e_020220
plain-language theorem explainer
For the six-index tuple (0,2,0,2,2,0) on Fin 4, the folded coupling numerator equals eight times the explicit kernel value at those indices. Gravity analysts cite it as one cell of the 4D Regge midpoint M2–TT identity table. The proof is a single kernel decision on concrete integer arithmetic.
Claim. At multi-index $(0,2,0,2,2,0)$ with each argument in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,2,0,2,2,0)=8\,Z_{\mathrm{explicit}}(0,2,0,2,2,0)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on $(\mathrm{Fin},4)^6$ are compared pointwise. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ folds a fixed coupling list, summing each contribution at those indices. The companion $Z_{\mathrm{explicit}}$ is a closed-form case table on the same six indices (sample values include $4$ and $-2$).
This module is chunk 2 of a 256-way split of the discrete identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$. Each chunk discharges a block of concrete tuples by kernel decision so the global assembly can finish by exhaustive case split on $\mathrm{Fin},4$.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,2,0,2,2,0)$ and checks integer equality. No intermediate lemmas are needed beyond the definitions of the fold-based numerator and the explicit case table; the kernel reduces the fold and the match to concrete numerals.
why it matters
Feeds the assembly theorem that states $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{explicit}}(a,b,c,d,i,j)$ for every six-tuple in $(\mathrm{Fin},4)^6$, proved by nested fin_cases. That global equality is the certified algebraic core of the 4D Regge midpoint M2–TT identity in the Gravity analysis stack.
The 256-decide chunking keeps each certificate small while covering the full discrete domain required downstream. Within Recognition Science gravity work this is pure combinatorial kernel certification; continuum or phi-ladder mass bookkeeping sits further out in the stack.
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