{\mathrm{num}}=8,Z$ that appears in the exact midpoint analysis of 4D Regge gravity. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution for each coupling term at the six indices. The comparison object $Z$ is an explicitly tabulated integer function on $(\mathbb{F}_4)^6$, with nonzero values only on a sparse set of index patterns (e.g. $4$ or $-2$ on selected pairs).
The local goal is not a continuum statement: it is exact integer equality at one concrete multi-index, so that a later assembly theorem can recombine all cells by exhaustive case split.
proof idea
One-line computational proof: decide. Both sides reduce to closed integer expressions once the six Fin-4 arguments are substituted, so the kernel decision procedure discharges equality with no lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and proves it by six nested fin_cases over $\mathbb{F}_4$, invoking one cell theorem per tuple. Without the full 256-cell cover, the midpoint mass-squared identity in the 4D Regge analysis remains uncertified at the integer level. This is pure discrete kernel bookkeeping inside the gravity analysis stack; it does not itself invoke the Recognition forcing chain (T5–T8) or the RCL, but it underwrites the exact algebraic step those continuum claims rely on when they specialize to the midpoint scheme.
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