e_000102
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (0,0,0,1,0,2) equals eight times the explicit integer kernel at those same indices. Gravity analysts cite it as one cell of the 256-case kernel that assembles m2Num = 8·explicitZ on (Fin 4)^6. Proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(0,0,0,1,0,2)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit integer kernel value: $m_2^{\mathrm{num}}(0,0,0,1,0,2)=8\,Z_{\mathrm{expl}}(0,0,0,1,0,2)$.
background
In the Regge exact-midpoint 4D analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at zero. The companion map $Z_{\mathrm{expl}}$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and defaults).
The module is chunk 0 of a 256-cell decide kernel whose sole job is to certify $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at every concrete multi-index. The present declaration is the cell for $(0,0,0,1,0,2)$.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (the left by evaluating the fold of contributions at fixed indices; the right by evaluating the explicit table and multiplying by eight), and the kernel checks equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{expl}}(a,b,c,d,i,j)$ and discharges the universal claim by exhaustive fin_cases on the six Fin-4 indices. Each chunk cell such as this one supplies a decide-closed ground instance the case split lands on.
In the broader gravity stack this identity is bookkeeping for the exact midpoint mass-squared / TT kernel in 4D Regge analysis: once numerator and explicit table match up to the universal factor eight, downstream curvature and propagator identities can quote a single closed form rather than a folded sum. It is pure finite combinatorial certification, not a continuum GR derivation.
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