e_020012
plain-language theorem explainer
For the six Fin-4 indices (0,2,0,0,1,2), the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge exact-midpoint M2TT identity. The proof is a single kernel decide on the integer equality.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,0,1,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator m2Num is defined by folding a fixed coupling list: it sums contribution terms contrib t a b c d i j over that list, yielding an integer for each six-tuple of Fin 4 indices. The companion table explicitZ is a closed-form integer kernel on the same six indices, specified by pattern-matching (e.g. certain diagonal pairs give 4, off-diagonal swaps give -2, and unmatched patterns default).
The module is chunk 2 of a 256-case kernel certification that m2Num = 8 · explicitZ pointwise. The local setting is pure integer arithmetic on a finite index set: no continuum limit or curvature hypothesis is in play here, only the discrete identity needed downstream for the midpoint M2TT assembly.
proof idea
One-line proof by decide. Lean reduces both sides of the integer equality for the concrete indices (0,2,0,0,1,2): the left-hand side evaluates the fold that defines m2Num, the right-hand side multiplies the looked-up explicitZ value by 8, and the kernel certifies they match.
why it matters
This atom is consumed by m2Num_eq_eight_explicitZ, which states the universal identity ∀ a b c d i j, m2Num a b c d i j = 8 * explicitZ a b c d i j and discharges it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies one concrete branch of that case split.
In the broader Recognition gravity stack, the identity underwrites the exact midpoint form of the 4D Regge M2TT kernel: once every index tuple is certified, the folded numerator may be replaced by the closed table, simplifying later curvature and continuum-matching arguments. It is bookkeeping rather than a new physical law, but without the full 256-case cover the assembly theorem does not close.
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