e_122010
plain-language theorem explainer
Equality of the Regge midpoint mass-squared numerator with eight times the explicit Z-kernel at multi-index (1,2,2,0,1,0). Gravity analysts cite it as one atomic case in the 4D midpoint M2TT identity. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,2,0,1,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the tabulated explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 6 of a kernel certification that the 4D Regge midpoint mass-squared numerator equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$ (four discrete directions).
Upstream, $m_2^{\mathrm{num}}$ is defined by folding a coupling list: it sums contrib terms over couplingZList at the six indices. The companion table explicitZ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.).
The local claim is one concrete instance of $m_2^{\mathrm{num}}=8\cdot Z$, discharged by computation rather than algebraic rewriting.
proof idea
One-line proof: by decide. Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed to $1,2,2,0,1,0$. Lean’s decision procedure compares the folded sum defining $m_2^{\mathrm{num}}$ against $8$ times the matching explicitZ clause. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembled theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one closes one kernel cell so the global identity is a finite certificate, not an open computation.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge M2TT kernel in 4D: it locks the numerator against the explicit Z-table before continuum or continuum-limit arguments. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is pure discrete kernel arithmetic supporting the gravity analysis layer.
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