e_221202
plain-language theorem explainer
Pointwise identity: the folded M2 numerator coupling at multi-index (2,2,1,2,0,2) equals eight times the explicit integer table value. Gravity analysts cite it as one of the 256 kernel decides that assemble into the universal m2Num = 8·explicitZ identity. The proof is a single kernel decide on two concrete Int expressions.
Claim. For indices $(a,b,c,d,i,j)=(2,2,1,2,0,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed-form entry: $m_2^{\mathrm{num}}(2,2,1,2,0,2)=8\,Z_{\mathrm{expl}}(2,2,1,2,0,2)$.
background
This module is chunk 10 of a 256-way case split establishing $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on the 4D Regge midpoint kernel. Indices run over $\mathrm{Fin},4$, i.e. the discrete 4-label set used for the exact midpoint TT identity in four dimensions.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The companion $Z_{\mathrm{expl}}$ is an explicit integer-valued table on the same six indices (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local claim is the single lattice point $(2,2,1,2,0,2)$ of that table identity. Sibling theorems cover the other points in the same chunk.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed, so the decidable equality procedure on Int discharges the goal with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and is proved by exhaustive fin_cases on all six indices. Each case lands on a chunk theorem of this form (or an equivalent decide).
In the gravity analysis stack this identity certifies that the folded numerator coupling on the 4D Regge exact-midpoint kernel matches the closed-form integer table up to the universal factor 8. That match is a computational backbone for the midpoint TT identity used in the discrete gravity sector; without the pointwise decides the assembly cannot close.
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