e_020222
plain-language theorem explainer
For the discrete index sextuple (0,2,0,2,2,2), the 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 the two integer definitions.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,2,2,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(0,2,0,2,2,2)$ equals $8\,Z(0,2,0,2,2,2)$, where $Z$ is the explicit integer kernel on six $\mathbb{F}_4$ indices.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six indices in $\mathbb{F}_4$ are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums a local contribution at each coupling triple. The comparison target is the explicit kernel $Z$, a piecewise integer function on the same six indices (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
This module is chunk 2 of a 256-cell decide table that checks $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration over $\mathrm{Fin},4$, not continuum GR: each cell is an integer identity at one multi-index.
proof idea
One-line computational proof: decide evaluates both sides of the equality at the concrete sextuple $(0,2,0,2,2,2)$. The left side runs the fold that defines $m_2^{\mathrm{num}}$; the right side multiplies the matching clause of $Z$ by eight. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all $(a,b,c,d,i,j)\in(\mathbb{F}_4)^6$, which discharges the full index range by nested fin_cases and hits this cell among the chunk-2 siblings. That global identity is the algebraic core of the Regge exact-midpoint M2–TT certificate in 4D gravity analysis inside the monolith. It is bookkeeping infrastructure rather than a forcing-chain landmark (T5–T8), but without the pointwise cells the assembly cannot close.
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