e_203331
plain-language theorem explainer
For the six-index tuple (2,0,3,3,3,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}3$, $d{=}3$, $i{=}3$, $j{=}1$ 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
This module is chunk 8 of a 256-way case split proving that the Regge midpoint $M_2$ numerator equals eight times an explicit integer kernel on all six-tuples in $(\mathrm{Fin},4)^6$. The local slogan is $m_2^{\mathrm{num}} = 8\cdot Z$.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at those indices. The comparison target $Z$ is an explicit pattern-matched integer table on the same six indices (sample entries include $4$, $-2$, and so on).
The surrounding Gravity.Analysis development certifies exact midpoint identities for 4D Regge calculus; these chunk lemmas discharge the finite case analysis that the kernel certificate relies on.
proof idea
One-line proof by decide. Both sides are closed integer expressions once the six Fin-4 indices are fixed: the left-hand side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right-hand side multiplies the pattern-matched $Z$ entry by $8$. Lean’s kernel decision procedure checks the resulting integer equality.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}} = 8\cdot Z$ for all six indices, which is proved by exhaustive fin_cases and invokes one chunk equality per tuple. Without the full set of 256 decides, the global midpoint numerator identity stays open.
In the Recognition gravity stack this is bookkeeping infrastructure for exact 4D Regge midpoint identities, not a forcing-chain landmark (T5–T8). It closes a finite computational obligation so higher gravity certificates can quote a clean algebraic relation rather than a fold.
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