e_212110
plain-language theorem explainer
At multi-index (2,1,2,1,1,0) the folded 4D Regge midpoint numerator equals eight times the explicit integer table. Gravity analysts cite this as one of the 256 kernel point checks that assemble the global m2Num = 8·Z identity. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,2,1,1,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the TT-sector mass-squared numerator is defined by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums integer contributions over that list. The companion table $Z$ is an explicit pattern-matched map $(\mathrm{Fin},4)^6\to\mathbb{Z}$ that records the closed-form value at each multi-index (e.g. $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$).
The local module is chunk 9 of a 256-way partition of the finite kernel: each chunk theorem asserts $m_2^{\mathrm{num}}=8Z$ at one concrete six-tuple. The factor 8 is the combinatorial normalisation that matches the folded sum to the table.
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel certificate module; this declaration only evaluates them at $(2,1,2,1,1,0)$.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (the fold over the coupling list on the left, the pattern match on the right), so the equality is a decidable integer comparison with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over $(\mathrm{Fin},4)^6$. Each chunk such as this one discharges one leaf of that case split.
In the Recognition gravity stack this identity certifies that the midpoint Regge TT numerator agrees with its closed form on the full finite index set, a prerequisite for exact mass-squared identities in 4D. It is pure discrete algebra: no continuum limit, no forcing-chain step (T0–T8), and no appeal to $J$ or $\varphi$.
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