e_020230
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,2,0,2,3,0) equals eight times the explicit integer table entry at the same indices. Gravity analysts cite it as one kernel cell in the 4D midpoint M2 TT identity. The proof is a single kernel decision on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,2,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(0,2,0,2,3,0)=8\,Z_{\mathrm{expl}}(0,2,0,2,3,0)$.
background
This module is chunk 2 of a 256-cell kernel certifying that the folded numerator of the 4D Regge exact-midpoint M2 TT identity equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, i.e. four discrete directions.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The companion $Z_{\mathrm{expl}}$ is an explicit pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on).
The local claim is the equality at one concrete six-tuple. Sibling theorems cover the other tuples in the chunk; the assembly theorem quantifies over all of $(\mathrm{Fin},4)^6$.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete integers once the six indices are fixed literals, so the decidable equality on $\mathbb{Z}$ discharges the goal with no lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by exhaustive fin_cases on all six indices, invoking one cell theorem per tuple. That global identity is the algebraic core of the 4D Regge exact-midpoint M2 TT certification in the Gravity analysis stack: it replaces a folded sum by a sparse explicit table, enabling exact midpoint curvature identities used downstream in the discrete gravity sector. No forcing-chain (T0–T8) step is touched directly; the result is pure discrete-geometry bookkeeping inside the RS gravity layer.
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