e_022233
plain-language theorem explainer
For the six-index slot (0,2,2,2,3,3) on Fin 4, the folded coupling numerator equals eight times the explicit kernel value. Gravity analysts cite it when auditing the 4D Regge midpoint M2 TT identity, one cell at a time. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator at $(0,2,2,2,3,3)$ equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,2,2,2,3,3)=8\,Z_{\mathrm{ex}}(0,2,2,2,3,3)$.
background
The ambient module is a chunk of the 4D Regge-exact midpoint analysis for the M2 TT identity. The claim under audit is the pointwise equality between a folded numerator and an explicit integer table, scaled by eight: $m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$ on all six-tuples in $(\mathrm{Fin},4)^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. $Z_{\mathrm{ex}}$ is the matching explicit integer function on the same six indices (sample values include $4$, $-2$, and other small integers on the diagonal and off-diagonal patterns). Chunk 2 packages 256 such kernel decides so the full identity can be assembled without a single giant proof term.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(0,2,2,2,3,3)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the folded numerator and the explicit kernel table.
why it matters
This cell is one of the chunk-2 witnesses feeding the universal statement that $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{ex}}(a,b,c,d,i,j)$ for every sextuple in $(\mathrm{Fin},4)^6$. That parent theorem is proved by exhaustive fin_cases on the six indices and lands in the Regge midpoint M2 TT identity pipeline for 4D gravity. In the Recognition Science gravity stack, such exact midpoint identities certify discrete curvature bookkeeping before continuum limits and phenomenological mass or coupling extractions. The declaration itself is pure bookkeeping: it closes one kernel cell so the assemble step can quote a complete cover.
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