e_202000
plain-language theorem explainer
Case check that the Regge midpoint mass-squared numerator at multi-index (2,0,2,0,0,0) equals eight times the explicit integer kernel Z there. Gravity analysts cite it as one of the 256 concrete kernel identities. The proof is a single kernel decision on fully instantiated integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,2,0,0,0)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module sits in the 4D Regge midpoint analysis of the mass-squared TT identity. The local goal, per the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ on one chunk of the $4^6=4096$ index space by 256 kernel decides.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at the six $\mathbb{F}_4$ indices. The comparison target $Z$ is an explicit integer-valued kernel on the same six indices, tabulated by pattern (e.g. diagonal blocks map to $4$, certain off-diagonal patterns to $-2$).
Both objects live in the kernel-certificate module imported here; this chunk only evaluates them at concrete points.
proof idea
One-line computational proof: after the six indices are fixed to $(2,0,2,0,0,0)$, both sides reduce to closed integer expressions, and decide discharges the equality in the integer ring. No algebraic lemmas are invoked beyond the definitions of the folded numerator and the explicit kernel table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathbb{F}_4)^6$ and proves it by exhaustive fin_cases on the six indices. Each chunk theorem such as this one supplies (or matches) one concrete case in that case split.
In the broader gravity stack this is bookkeeping for the exact midpoint Regge TT mass-squared identity in 4D: the factor of eight relates the summed coupling expansion to the closed-form kernel used downstream. It does not itself touch the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete-kernel certification inside the gravity analysis layer.
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