e_120201
plain-language theorem explainer
For the Fin-4 sextuple (1,2,0,2,0,1), the folded coupling numerator equals eight times the explicit integer kernel value. Gravity analysts certifying the 4D Regge exact-midpoint M2 TT identity cite this as one decided case among the 256-point kernel. Proof is a single kernel decide on closed integer expressions.
Claim. For indices $a=1$, $b=2$, $c=0$, $d=2$, $i=0$, $j=1$ in $\{0,1,2,3\}$, 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 6 of a 256-case kernel establishing pointwise that the folded numerator equals eight times an explicit integer table, inside the 4D Regge exact-midpoint M2 TT identity analysis.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution map over a fixed coupling list, yielding an integer for every six-tuple of indices in $\mathrm{Fin},4$. The companion table $Z$ is a sparse pattern-matched function on the same six indices, returning small integers (e.g. $4$, $-2$) on the supported patterns and (implicitly) zero elsewhere.
The local goal is pure arithmetic bookkeeping: discharge one concrete sextuple so a later assembler can quote the full $\forall$ identity.
proof idea
One-line wrapper: the decide tactic. After substituting the six concrete $\mathrm{Fin},4$ values, both sides reduce to closed integer expressions via the definitions of the folded numerator and the explicit kernel table, so the kernel checks equality with no further lemmas.
why it matters
Feeds the assembler theorem that states $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for every six-tuple in $\mathrm{Fin},4$, proved there by exhaustive fin_cases. That $\forall$ identity is a certification step for the Regge exact-midpoint M2 TT identity in 4D gravity analysis. The chunk itself is discrete arithmetic support, not a new physical law; it closes one cell of the 256-case kernel named in the module doc.
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