e_033332
plain-language theorem explainer
For the Fin-4 index tuple (0,3,3,3,3,2), the folded M2 coupling numerator equals eight times the explicit integer Z-table entry. Gravity analysts certifying the 4D Regge exact midpoint identity cite this as one of 256 kernel-decided cells. Proof is a single decide on concrete integer equality.
Claim. For indices $a=0$, $b=3$, $c=3$, $d=3$, $i=3$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint gravity analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $N$ is defined by folding a contribution function over a fixed coupling list, summing integer increments for each coupling term. The companion map $Z$ is an explicit pattern-matched table returning fixed integers (examples in the kernel cert include $4$, $-2$, and similar small values) on those same six indices.
The local module is chunk 3 of a 256-cell partition: each cell asserts $N=8Z$ at one concrete index sextuple. Upstream, both $N$ and $Z$ are supplied by the kernel-cert module; this chunk only evaluates them at fixed arguments.
proof idea
One-line kernel proof: decide. With all six arguments literal $\mathrm{Fin},4$ values, both sides reduce to concrete integers (the fold for $N$, the table clause for $Z$), and the decision procedure checks equality.
why it matters
Parent theorem is the assembled identity $\forall a,b,c,d,i,j,, N=8Z$, proved by nested fin_cases that dispatch each of the $4^6$ cells to a chunk lemma of this form. Without the cell equalities, the assemble step cannot close. The global numerator identity is part of the certified 4D Regge midpoint M2/TT stack in the gravity analysis layer; it is bookkeeping for the discrete curvature coupling, not a continuum Einstein equation by itself.
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