e_022220
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (0,2,2,2,2,0) equals eight times the explicit integer table at that same point. Gravity analysts cite it when assembling the full 4D Regge midpoint m2-numerator identity over all Fin-4 indices. The proof is a single kernel decide on two concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,2,2,2,2,0)$ with each coordinate in $\mathbb{F}_4$, the folded coupling sum $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint analysis: it discharges a block of the 256 kernel pointwise checks that the folded numerator equals eight times a closed-form integer table. The local slogan is "$m_2^{\mathrm{num}}=8\cdot Z$, chunk 2 (256 kernel decides)."
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at the six Fin-4 indices. The table $Z$ is an explicit pattern-matched map $(\mathbb{F}_4)^6\to\mathbb{Z}$ with sparse nonzero entries (for example $4$ on diagonal-type pairs and $-2$ on certain off-diagonal swaps).
Both objects live in the kernel certificate module imported here; the chunk theorems only evaluate them at fixed indices.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete integers once the six indices are literals, so the equality is a decidable Int comparison with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathbb{F}_4)^6$ by exhaustive fin_cases and invokes each chunk equality such as this one. That global identity is the algebraic core of the Regge exact-midpoint $m_2$ TT certificate in four dimensions: it replaces a summed coupling expression by eight times a sparse closed form, enabling later gravity-analysis reductions. It is pure discrete algebra on Fin-4 indices, not a continuum GR claim, and sits inside the Gravity analysis layer rather than the T0–T8 forcing chain.
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