e_021000
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (0,2,1,0,0,0) equals eight times the explicit Z-table entry. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT kernel identity. The proof is a single kernel decide on closed integer arithmetic for that sextuple.
Claim. For indices $(a,b,c,d,i,j)=(0,2,1,0,0,0)$ in $\mathrm{Fin}\,4$, 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
This module is chunk 2 of a 256-way kernel split proving $m_2^{\mathrm{num}}=8\cdot Z$ on every sextuple of $\mathrm{Fin},4$ indices. The setting is the exact midpoint analysis of the 4D Regge M2–TT identity used in the gravity stack.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six indices. The comparison table $Z$ is an explicit integer function on six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and other small integers on the listed patterns).
Upstream, both objects live in the kernel certificate module; this chunk only evaluates one concrete point of the equality they jointly define.
proof idea
One-line proof: decide. Lean reduces both sides of the integer equality at the fixed indices $(0,2,1,0,0,0)$—the fold that defines the numerator and the pattern match that defines the explicit table—and checks they agree. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices and discharge of each point by a chunk identity of this form.
In the Recognition gravity analysis, that global identity is the certified algebraic core of the exact midpoint M2–TT relation in 4D Regge calculus. Closing every chunk removes scaffolding from the kernel certificate and lets the assemble step stand as a proved, sorry-free forall.
No direct T0–T8 or RCL citation; this is pure discrete-kernel bookkeeping under the gravity domain.
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