e_310030
plain-language theorem explainer
For the six-index tuple (3,1,0,0,3,0) on Fin 4, the folded midpoint M2 numerator equals eight times the explicit integer kernel Z. Gravity analysts cite it as one atomic case in the 256-point kernel identity that underwrites the 4D Regge midpoint TT check. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=0$, $i=3$, $j=0$ in $\mathrm{Fin}\,4$, the summed 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
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer weight depending on the six indices, and the fold starts at zero. The comparison target is an explicit piecewise integer kernel $Z$, tabulated by pattern (e.g. diagonal-type slots map to $4$, certain off-diagonal slots to $-2$).
The local module is chunk 13 of a 256-case kernel certification: each chunk theorem pins one concrete multi-index equality $m_2^{\mathrm{num}}=8Z$. The factor eight is the global normalization relating the folded coupling sum to the closed-form kernel used in the midpoint TT identity.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted into the fold definition of the numerator and the pattern-match definition of the explicit kernel; the kernel then checks integer equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple by exhaustive fin_cases on each index. That universal equality is the certified bridge between the folded coupling definition and the explicit kernel table in the 4D Regge midpoint M2 TT analysis. Without each chunk (including this one) the assembly cannot close. It is pure discrete linear-algebra bookkeeping inside the gravity sector, not a continuum GR claim.
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