e_020011
plain-language theorem explainer
For the Fin-4 multi-index (0,2,0,0,1,1), the folded coupling numerator m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2 TT identity. The proof is a single kernel decide on two concrete integers.
Claim. For indices $a=0$, $b=2$, $c=0$, $d=0$, $i=1$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of a 256-cell kernel certification that the 4D Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times an explicit integer table $Z$ on $\mathrm{Fin},4^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each term's contribution at those six indices. The companion table $\mathrm{explicitZ}$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on for distinguished index patterns).
The local goal is pointwise equality $m_2^{\mathrm{num}}=8\cdot Z$ at every sextuple; each chunk theorem discharges one concrete cell so the assembler can finish by exhaustive fin_cases.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values once the six indices are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern match for $Z$, so the kernel closes the equality with no further lemmas.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases, each leaf a chunk theorem of this form. That global identity is the algebraic core of the 4D Regge exact-midpoint M2 TT certification in the Gravity analysis stack. It is bookkeeping support for the discrete curvature side of Recognition gravity, not a forcing-chain (T0–T8) step; it closes one cell of the 256-decide kernel so the midpoint identity can be cited without residual case splits.
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