e_013132
plain-language theorem explainer
For simplex labels (0,1,3,1,3,2) in Fin 4, the folded midpoint Regge mass-squared numerator equals eight times the explicit integer kernel Z. Gravity analysts building the exact 4D midpoint M2 TT identity cite this atomic case among the 256 kernel decides. Proof is a single decide on closed integer arithmetic of the fold versus the lookup table.
Claim. For indices $a=0$, $b=1$, $c=3$, $d=1$, $i=3$, $j=2$ 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 value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices appear. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib kernel, summed from zero. The companion map $Z$ is an explicit piecewise table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
The module is chunk 1 of a 256-case kernel certification whose sole goal is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$. The present declaration is one concrete six-tuple in that grid. Upstream, both maps live in the KernelCert module; downstream assembly will quantify over all of $\mathrm{Fin},4^6$.
proof idea
One-line computational proof: decide. Lean reduces both sides of the integer equality. The left-hand side evaluates the fold that defines the numerator at $(0,1,3,1,3,2)$; the right-hand side multiplies the table value of $Z$ at the same indices by eight. No lemmas are invoked beyond the definitions of the two maps and decidable equality on Int.
why it matters
This atom feeds the universal statement m2Num_eq_eight_explicitZ, which asserts $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for every six-tuple in $\mathrm{Fin},4$ by exhaustive fin_cases. That identity is the algebraic core of the exact midpoint M2 TT certification in 4D Regge gravity analysis inside the monolith.
Within Recognition Science gravity work, clearing the numerator-versus-kernel factor of eight is bookkeeping needed before continuum or continuum-limit claims about the midpoint mass-squared operator can be stated without residual combinatorial slack. The declaration itself is not a physics law; it is a certified cell in the forcing of that combinatorial identity.
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