e_310031
plain-language theorem explainer
For the six Fin-4 indices (3,1,0,0,3,1), the midpoint Regge numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one cell of the 4D identity m2Num = 8·explicitZ. The proof is a single kernel decide on that concrete sextuple.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,0,3,1)$ 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, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at a sextuple of $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map $Z$ (explicitZ) is a sparse closed-form table on the same six indices, returning small integers such as $\pm 2,,4$ on the nonzero support and (implicitly) zero elsewhere.
The local module is chunk 13 of a 256-cell kernel certification: the claim $m_2^{\mathrm{num}}=8Z$ is discharged pointwise by decide on each concrete sextuple rather than by a single symbolic identity. The ambient goal is the exact midpoint $M_2$ TT identity in four dimensions, of which this equality is the numerator half.
proof idea
One-line proof: by decide. Lean reduces both sides at the fixed indices $(3,1,0,0,3,1)$—the fold defining $m_2^{\mathrm{num}}$ and the pattern-match defining $Z$—and checks integer equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on all six $\mathrm{Fin},4$ arguments, each case landing on a chunk theorem of this form. Without the pointwise cells, the universal identity (and thus the exact midpoint TT numerator identity in 4D Regge calculus) does not close. In the Recognition gravity stack this is pure algebraic certification infrastructure for the discrete curvature side, not a forcing-chain (T0–T8) step.
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