e_010332
plain-language theorem explainer
For the six-index slot (0,1,0,3,3,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2–TT identity certification. The proof is a single kernel decide on two concrete integers.
Claim. For indices $a{=}0,b{=}1,c{=}0,d{=}3,i{=}3,j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table entry: $m_2^{\mathrm{num}}(0,1,0,3,3,2)=8\,Z_{\mathrm{expl}}(0,1,0,3,3,2)$.
background
This module is chunk 1 of a 256-cell kernel certification that the 4D Regge exact-midpoint M2 numerator agrees with an explicit integer table, pointwise on $\mathrm{Fin},4^6$. The local claim is the scalar identity $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ at one multi-index.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The companion $Z_{\mathrm{expl}}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.). Both live in the kernel-cert module imported here.
The surrounding gravity analysis treats these equalities as the discrete algebraic content of the midpoint M2–TT identity in four dimensions, before any continuum or continuum-limit statements.
proof idea
One-line proof: by decide. Lean reduces both sides at the concrete indices $(0,1,0,3,3,2)$ to integers (the fold of m2Num versus 8 * explicitZ) and discharges equality in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent assembly theorem m2Num_eq_eight_explicitZ quantifies over all six $\mathrm{Fin},4$ indices and proves $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ everywhere by nested fin_cases; each leaf is one of these e_* decides. This cell is the $(0,1,0,3,3,2)$ leaf in chunk 1 of that case split.
In the Recognition gravity stack the identity is bookkeeping for the exact midpoint Regge M2–TT relation in $D=4$, not a continuum Einstein equation. It does not itself invoke the forcing chain (T5 J-uniqueness, T8 $D=3$) or the mass ladder; those enter only if a later bridge maps this discrete identity into continuum RS gravity. Closing the full fin_cases tree removes the last scaffolding between the folded numerator and the explicit table.
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