e_201321
plain-language theorem explainer
For the six-index tuple (2,0,1,3,2,1) on Fin 4, the folded Regge midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one cell of the 4D M2–TT identity table. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2,b=0,c=1,d=3,i=2,j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint 4D analysis, the numerator $m_2$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the six Fin-4 indices. The companion map $Z$ is an explicit integer-valued kernel on the same six indices, given by a finite pattern match (e.g. selected pairs evaluate to $4$, $-2$, and so on).
The local module is chunk 8 of a 256-cell decide table whose sole job is to certify $m_2=8Z$ pointwise. Upstream, both $m_2$ and $Z$ live in the kernel certificate module; this chunk only evaluates one concrete multi-index.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted into the fold definition of $m_2$ and the pattern match for $Z$, so the kernel closes the equality with no further lemmas.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,, m_2=8Z$ by exhaustive fin_cases on all six indices; each case invokes one chunk equality such as this one. The identity is the algebraic backbone of the Regge midpoint M2–TT certificate in the gravity analysis stack. It does not itself touch the T0–T8 forcing chain, but it is part of the discrete geometric bookkeeping that supports RS gravity numerics.
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