e_112201
plain-language theorem explainer
Single multi-index identity: the folded Regge midpoint numerator at (1,1,2,2,0,1) equals eight times the explicit integer table entry. Gravity analysts assembling the 4D M2 TT kernel cite it as one of 256 decide-chunks. The proof is a pure kernel decision on concrete integers.
Claim. For indices $a{=}1$, $b{=}1$, $c{=}2$, $d{=}2$, $i{=}0$, $j{=}1$ in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator of a mass-squared (M2) TT coupling is defined by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and the fold starts at zero. That sum is the object compared against a hand-written integer table explicitZ, which assigns a sparse pattern of values such as $4$, $-2$, and so on to selected multi-indices and (implicitly) zero elsewhere.
This module is chunk 5 of a 256-way case split that discharges the pointwise claim numerator $= 8\cdot Z$ by kernel decision. The local setting is pure finite arithmetic on Fin 4, not continuum GR: every index runs over four discrete labels, so the identity is a concrete integer equality rather than a symbolic tensor identity.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides at the fixed six-tuple $(1,1,2,2,0,1)$: the left-hand fold over the coupling list and the right-hand lookup $8\cdot Z(1,1,2,2,0,1)$, then checks integer equality. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in Fin 4 by exhaustive fin_cases and invokes one decide-chunk per cell. Closing all 256 chunks certifies that the folded Regge midpoint numerator is exactly eight times the sparse explicit Z table, a bookkeeping step inside the Gravity analysis path toward the exact midpoint M2 TT identity in 4D. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is infrastructure for the discrete curvature/coupling side of the RS gravity stack.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.