e_210312
plain-language theorem explainer
For the Fin-4 index tuple (2,1,0,3,1,2), the folded coupling numerator equals eight times the explicit kernel table entry. Gravity analysts assembling the exact 4D midpoint M2 TT identity cite this as one atomic case among the full 4^6 grid. The proof is a single kernel `decide` on concrete integer equality.
Claim. For indices $a=2$, $b=1$, $c=0$, $d=3$, $i=1$, $j=2$ in $\{0,1,2,3\}$, the folded integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the tabulated kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator folds a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts from 0. The comparison target is an explicit sparse table that hard-codes the nonzero kernel values (examples in the table include $4$ and $-2$ on selected patterns) and returns an integer for every index sextuple.
This module is chunk 9 of the identity "numerator = 8 · explicit table", discharged by 256 kernel decides. The local claim is the single grid point $(2,1,0,3,1,2)$ inside that chunk. Upstream, both maps are pure definitions: the fold over couplings, and the pattern-matched table.
proof idea
One-line computational discharge: decide. Lean evaluates both sides to concrete integers (the fold of contributions on the left, the table lookup on the right) and checks equality in the kernel. No algebraic lemmas or case splits appear in this atom.
why it matters
Parent theorem is the universal statement that the folded numerator equals eight times the explicit table for every Fin-4 sextuple. That assembler introduces the six indices and runs fin_cases on each, so each grid point such as this one is a required leaf. The global identity is the algebraic core of the Regge exact midpoint M2 TT identity in 4D: it certifies that the discrete numerator matches eight copies of the sparse Z kernel, which is the bookkeeping step before continuum or physical identification. Within the gravity side of the framework it is pure finite combinatorics, not a forcing-chain (T0–T8) step.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.