e_132003
plain-language theorem explainer
For the six Fin-4 indices (1,3,2,0,0,3), the midpoint Regge numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D TT-identity table. The proof is a pure decide on the closed integer expressions.
Claim. For indices $a=1$, $b=3$, $c=2$, $d=0$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling 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 $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at zero. The comparison target is an explicit piecewise integer function $Z$ on the same six indices, tabulated by pattern (e.g. diagonal blocks yield $4$, certain off-diagonal pairings yield $-2$).
The module is chunk 7 of a 256-cell decide grid that exhausts all index sextuples. The local claim is the scalar identity $m_2^{\mathrm{num}}=8Z$ at one fixed sextuple. Upstream definitions supply both sides: the fold for the numerator and the case table for $Z$.
proof idea
One-line computational proof: decide evaluates both closed integer sides at the concrete indices $(1,3,2,0,0,3)$ and checks equality. No lemmas are invoked beyond the definitions of the numerator fold and the explicit kernel table; the kernel certificate module already reduces both expressions to ground integers.
why it matters
Feeds the universal assembly theorem $m_2^{\mathrm{num}}=8Z$ for every sextuple in $\mathrm{Fin},4$, which is proved by nested fin_cases that dispatch to these chunk cells. That identity is the algebraic core of the exact midpoint TT kernel certificate in 4D Regge gravity analysis: it converts a summed coupling expansion into a sparse explicit table, enabling later continuum or continuum-limit comparisons. Within Recognition Science gravity work it is bookkeeping, not a forcing-chain step, but it closes a finite decide obligation on the discrete curvature side.
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