e_310311
plain-language theorem explainer
For the fixed Fin-4 indices (3,1,0,3,1,1), the folded midpoint mass-squared numerator equals eight times the explicit integer kernel. Gravity analysts building the 4D Regge exact-midpoint M2TT identity cite this as one atomic case among the 256 kernel decides. The proof is a single kernel decision on concrete integers.
Claim. For $a=3$, $b=1$, $c=0$, $d=3$, $i=1$, $j=1$ 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
This module is one chunk of the discrete certification that the midpoint mass-squared numerator agrees with eight times an explicit six-index integer table on $\mathrm{Fin},4$. The numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution map over a fixed coupling list; the explicit kernel $Z$ is a pattern-matched $\mathrm{Int}$-valued lookup on six $\mathrm{Fin},4$ arguments.
The local setting is pure finite arithmetic in the Gravity.Analysis stack for the Regge exact-midpoint M2TT identity in 4D. Chunk 13 supplies a block of the 256 kernel decides that exhaust the index space before assembly into a single quantified statement.
proof idea
One-line proof by decide. Both sides of the equality are closed integer terms once the six indices are fixed, so the kernel evaluates $m_2^{\mathrm{num}}(3,1,0,3,1,1)$ and $8\cdot Z(3,1,0,3,1,1)$ and checks they coincide. No lemmas beyond the definitions of the numerator fold and the explicit table are required.
why it matters
The parent theorem is the assembled identity that $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for every sextuple in $\mathrm{Fin},4$. That result proceeds by nested fin_cases and lands on atomic equalities such as this one. Closing each concrete cell is what turns the explicit kernel table into a usable global numerator identity inside the Regge midpoint analysis. The declaration itself is infrastructure, not a forcing-chain landmark; it exists so the assemble step has a discharged case rather than an open goal.
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