e_202023
plain-language theorem explainer
For index tuple (2,0,2,0,2,3), the Regge midpoint mass-squared numerator equals eight times the explicit 4D kernel entry. Gravity analysts building the exact midpoint TT identity cite this as one decided cell in chunk 8 of the kernel table. The proof is a single computational decide on the unfolded integer equality.
Claim. For the six indices $(2,0,2,0,2,3)\in(\mathrm{Fin}\,4)^6$, the midpoint mass-squared numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(2,0,2,0,2,3)=8\,Z(2,0,2,0,2,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of coupling contributions: it sums contrib over couplingZList at the given indices. The comparison table $Z$ is the piecewise explicit kernel explicitZ, a finite case table of small integers (entries such as $4$, $-2$, and so on).
This module is chunk 8 of the cell-by-cell certification that $m_2^{\mathrm{num}}=8\cdot Z$ on the full $4^6$ index space. The local setting is pure integer arithmetic after the continuum TT identity has been reduced to a finite kernel check; no continuum estimates remain inside the chunk.
proof idea
One-line computational proof: decide unfolds m2Num and explicitZ at the concrete indices $(2,0,2,0,2,3)$ and checks the resulting integer equality. No lemmas beyond the two kernel definitions are invoked; the goal is closed by Lean's decision procedure on closed integer arithmetic.
why it matters
Parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple and discharges the universal quantifier by exhaustive fin_cases on each index. This declaration is one named cell in that case tree (chunk 8), so the global equality inherits its truth from the decide kernel.
In the Recognition gravity stack the identity is the algebraic core of the exact midpoint Regge $M_2$ TT check in 4D: once every cell matches, the discrete curvature/mass-squared numerator is certified to be eight times the explicit kernel, with no residual combinatorial gap. It does not itself touch T0–T8 or the J-cost forcing chain; it sits downstream in the gravity analysis layer that consumes those structural facts.
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