e_120013
plain-language theorem explainer
For the Fin-4 index sextuple (1,2,0,0,1,3), the folded coupling numerator equals eight times the explicit integer kernel value. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one atomic point of the 256-way kernel check. The proof is a single decide on concrete integer arithmetic.
Claim. For indices $a=1$, $b=2$, $c=0$, $d=0$, $i=1$, $j=3$ 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 $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums integer contributions over six indices in $\mathrm{Fin},4$. The companion map $Z$ is a sparse explicit table of integers on the same six-index domain (nonzero only on a short list of patterns such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
This module is chunk 6 of a 256-way case split whose sole job is to verify $m_2^{\mathrm{num}}=8\cdot Z$ at every concrete sextuple. The parent assembly theorem then exhausts all $\mathrm{Fin},4$ arguments by nested fin-cases and stitches the chunk equalities into a single universal statement.
proof idea
One-line wrapper proved by decide. Both sides evaluate to concrete integers: the left via the fold of contrib over the coupling list, the right via the match table for the explicit kernel. Equality of those integers is decidable, so the tactic closes the goal with no further lemmas.
why it matters
This declaration is one atomic witness consumed by 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 sextuple of $\mathrm{Fin},4$ indices. That global integer identity is the numerical core of the Regge exact-midpoint M2TT certificate in 4D: it replaces floating-point or schematic checks with a fully decided equality between the folded numerator and eight times the explicit kernel. Downstream gravity analysis in the monolith relies on the assembled theorem rather than on any single chunk, but the chunks are what make the fin-cases proof go through.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.