e_200232
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (2,0,0,2,3,2) equals eight times the explicit Z-table entry. Gravity analysts cite it as one of 256 kernel decides that assemble into the universal m2Num = 8·explicitZ law. Proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,0,2,3,2)$ in $(\mathbb{F}_4)^6$, 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
In the 4D Regge exact-midpoint analysis, two integer kernels sit side by side. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list and summing a local contribution at each six-index slot in $(\mathbb{F}_4)^6$. The companion table $Z$ is an explicit pattern-matched integer function on the same domain (typical values $\pm 2,,4$, and sparse support).
The module is chunk 8 of a 256-way partition of that domain. Its sole job is to certify $m_2^{\mathrm{num}}=8\cdot Z$ at each concrete multi-index in the chunk, so a later assembly theorem can reassemble the universal identity by exhaustive case split on $\mathbb{F}_4$.
Upstream, $m_2^{\mathrm{num}}$ is the fold of couplingZList contributions, and $Z$ is the closed-form table those contributions are claimed to match after the factor of eight.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(2,0,0,2,3,2)$ and checks integer equality. No algebraic lemmas are invoked; the kernel definitions reduce to concrete Int arithmetic that the decision procedure discharges.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases over $\mathbb{F}_4$. Each chunk decide such as this one is a leaf of that case tree (or an equivalent precomputed witness the assembler relies on).
In the broader gravity stack, the identity is the algebraic certificate that the midpoint mass-squared numerator is exactly eight copies of the sparse explicit $Z$-kernel. That normalization is what lets later Regge/TT identities treat $Z$ as the canonical integer skeleton rather than the folded sum. It is pure discrete kernel bookkeeping, not a continuum GR claim, but it is load-bearing for the exact 4D midpoint analysis.
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