e_033111
plain-language theorem explainer
For the discrete index tuple (0,3,3,1,1,1), the midpoint Regge numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global 4D midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}3$, $c{=}3$, $d{=}1$, $i{=}1$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions at each multi-index. The comparison target is an explicit piecewise integer table $Z$ (values such as $4$, $-2$, and so on on selected index patterns).
The module is chunk 3 of a 256-case kernel certification: each pointwise equality $m_2^{\mathrm{num}}=8Z$ is discharged separately so the global identity can be assembled by exhaustive case split on the six indices. The local setting is pure discrete algebra over $\mathbb{Z}$, not continuum curvature.
proof idea
One-line decide proof. Both sides reduce to concrete integers once the six Fin 4 arguments are substituted: the fold defining $m_2^{\mathrm{num}}$ evaluates, the pattern match defining $Z$ evaluates, and Lean checks $n=8m$ by kernel computation. No lemmas beyond the two definitions are invoked.
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 fin_cases on all six indices and dispatch to these chunk lemmas. That global identity is the algebraic core of the exact midpoint M2–TT relation in the 4D Regge analysis used on the gravity side of the Recognition stack. It is bookkeeping infrastructure rather than a forcing-chain step (T0–T8), but without the pointwise certificates the continuum-facing curvature identities cannot be grounded in the discrete kernel.
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