e_200312
plain-language theorem explainer
Pointwise identity: the Regge midpoint coupling numerator at multi-index (2,0,0,3,1,2) equals eight times the explicit integer table at those indices. Gravity analysts cite it when assembling the global m2Num = 8·explicitZ certificate over (Fin 4)^6. Proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,0,0,3,1,2)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(2,0,0,3,1,2)=8\,Z_{\mathrm{expl}}(2,0,0,3,1,2)$.
background
In the 4D Regge exact-midpoint M2 TT analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each term's contribution at those indices. The comparison target $Z_{\mathrm{expl}}$ is an explicit case-table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
The module is chunk 8 of a 256-way split of the kernel: each chunk discharges a block of concrete index tuples so the global identity can be assembled without a single monolithic decide. Local setting: prove $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ pointwise on this block.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(2,0,0,3,1,2)$ and checks integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (list fold of contributions) and $Z_{\mathrm{expl}}$ (explicit table) are required; the kernel reduces the closed arithmetic directly.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{expl}}(a,b,c,d,i,j)$ by exhausting Fin-4 cases. That global certificate is part of the Regge exact-midpoint M2 TT identity chain in the gravity analysis layer: it replaces a summed coupling expression by a sparse explicit integer kernel, enabling exact algebraic identities downstream rather than numeric checks.
Within Recognition Science gravity work this is bookkeeping infrastructure, not a forcing-chain landmark (T5–T8). It closes one of 256 decide obligations so the midpoint identity can be cited as a proved Lean theorem rather than scaffolding.
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