e_033222
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (0,3,3,2,2,2) equals eight times the explicit integer table at those indices. Gravity analysts cite it when assembling the full Regge midpoint M2 numerator identity over (Fin 4)^6. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(0,3,3,2,2,2)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $m_2^{\mathrm{num}}(0,3,3,2,2,2)=8\,Z_{\mathrm{expl}}(0,3,3,2,2,2)$.
background
This module is chunk 3 of a 256-case kernel certification that the Regge-exact midpoint M2 numerator equals eight times an explicit integer table on every sextuple of indices in $\mathbb{F}_4$.
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 contribution term at those six indices. The explicit table $Z_{\mathrm{expl}}$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and other small integers on distinguished index patterns).
The local claim is one concrete sextuple in that exhaustive grid: indices $(0,3,3,2,2,2)$.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete integers once the fold for the numerator and the pattern match for the explicit table are evaluated at the fixed indices $(0,3,3,2,2,2)$, so the equality is a closed numeric check.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and discharges the quantifier by fin_cases on all six indices. Each chunk lemma such as this one supplies one grid point so the global numerator identity is a pure finite check rather than an open algebraic obligation.
In the gravity analysis stack this certifies the midpoint M2 TT identity kernel in 4D Regge calculus bookkeeping. It is infrastructure for the exact discrete curvature side, not a continuum GR theorem and not a forcing-chain (T0–T8) step.
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