e_201221
plain-language theorem explainer
For the Fin-4 index tuple (2,0,1,2,2,1), the folded midpoint mass-squared numerator equals eight times the explicit kernel integer. Gravity analysts certifying the Regge exact midpoint M2TT identity cite this as one finite case among the 256 kernel decides. Proof is a single native decide on the unfolded integer equality.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,2,2,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint gravity analysis, the mass-squared numerator is the integer
$$m_2^{\mathrm{num}}(a,b,c,d,i,j)=\sum_{t\in\mathrm{couplingZList}}\mathrm{contrib}(t;a,b,c,d,i,j),$$
obtained by folding a fixed contribution map over a static coupling list. The companion table $Z$ (explicitZ) is a six-argument pattern match on $\mathrm{Fin},4$ that returns the closed-form integer expected for each index tuple (for example $Z(0,0,1,2,2,1)=-2$).
This module is chunk 8 of the kernel certification: it discharges one concrete six-tuple among the 256 decide obligations that together underwrite $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The local setting is pure finite arithmetic on $\mathrm{Fin},4$; no continuum limit is taken here.
proof idea
One-line wrapper: by decide. After unfolding m2Num (the fold of contrib over couplingZList) and explicitZ (the pattern-match table), both sides reduce to concrete integers for the fixed arguments $(2,0,1,2,2,1)$. Lean’s kernel decision procedure checks the equality with the factor $8$. No lemmas beyond the two definitions are invoked.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every finite case by nested fin_cases on the six $\mathrm{Fin},4$ indices and invokes this decide (among its siblings) to conclude
$$\forall a,b,c,d,i,j,\quad m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j).$$
That universal identity is the algebraic core of the Regge exact-midpoint M2TT 4D kernel certificate in the gravity stack. It sits downstream of the discrete coupling bookkeeping and upstream of any continuum or phenomenological reading of the midpoint mass-squared operator. The chunked decide layout keeps each kernel obligation tiny and independently checkable.
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