e_200331
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (2,0,0,3,3,1) equals eight times the explicit integer table entry. Gravity analysts cite it as one of 256 kernel cells assembling the global m2Num = 8·explicitZ fact for the 4D Regge midpoint M2 TT identity. Proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,0,3,3,1)$ 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 table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the M2 TT identity is reduced to an integer equality between two six-index tables on $\mathbb{F}_4$. The numerator side $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the given multi-index. The closed form side is an explicit pattern-matched integer table $Z$ on the same six $\mathrm{Fin},4$ arguments (sample clauses include $Z(0,0,1,1,2,2)=4$ and several $-2$ off-diagonal patterns).
This module is chunk 8 of the 256 kernel decides that discharge $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration: every index lives in a four-element type, so each cell is a concrete integer equality.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed multi-index $(2,0,0,3,3,1)$. The left side runs the fold that defines the numerator; the right side looks up (or computes) $8$ times the explicit table entry. No lemmas are invoked beyond the definitions of the two tables.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on all six indices, each case landing on one chunk cell such as this one. That global identity is the certified algebraic core of the 4D Regge midpoint M2 TT analysis in the Gravity domain. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is infrastructure for the discrete curvature/mass side of the RS gravity stack.
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