e_033033
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,3,3,0,3,3) equals eight times the explicit integer kernel table at that same index. Gravity analysts assembling the exact midpoint M2 TT identity in 4D cite these chunk lemmas. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For the multi-index $(0,3,3,0,3,3)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer table: $m_2^{\mathrm{num}}(0,3,3,0,3,3)=8\,Z_{\mathrm{expl}}(0,3,3,0,3,3)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2 TT identity certification. The local goal, stated in the module header, is to discharge $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on a block of the six-index kernel by pure decision.
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 $\mathrm{Fin},4$ slots. The comparison target $Z_{\mathrm{expl}}$ is an explicit integer-valued table on $(\mathrm{Fin},4)^6$, with sparse nonzero pattern (e.g. $4$ on matched pairs such as $(0,0,1,1,2,2)$, $-2$ on crossed pairs).
Both definitions live in the kernel certificate module imported here. The present lemma fixes one concrete six-tuple inside chunk 3.
proof idea
One-line computational proof: decide. After substituting the concrete indices $0,3,3,0,3,3$, both sides reduce to closed integer expressions (the fold of m2Num versus the pattern-match of explicitZ scaled by 8). Lean’s decision procedure checks integer equality; no algebraic rewriting or intermediate lemmas are required.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ by exhaustive fin_cases on the six indices. Each chunk lemma such as this one closes one leaf of that case tree.
In the Recognition gravity stack, the equality $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ is the certified algebraic core of the Regge exact-midpoint M2 TT identity in 4D. It converts a folded coupling sum into a sparse closed table, which downstream curvature and continuum-limit arguments consume. The result is pure discrete kernel arithmetic; it does not itself invoke the T0–T8 forcing chain, but it sits inside the gravity analysis that those foundations support.
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