e_321120
plain-language theorem explainer
For the six-tuple of Fin-4 indices (3,2,1,1,2,0), the folded coupling numerator equals eight times the explicit integer kernel value. Gravity analysts assembling the pointwise 4D Regge midpoint M2TT identity cite this as one decided case among the 256 kernel checks. The proof is a single computational decide on concrete integers.
Claim. For indices $a=3$, $b=2$, $c=1$, $d=1$, $i=2$, $j=0$ ranging in $\mathrm{Fin}\,4$, 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 exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib map at a six-tuple of $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion explicit kernel $Z$ is a sparse pattern match on those same six indices, returning closed integers such as $4$ or $-2$ on the supported patterns and (implicitly) handling the rest by the match.
This module is chunk 14 of the 256 kernel decides that establish $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. Both definitions live in the KernelCert import; the local setting is purely the finite enumeration of index tuples needed before assembly.
proof idea
One-line computational proof. After unfolding the definitions of the folded numerator and the explicit kernel at the concrete indices $(3,2,1,1,2,0)$, both sides reduce to closed integers in $\mathbb{Z}$; decide checks the equality $m_2^{\mathrm{num}}=8Z$ by kernel evaluation. No intermediate lemmas are invoked.
why it matters
This case is one brick in the exhaustive assembly theorem that states $\forall(a,b,c,d,i,j:\mathrm{Fin},4),; m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$. That parent result proceeds by nested fin_cases on all six indices and discharges each leaf by a chunk equality of this form. The global identity is part of the Regge midpoint M2TT analysis in the gravity sector; it certifies that the folded coupling numerator is exactly eight times the explicit kernel on the entire finite index space, which is the algebraic content needed before any continuum or physical reading of the midpoint identity.
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