e_122101
plain-language theorem explainer
At multi-index (1,2,2,1,0,1) the folded numerator coupling equals eight times the explicit integer Z-table entry. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of 256 kernel point checks. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,2,1,0,1)$ in $(\mathbb{F}_4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator coupling $m_2^{\mathrm{num}}$ is the fold of a fixed contribution list over a coupling table: it sums every local contribution at the given multi-index. The explicit table $Z$ is a sparse pattern-matched integer function on the same domain (nonzero only on a short list of index patterns, with values such as $\pm 2,,4$).
The local module is chunk 6 of a 256-way kernel split: each chunk discharges a block of concrete sextuples so that the global identity $m_2^{\mathrm{num}}=8Z$ can be assembled by exhaustive fin_cases. The factor 8 is the structural multiplicity relating the folded sum to the closed-form table.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(1,2,2,1,0,1)$ and checks integer equality. No lemmas are invoked; the kernel reduces the fold defining $m_2^{\mathrm{num}}$ and the pattern match defining $Z$ to concrete Int values and confirms $m_2^{\mathrm{num}}=8Z$ at that point.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j),,m_2^{\mathrm{num}}=8Z$ by ranging over all six $\mathrm{Fin},4$ indices and dispatching each concrete case to a chunk theorem of this form. Without the full 256-point cover, the closed-form replacement of the folded numerator by the explicit $Z$ table fails, and downstream Regge midpoint M2–TT identities cannot substitute the sparse table for the sum. This is pure gravity-analysis scaffolding inside the 4D kernel certification path, not a forcing-chain (T0–T8) step.
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