e_010130
plain-language theorem explainer
For the multi-index (0,1,0,1,3,0), the discrete 4D midpoint numerator m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite this as one cell of the 256-case kernel identity that underwrites the exact Regge midpoint M2TT relation. The proof is a single kernel decide on both integer sides.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,1,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-way case split proving $m_2^{\mathrm{num}}=8\cdot Z$ on every 4D multi-index. The ambient setting is the exact midpoint identity for the Regge/TT discrete curvature kernel in four dimensions.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a local contribution at each coupling triple for the six $\mathrm{Fin},4$ indices. The comparison object $Z$ is an explicit integer-valued table on the same six indices (sample entries include $4$, $-2$, and other small integers on the diagonal and off-diagonal patterns).
The full quantified identity is assembled downstream by exhausting all $\mathrm{Fin},4$ cases; each chunk theorem such as this one discharges one concrete tuple.
proof idea
One-line proof by decide. Both sides are closed integer expressions once the six indices are fixed: the left side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right side looks up $8\cdot Z$ at $(0,1,0,1,3,0)$. Lean’s kernel decision procedure checks the two integers are equal. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ and $Z$ are required.
why it matters
This cell is consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by nested fin_cases over all six indices. That universal identity is the algebraic backbone of the exact 4D Regge midpoint M2TT kernel certificate: it replaces a summed coupling expression by a sparse explicit integer table, making later curvature and gravity identities computable by table lookup rather than by re-folding couplings.
In the broader Recognition gravity stack, such exact discrete identities sit under continuum limits and continuum matching arguments; they do not themselves invoke the T0–T8 forcing chain, but they keep the discrete geometric side free of numerical approximation.
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