e_312100
plain-language theorem explainer
Pointwise identity: the folded M2 numerator coupling at multi-index (3,1,2,1,0,0) equals eight times the explicit Z-table value there. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite it as one of the 256 kernel cases. Proof is a single `decide` after unfolding the fold and the table.
Claim. For the multi-index $(a,b,c,d,i,j)=(3,1,2,1,0,0)$ in $(\mathrm{Fin}\,4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the M2 numerator is defined by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and $m_2^{\mathrm{num}}$ is their sum. The companion object is an explicit integer table $Z$ on the same six-index domain, given by a finite pattern-match (e.g. diagonal-type slots map to $4$, mixed slots to $-2$, and so on).
The local module is chunk 13 of a 256-way case split whose sole job is to certify $m_2^{\mathrm{num}}=8\cdot Z$ at every multi-index. Upstream, both the fold definition and the table live in the kernel-certificate module; this chunk only evaluates one concrete point.
proof idea
One-line kernel decision: unfold the fold that defines the numerator and the pattern-match that defines the explicit table at $(3,1,2,1,0,0)$, reduce both sides to concrete integers, and let decide close the equality. No lemmas beyond definitional reduction are required.
why it matters
Feeds the assembler theorem that states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$. That assembler runs fin_cases on all six indices and dispatches each cell to a chunk lemma of this form; without the cell equalities the universal identity does not close.
In the broader gravity stack this identity is bookkeeping for the Regge midpoint M2–TT certificate in 4D: it replaces a summed coupling expression by a sparse explicit table scaled by eight, which later steps treat as a closed algebraic object. It does not itself touch the RS forcing chain (T5–T8) or the J-cost; it is pure discrete tensor arithmetic supporting the continuum limit argument.
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