e_200103
plain-language theorem explainer
For the six-index slot (2,0,0,1,0,3) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single decide on concrete integer arithmetic.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}0$, $d{=}1$, $i{=}0$, $j{=}3$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at six Fin-4 indices, and the fold starts from 0. The companion map explicitZ is a closed-form integer table on the same six indices (nonzero only on a sparse set of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and sign-flipped off-diagonal pairs $\mapsto -2$).
The local module is chunk 8 of a 256-way case split: the claim is the single equality $m_2^{\mathrm{num}}=8\cdot Z$ at one concrete multi-index. The ambient goal is the universal identity over all of $(\mathrm{Fin},4)^6$, assembled later by exhaustive fin_cases.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are literals, so decide discharges the equality by evaluation of the fold defining $m_2^{\mathrm{num}}$ against the pattern match defining $Z$, with no lemmas beyond computational reduction.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases over Fin 4, invoking one chunk theorem per cell. That global identity is the algebraic certificate that the midpoint TT numerator is exactly eight copies of the explicit kernel, a step in the Regge-exact gravity analysis inside the Recognition Science gravity stack. It does not itself touch the forcing chain (T5–T8) or the RCL; it is pure discrete kernel bookkeeping needed before continuum or phenomenological claims.
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