e_201303
plain-language theorem explainer
For the six-index slot (2,0,1,3,0,3) on Fin 4, the folded Regge coupling numerator equals eight times the tabulated explicit kernel entry. Gravity analysts cite it as one atomic case in the 4D midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. At multi-index $(a,b,c,d,i,j)=(2,0,1,3,0,3)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the given indices. The comparison target is explicitZ, a sparse case table of small integers (entries such as $4$, $-2$, and so on) that packages the closed-form kernel of the M2 TT identity.
This module is chunk 8 of that comparison: it discharges a block of the $4^6$ concrete index tuples by kernel decision, under the local claim that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The present declaration is the single tuple $(2,0,1,3,0,3)$.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side by evaluating the fold that defines the numerator, the right-hand side by looking up the matching clause of the explicit kernel table (or the default zero). Lean’s kernel checks integer equality; no lemmas are invoked.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in Fin 4, proved by exhaustive fin_cases on all six indices. Each atomic equality such as this one is a leaf of that case split. Closing the full pointwise match certifies that the folded Regge coupling numerator is exactly eight times the explicit midpoint kernel, which is the algebraic content of the 4D M2 TT identity in this analysis stack. It is bookkeeping inside the gravity side of the Recognition framework, not a forcing-chain step (T0–T8), but it hardens the discrete curvature identities used downstream.
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