e_220130
plain-language theorem explainer
Pointwise check that the midpoint Regge m2-numerator equals eight times the explicit integer Z-kernel at multi-index (2,2,0,1,3,0). Gravity analysts cite the assembled universal identity; this entry is one of 256 kernel discharges in chunk 10. The proof is a single decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,2,0,1,3,0)$ with each coordinate 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
In the 4D Regge midpoint analysis the TT-sector identity is reduced to an equality between two integer-valued kernels on six indices in $\mathrm{Fin},4$. The numerator side $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The comparison side is an explicitly tabulated integer function $Z$ on the same six-tuple domain (nonzero only on a sparse set of index patterns).
The module is chunk 10 of a 256-way case split: every concrete six-tuple is discharged separately so the universal statement can be reassembled by exhaustive fin_cases. Local setting: prove $m_2^{\mathrm{num}}=8\cdot Z$ pointwise by kernel decide.
proof idea
One-line computational discharge: both sides evaluate to concrete integers at the fixed indices $(2,2,0,1,3,0)$, and decide checks equality in Int. No lemmas beyond the definitions of the folded numerator and the explicit $Z$ table are required.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, invoking one chunk theorem per cell. That universal identity is the algebraic core of the exact midpoint M2 TT identity in 4D Regge calculus inside the Gravity analysis stack. Without each pointwise kernel fact the case split cannot close. Landmark link is structural (discrete curvature / TT projection bookkeeping), not a T0–T8 forcing step.
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