e_220103
plain-language theorem explainer
For the six Fin-4 indices (2,2,0,1,0,3), the discrete midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,b{=}2,c{=}0,d{=}1,i{=}0,j{=}3$ in $\mathrm{Fin}\,4$, the folded midpoint 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, two integer-valued maps on six $\mathrm{Fin},4$ indices appear. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list and summing local contributions at those indices. The companion map $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold.
The local module is chunk 10 of a 256-cell decide grid: each cell fixes one concrete 6-tuple and asserts $m_2^{\mathrm{num}}=8Z$ at that point. The factor eight is the global normalization relating the folded sum to the tabulated kernel across the full identity.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six indices are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the case table for $Z$; the kernel checker confirms equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every 6-tuple by exhaustive fin_cases and dispatches each cell to a chunk lemma of this form. That global equality is the algebraic core of the exact midpoint M2TT identity in the 4D Regge gravity analysis: it certifies that the discrete numerator is exactly eight times the explicit kernel, with no residual terms on the finite index set.
Within Recognition Science gravity work this is bookkeeping infrastructure rather than a forcing-chain landmark (T0–T8). It closes one of the 256 decide obligations needed before continuum or continuum-limit statements can trust the midpoint kernel.
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