e_021211
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,2,1,2,1,1) equals eight times the explicit integer table at that same point. Gravity analysts cite it as one cell of the 4D midpoint M2–TT kernel certification. The proof is a single kernel decide on concrete Int values.
Claim. For the multi-index $(0,2,1,2,1,1)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit table entry: $m_2^{\mathrm{num}}(0,2,1,2,1,1)=8\,Z_{\mathrm{expl}}(0,2,1,2,1,1)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six indices in $\mathbb{F}4$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a local contribution at each list entry for the given sextuple. The companion map $Z{\mathrm{expl}}$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on on selected index patterns).
The module is chunk 2 of a 256-cell kernel certification whose global claim is $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at every multi-index. Each chunk theorem pins one concrete sextuple so the assembly proof can finish by exhaustive fin_cases rather than symbolic algebra on the fold.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (the left via the fold definition of the numerator, the right via the pattern table and a factor of eight), and the decidable equality checker discharges the goal with no further lemmas.
why it matters
This cell is consumed by the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathbb{F}_4$ and proves it by six nested fin_cases over the finite index set, invoking one pointwise decide per cell. Closing the numerator-versus-table match is a bookkeeping step inside the broader Regge midpoint M2–TT identity certification in the gravity analysis stack. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure discrete kernel arithmetic supporting the continuum gravity side.
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