e_010233
plain-language theorem explainer
For the six-index tuple (0,1,0,2,3,3) on Fin 4, the summed midpoint coupling numerator equals eight times the explicit kernel value. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0$, $b=1$, $c=0$, $d=2$, $i=3$, $j=3$ in $\mathrm{Fin}\,4$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample values include $4$, $-2$, and so on for distinguished patterns).
The module is chunk 1 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each chunk theorem pins one concrete sextuple; the present case is $(0,1,0,2,3,3)$. Downstream assembly will reassemble these decides into the universal identity over all indices.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed Fin-4 sextuple $(0,1,0,2,3,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table.
why it matters
Feeds the universal statement that the midpoint numerator equals eight times the explicit kernel for every six-tuple of Fin-4 indices. That parent theorem proceeds by exhaustive fin_cases on all six arguments and dispatches each leaf to a chunk decide of this form.
In the Recognition gravity stack this is bookkeeping for the exact 4D Regge midpoint M2/TT identity: once every kernel cell is certified, the algebraic identity used in the continuum or discrete curvature analysis is free of residual case obligations. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure finite combinatorial support for the gravity analysis layer.
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