e_012303
plain-language theorem explainer
For the index sextuple (0,1,2,3,0,3), the folded midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2 identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0,b{=}1,c{=}2,d{=}3,i{=}0,j{=}3$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding the coupling list equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample clauses assign values such as $4$, $-2$, and so on).
The module is chunk 1 of a 256-cell decide grid that discharges $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration: every index runs over four values, so the identity is a finite table of integer equalities rather than an analytic continuum argument.
proof idea
One-line decide on the fully concrete sextuple $(0,1,2,3,0,3)$. Lean reduces both sides to closed integers via the fold definition of $m_2^{\mathrm{num}}$ and the pattern match of $Z$, then checks equality in the kernel. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8Z$ for all six indices, which runs fin_cases over $a,b,c,d,i,j$ and lands on cells such as this one. That global identity is the algebraic core of the Regge exact midpoint $M_2$ TT identity in 4D gravity analysis inside the monolith. It is bookkeeping infrastructure for the discrete curvature side, not a forcing-chain landmark (T5–T8), but it closes a finite certificate needed before continuum or continuum-limit claims can be stated cleanly.
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