e_212202
plain-language theorem explainer
For the six-index slot (2,1,2,2,0,2) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic decide cell in the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=1$, $c=2$, $d=2$, $i=0$, $j=2$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at six Fin-4 indices, and the fold starts from 0. The companion map $Z$ (explicitZ) is a sparse lookup table on the same six indices, returning small integers such as $4$, $-2$, and so on at the nonzero support.
The local module is chunk 9 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell fixes one concrete 6-tuple in $(\mathrm{Fin},4)^6$ so that decide can discharge the integer equality without quantifying over indices.
Upstream, $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel certificate module; no analytic hypotheses are carried into this cell.
proof idea
One-line tactic proof: decide. Lean evaluates both sides at the concrete indices $(2,1,2,2,0,2)$, reduces the fold that defines $m_2^{\mathrm{num}}$ and the pattern match that defines $Z$, and checks the resulting integer equality $m_2^{\mathrm{num}}=8Z$.
why it matters
This cell is consumed by the assembler theorem $m_2^{\mathrm{num}}=8Z$ for all six Fin-4 indices, which runs nested fin_cases and lands on the per-tuple decide lemmas (including this one). That global identity is part of the exact midpoint M2/TT certification in the 4D Regge gravity analysis.
Within Recognition Science gravity work, such kernel identities underwrite discrete curvature and coupling bookkeeping on the eight-tick, three-spatial-dimension scaffold (T7–T8). The chunk does not itself derive Einstein equations; it closes one arithmetic obligation in the midpoint expansion.
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