e_201010
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at multi-index (2,0,1,0,1,0) equals eight times the explicit integer kernel at those indices. Gravity analysts cite it as one cell of the 256-case kernel that assembles m2Num = 8·explicitZ. The proof is a single kernel decide on concrete Int arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,0,1,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(2,0,1,0,1,0)=8\,Z_{\mathrm{expl}}(2,0,1,0,1,0)$.
background
This module is chunk 8 of a 256-cell decide grid proving $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on every 6-tuple of $\mathrm{Fin},4$ indices. The setting is the exact midpoint identity for the 4D Regge/TT mass-squared numerator in the Gravity analysis stack.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing the local contribution of each coupling term at those indices. The companion $Z_{\mathrm{expl}}$ is an explicit integer-valued kernel on the same six $\mathrm{Fin},4$ arguments (pattern-matched cases such as $(0,0,1,1,2,2)\mapsto 4$ and mixed sign entries like $-2$).
The full quantified statement is assembled downstream by exhausting all $4^6$ index combinations; each chunk theorem discharges one concrete cell.
proof idea
One-line kernel proof: by decide. Both sides reduce to closed integer expressions once the six indices are literals, so the decidable equality procedure on Int closes the goal with no lemmas or case splits inside this declaration.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall(a,b,c,d,i,j:\mathrm{Fin},4),, m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by six nested fin_cases over the index space. That global identity is the certified bridge from the folded coupling definition to the explicit kernel used in the Regge exact-midpoint M2/TT analysis.
Within Recognition Science gravity work, these kernel cells underwrite the discrete curvature/mass-squared bookkeeping on the 4D lattice side of the forcing chain (spatial $D=3$ plus the discrete tick structure). The chunk exists only to keep the 256 decides modular and machine-checkable.
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