e_202010
plain-language theorem explainer
For the six-index slot (2,0,2,0,1,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2=8Z certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2,b=0,c=2,d=0,i=1,j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The comparison target explicitZ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on on the diagonal and near-diagonal slots).
The module is one chunk of the exhaustive check that $m_2^{\mathrm{num}}=8\cdot Z$ holds on every six-tuple. Module doc frames the work as "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)". Sibling theorems cover the neighboring index patterns in the same chunk.
proof idea
One-line decide. Both sides evaluate to concrete Int values once the six Fin-4 arguments are fixed: the fold that defines $m_2^{\mathrm{num}}$ runs over a finite list, and explicitZ is a closed pattern match. Lean’s kernel decision procedure discharges the resulting integer equality with no lemmas and no case split inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ by fin_cases on each coordinate. That global certificate is the algebraic backbone of the Regge-exact midpoint $M_2$–TT identity in 4D gravity analysis inside the monolith.
Within Recognition Science this sits in the gravity domain: exact discrete curvature bookkeeping on the eight-tick / $D=3$ lattice side, not a continuum GR postulate. Closing every chunk removes scaffolding from the $M_2=8Z$ bridge and keeps the midpoint identity fully machine-checked.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.