e_113003
plain-language theorem explainer
At multi-index (1,1,3,0,0,3), the folded M2 numerator equals eight times the explicit Z table entry. Gravity analysts cite it as one kernel cell in the 4D Regge midpoint M2–TT identity. The proof is a single `decide` on concrete integers from the coupling fold and the Z lookup.
Claim. For indices $(a,b,c,d,i,j)=(1,1,3,0,0,3)$ 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 value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices appear. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer contrib and the accumulator starts at 0. The companion table $Z$ is an explicit case-split on the six indices, returning small integers such as $4$, $-2$, and so on.
The local module is chunk 5 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell is a concrete equality at one multi-index; the present declaration is the cell $(1,1,3,0,0,3)$.
Upstream, both maps live in the kernel certificate module: $m_2^{\mathrm{num}}$ as the fold over couplingZList, and $Z$ as the pattern-matched table explicitZ.
proof idea
One-line computational proof: decide. With all six indices ground in $\mathrm{Fin},4$, both sides reduce to concrete Int values (the fold of contrib versus eight times the matching explicitZ clause), and the kernel decides the equality.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full pointwise identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on the six indices. This cell is one of the 256 kernel decides that discharge those cases (chunk 5).
In the gravity stack, the identity is bookkeeping for the Regge midpoint M2–TT coupling in 4D: it certifies that the folded numerator matches the closed-form Z table, so later curvature and mass-squared identities can quote a single explicit integer rather than a fold. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel arithmetic supporting the continuum gravity layer.
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