e_103131
plain-language theorem explainer
For the six Fin-4 indices (1,0,3,1,3,1), the folded coupling numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts assembling the full 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel cases. The proof is a single decide, discharging the equality by native integer arithmetic on the explicit fold.
Claim. For indices $a=1$, $b=0$, $c=3$, $d=1$, $i=3$, $j=1$ in $\mathrm{Fin}\,4$, the coupling numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2$ is defined by folding a fixed coupling list: each term contributes an integer weight depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion map explicitZ is a piecewise integer table on the same six indices (typical nonzero values $\pm 2,,4$), intended as the closed form of that fold divided by eight.
This module is chunk 4 of the 256-case kernel certification that $m_2=8\cdot Z$ pointwise. The local setting is pure finite enumeration over $\mathrm{Fin},4^6$, with no continuum limit or curvature hypothesis yet in play; the identity is algebraic bookkeeping for the discrete gravity kernel.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers. The left side runs the fold that defines m2Num at $(1,0,3,1,3,1)$; the right side multiplies the matching explicitZ clause by eight. Lean’s kernel confirms the two integers coincide, with no tactic branching or lemmas beyond the definitions of m2Num and explicitZ.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 sextuple by nested fin_cases; each leaf is one of these chunk theorems (here e_103131). Establishing $m_2=8Z$ everywhere certifies that the explicit kernel table is faithful to the coupling fold, which is the algebraic core of the Regge midpoint M2–TT identity in four dimensions.
Within Recognition Science gravity work this is scaffolding for discrete curvature bookkeeping, not a continuum Einstein equation. It does not invoke the forcing chain (T5–T8), RCL, or the phi-ladder; those enter only once the certified kernel is wired into continuum or continuum-limit statements.
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