e_012111
plain-language theorem explainer
Kernel equality: the Regge midpoint mass-squared numerator at multi-index (0,1,2,1,1,1) equals eight times the explicit integer kernel Z at those indices. Gravity analysts assembling the 4D midpoint M2TT identity cite these 256 chunk lemmas. Proof is a single kernel decide on concrete Fin-4 data.
Claim. For indices $(a,b,c,d,i,j)=(0,1,2,1,1,1)$ in $\mathrm{Fin}\,4$, the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-case kernel certification that the folded numerator of the 4D Regge midpoint mass-squared form agrees with eight times a closed-form integer table. The ambient setting is discrete gravity analysis: midpoint evaluations of a Regge-type quadratic form on a 4-index combinatorial skeleton.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating integer contributions at the six $\mathrm{Fin},4$ slots. The comparison object $Z$ is an explicit case table $\mathrm{Fin},4^6\to\mathbb{Z}$ (sample entries include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$). The claim is the pointwise identity at one concrete multi-index.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(0,1,2,1,1,1)$. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table value of $Z$ by eight. Both sides are concrete integers, so the equality is discharged by computation with no further lemmas.
why it matters
Parent assembly theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$, proved by exhaustive fin_cases on all six indices. Each chunk lemma such as this one supplies one branch of that case split. Closing the full identity certifies that the midpoint Regge M2TT numerator is exactly eight times the explicit kernel, a computational step inside the 4D discrete-gravity analysis stack. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure integer kernel bookkeeping supporting the gravity side.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.