e_030231
plain-language theorem explainer
Pointwise identity: the midpoint M2 numerator at index tuple (0,3,0,2,3,1) equals eight times the explicit Z-kernel value there. Gravity analysts cite it as one cell of the 4D Regge midpoint kernel table. The proof is a single kernel `decide` on concrete integers.
Claim. For the six $4$-valued indices $(a,b,c,d,i,j)=(0,3,0,2,3,1)$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of the exhaustive table proving $m_2^{\mathrm{num}}=8\cdot Z$ on all $4^6$ index tuples in the 4D Regge exact-midpoint M2TT identity. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each six-index slot. The comparison target $Z$ is an explicit integer-valued kernel on $\mathrm{Fin},4^6$, given by a finite pattern match (e.g. diagonal blocks map to $4$, certain off-diagonal pairs to $-2$).
The local setting is pure integer arithmetic certification: no continuum limit, no physical units. Upstream, both sides are closed definitions in the kernel certificate module; this chunk only discharges concrete equalities.
proof idea
One-line kernel proof: decide evaluates both integer sides at the fixed tuple $(0,3,0,2,3,1)$ and checks equality. No lemmas are invoked beyond the elaborator reducing m2Num (the fold) and explicitZ (the match) to numerals.
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 six nested fin_cases over the $4$-valued indices. Each chunk cell such as this one is a leaf of that case tree. In the gravity stack this identity is the algebraic core of the exact midpoint M2TT relation in 4D Regge calculus; it is bookkeeping, not a new physical law, but without the full table the continuum identification cannot be certified inside Lean.
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