e_000303
plain-language theorem explainer
Atomic kernel equality: the midpoint M2 numerator at multi-index (0,0,0,3,0,3) equals eight times the explicit Z-table value there. Gravity analysts cite it only as one cell of the 4D Regge midpoint identity. The proof is a single decide on concrete Fin-4 indices and integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(0,0,0,3,0,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 0 of a 256-cell kernel certifying the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on $\mathrm{Fin},4^6$, in the 4D Regge exact-midpoint TT analysis.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at 0. The companion $Z$ is an explicit integer-valued table on the same six $\mathrm{Fin},4$ arguments (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
The local goal is not the universal statement, only the single sextuple $(0,0,0,3,0,3)$.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table lookup by 8. Both land on the same integer, so the equality is discharged by the kernel decision procedure with no lemmas.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every index cell into the universal identity
$\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$
by exhaustive fin_cases on the six $\mathrm{Fin},4$ arguments. This declaration is the cell for $(0,0,0,3,0,3)$ inside chunk 0 of that kernel.
In the broader gravity stack the identity is bookkeeping for the exact midpoint M2 TT contraction in 4D Regge calculus: it converts a folded coupling sum into a sparse explicit table, which later feeds curvature and continuum-limit arguments. It does not itself touch the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete-gravity algebra.
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