e_131003
plain-language theorem explainer
For the index sextuple (1,3,1,0,0,3) on Fin 4, the midpoint mass-squared numerator equals eight times the explicit Z-kernel value. Gravity analysts cite it as one of the 256 decided cells in the Regge exact midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}1,b{=}3,c{=}1,d{=}0,i{=}0,j{=}3$ in $\mathbb{F}_4$, the midpoint mass-squared numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit $Z$-kernel entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-cell kernel certification that the 4D Regge midpoint mass-squared numerator equals eight times an explicit integer kernel. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the midpoint TT identity.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a closed-form integer table on six $\mathrm{Fin},4$ arguments (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
Local goal of the chunk: discharge one concrete sextuple so the assembler can rebuild the universal identity by exhaustive case split.
proof idea
One-line computational proof: by decide. Lean evaluates both sides as concrete Int values (the fold defining the numerator versus eight times the matching explicitZ clause) and accepts equality by the kernel decision procedure. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$ and proves it by six nested fin_cases over the 256 cells. Each chunk theorem such as this one is a decided cell in that grid.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge M2–TT relation in 4D: once numerator and explicit kernel match everywhere, downstream curvature and mass-ladder arguments can treat the factor-of-eight normalization as settled rather than symbolic. It does not itself invoke the forcing chain (T5–T8) or the J-cost; it is pure discrete tensor algebra supporting those layers.
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