e_311200
plain-language theorem explainer
For the six-index tuple (3,1,1,2,0,0) in Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel point checks. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,2,0,0)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit kernel integer: $m_2^{\mathrm{num}}(3,1,1,2,0,0)=8\,Z_{\mathrm{expl}}(3,1,1,2,0,0)$.
background
The ambient module is a chunk of the 4D Regge exact-midpoint M2–TT identity certification. The numerator m2Num is defined by folding a fixed coupling list: start at 0 and add each term's contribution at the six Fin-4 indices. The comparison target explicitZ is a piecewise integer table on those same six indices (typical values 4, -2, and so on for the nonzero patterns).
Local setting (module doc): prove m2Num = 8 · explicitZ pointwise, chunk 13 of the 256 kernel decides. The full universal statement is assembled downstream by exhausting all Fin-4 sextuples. This declaration is the single point $(3,1,1,2,0,0)$.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values (the fold of m2Num at this tuple versus eight times the matching explicitZ clause), and the decidable equality checker discharges the goal with no further lemmas.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by nested fin_cases over all six Fin-4 indices. Each leaf is one of these chunk theorems (siblings e_310000, …). Closing the pointwise identity is a bookkeeping step toward the exact midpoint M2–TT relation in the 4D Regge gravity analysis inside Recognition Science; it is pure discrete kernel arithmetic, not a continuum GR claim.
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