e_310203
plain-language theorem explainer
For the six Fin-4 indices (3,1,0,2,0,3), the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=2$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map $Z$ is an explicit piecewise integer table on the same six indices (sample values include $4$, $-2$, and so on for distinguished index patterns).
The local module is chunk 13 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ holds pointwise. The ambient goal is an exact midpoint identity for the M2 tensor structure in discrete gravity, reduced to finite integer arithmetic on $\mathrm{Fin},4$.
proof idea
One-line decide proof. Both sides are closed integer expressions once the six concrete indices are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the piecewise table for $Z$, so the kernel evaluates the equality directly.
why it matters
Feeds the universal statement $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ for all six $\mathrm{Fin},4$ indices, assembled by exhaustive fin_cases over the 256 cells. That identity is the algebraic backbone of the Regge exact-midpoint M2/TT certification in the gravity analysis stack. It is bookkeeping rather than a forcing-chain landmark (T0–T8), but it locks the discrete curvature numerator to a sparse explicit kernel needed for downstream continuum-limit and continuum-matching arguments.
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