e_200321
plain-language theorem explainer
For the six-index tuple (2,0,0,3,2,1) on Fin 4, the folded Regge coupling numerator equals eight times the explicit kernel integer. Gravity analysts cite it as one concrete cell of the 4D midpoint M2TT identity. The proof is a single kernel decide on two closed integer expressions.
Claim. On indices $(a,b,c,d,i,j)=(2,0,0,3,2,1)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(2,0,0,3,2,1)=8\,Z(2,0,0,3,2,1)$.
background
This module is chunk 8 of a 256-way split of the kernel identity $m_2^{\mathrm{num}}=8\cdot Z$ for the 4D Regge exact-midpoint M2TT analysis. Indices run in $\mathbb{F}_4$ (four values each), so the full statement is a finite table of integer equalities.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at those six indices. The comparison side $Z$ is an explicit six-argument integer table (pattern-matched on the Fin 4 coordinates), recording the closed-form kernel entries used in the midpoint identity.
Local setting (module doc): prove $m_2^{\mathrm{num}}=8\cdot Z$ cell-by-cell via kernel decides, then reassemble.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values (the fold over the coupling list on the left; the pattern-match table on the right), so the kernel closes the equality with no lemmas or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):\mathbb{F}_4^6$ by exhausting Fin cases and discharging each cell. That global equality is the certified numerator form of the 4D Regge exact-midpoint M2TT kernel identity used in the gravity analysis stack.
Within Recognition Science gravity work, these kernel certificates lock the discrete curvature/coupling bookkeeping before continuum or phenomenological claims. This chunk is pure finite arithmetic; it does not itself invoke the forcing chain (T5–T8) or the J-cost, but it is infrastructure those geometric layers sit on once the Regge midpoint calculus is wired in.
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