e_330331
plain-language theorem explainer
Pointwise identity: the folded midpoint numerator at multi-index (3,3,0,3,3,1) equals eight times the explicit integer table entry. Gravity analysts cite it as one of 256 kernel cells assembling the 4D Regge midpoint M₂TT numerator identity. The proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,3,0,3,3,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(3,3,0,3,3,1)=8\,Z_{\mathrm{expl}}(3,3,0,3,3,1)$.
background
This module is chunk 15 of a 256-cell kernel certifying that the folded 4D Regge midpoint numerator agrees with an explicit closed-form integer table, up to the universal factor 8. The ambient setting is exact midpoint analysis of the M₂TT identity in discrete gravity.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: sum the local contribution of each coupling term at the six Fin-4 indices. The comparison object $Z_{\mathrm{expl}}$ is a hand-written integer table on $(\mathrm{Fin},4)^6$, with nonzero entries such as $4$ on matched pairs like $(0,0,1,1,2,2)$ and $-2$ on crossed pairs like $(0,0,1,2,1,2)$.
The full universal statement is assembled downstream by exhausting all six indices; each chunk discharges a block of concrete cells.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(3,3,0,3,3,1)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the fold $m_2^{\mathrm{num}}$ and the table $Z_{\mathrm{expl}}$.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for all six Fin-4 indices, which is the numerator half of the exact 4D Regge midpoint M₂TT identity. That identity is infrastructure for the discrete-gravity side of Recognition Science, where curvature and coupling bookkeeping must match closed forms before continuum or phenomenological limits are taken. The chunking (256 decides) keeps the kernel certificates small and independently checkable; this cell is one concrete brick in that wall.
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