e_301002
plain-language theorem explainer
At multi-index (3,0,1,0,0,2), the folded M2 numerator coupling equals eight times the explicit integer Z-table entry. Gravity analysts cite it as one of 256 kernel cells in the Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,0,1,0,0,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer lookup table.
background
In the 4D Regge midpoint analysis, the M2 numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: sum the local contributions of each coupling term at a sextuple of $\mathrm{Fin},4$ indices. The companion table $Z$ is an explicit piecewise integer function on the same index domain (sample values include $4$, $-2$, and so on at listed patterns).
The module is chunk 12 of a 256-cell kernel certificate that $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell pins one concrete sextuple so the global identity can be assembled by exhaustive case split on $\mathrm{Fin},4$.
Upstream, $m_2^{\mathrm{num}}$ is the fold over couplingZList; $Z$ is the closed-form integer table used as the target of the identity.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(3,0,1,0,0,2)$ and checks integer equality. No lemmas are invoked beyond the definitions of the fold and the explicit table.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all $\mathrm{Fin},4$ sextuples, which case-splits on every index and discharges each cell by a chunk theorem of this form. That global equality is part of the certified algebraic core of the Regge exact midpoint M2–TT identity in 4D, a gravity-side support lemma in the Recognition Science stack (discrete curvature / continuum matching bookkeeping). It does not itself touch T5–T8 or the RCL; it is infrastructure for the discrete gravity side.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.