e_311203
plain-language theorem explainer
For the six-index tuple (3,1,1,2,0,3) on Fin 4, the folded M2 numerator equals eight times the explicit kernel value Z. Gravity analysts cite it as one atomic cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=1$, $d=2$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the M2 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, the M2 numerator at six Fin-4 indices is defined by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at zero. The companion object is an explicit integer kernel $Z$ on the same six indices, given by a finite case table (e.g. selected pairs evaluate to $4$, $-2$, and so on).
The local module is chunk 13 of the exhaustive check that the numerator equals eight times that kernel at every index. The ambient goal is a certified algebraic identity between the assembled M2 numerator and the explicit $Z$ table used in the midpoint TT analysis.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(3,1,1,2,0,3)$ and checks integer equality. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the looked-up explicitZ entry by eight. No lemmas beyond the two definitions are required.
why it matters
This cell is one of the finite cases discharged inside m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices by nested fin_cases. That assembly theorem is the certified bridge from the folded numerator definition to the explicit kernel table in the 4D Regge midpoint M2–TT identity stack. Without each chunk equality, the global ∀ statement does not close. The result is pure discrete gravity bookkeeping inside Recognition Science’s geometric analysis layer; it does not itself invoke the T0–T8 forcing chain or the J-cost calculus.
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