e_132201
plain-language theorem explainer
Pointwise kernel check: the folded M2 numerator at multi-index (1,3,2,2,0,1) equals eight times the explicit Z-table value there. Gravity analysts cite it when assembling the exact 4D Regge midpoint M2TT identity. The proof is a single native decide on concrete Fin-4 integers.
Claim. At the multi-index $(a,b,c,d,i,j)=(1,3,2,2,0,1)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on $(\mathrm{Fin},4)^6$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a local contribution at each list entry for the six indices. The comparison target is an explicit sparse table $Z$ on the same six indices, with only a handful of nonzero integer patterns (e.g. $4$, $-2$).
The module is chunk 7 of a 256-way kernel split: each chunk discharges a batch of concrete index tuples so the universal identity can be assembled by exhaustive case split. The local claim is exactly one such tuple.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed. The tactic decide runs the kernel decision procedure on that closed integer equality and closes the goal; no algebraic rewriting is required.
why it matters
The parent theorem is the universal statement that $m_2^{\mathrm{num}}=8Z$ for every six-tuple in $(\mathrm{Fin},4)^6$. That parent is proved by nested fin_cases on all six indices; each leaf is one of these chunk equalities. Closing the leaves is what turns the explicit table into a certified closed form for the folded numerator in the exact 4D Regge midpoint M2TT identity. Within Recognition gravity, this is bookkeeping infrastructure for the discrete curvature/mass-squared side, not a new dynamical law.
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