e_130330
plain-language theorem explainer
For the six-index tuple (1,3,0,3,3,0) on Fin 4, the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. 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 $(a,b,c,d,i,j)=(1,3,0,3,3,0)$ in $\mathrm{Fin}\,4$, the integer numerator $m_2(a,b,c,d,i,j)$ obtained by folding coupling contributions equals $8$ times the explicit kernel entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices appear. The numerator $m_2$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the given multi-index. The companion map explicitZ is a sparse lookup table of small integers (entries such as $4$, $-2$, and so on) that records the closed-form kernel.
The local module is chunk 7 of a 256-cell case split whose sole job is to certify $m_2=8\cdot Z$ pointwise. The ambient goal is an exact midpoint identity for the M2–TT sector of discrete gravity, reduced to finite arithmetic on Fin 4.
proof idea
One-line proof by decide. Both sides evaluate to concrete integers once the six Fin-4 arguments are substituted: the left-hand side runs the fold that defines m2Num, the right-hand side multiplies the matching explicitZ clause by 8. The kernel checks equality of the resulting integers.
why it matters
This cell is consumed by m2Num_eq_eight_explicitZ, which assembles the universal statement $\forall a,b,c,d,i,j,; m_2=8\cdot Z$ by exhaustive fin_cases on all six indices. That universal equality is the algebraic backbone of the exact midpoint M2–TT identity in 4D Regge calculus inside the Gravity analysis stack. It does not itself invoke the Recognition forcing chain (T5–T8) or the J-cost; it is pure discrete-kernel bookkeeping that later gravity identities rely on.
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