e_330320
plain-language theorem explainer
For the Fin-4 multi-index (3,3,0,3,2,0), the folded M2 numerator equals eight times the explicit kernel integer Z. Gravity analysts certifying the 4D Regge midpoint M2-TT identity cite this as one cell in a 256-decide chunk. The proof is a single kernel decide on concrete integers.
Claim. For $a=3$, $b=3$, $c=0$, $d=3$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling sum $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module sits in the 4D Regge midpoint gravity analysis. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: at each six-tuple of $\mathrm{Fin},4$ indices it accumulates integer contributions. The companion table $Z$ is an explicit sparse map from the same six-tuples to small integers (typical nonzero values $\pm 2,\pm 4$).
The local goal, stated in the module header, is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$, discharged by kernel decides in chunks of 256. Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel-certificate module; no analytic hypotheses are attached.
The present declaration fixes one concrete cell of that table, namely the index string $3,3,0,3,2,0$. Sibling lemmas cover the other cells of the same chunk.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete Int values (the fold over the coupling list on the left, the pattern-match table on the right), and the kernel equality checker accepts them. No lemmas are invoked beyond the definitions of the numerator fold and the explicit $Z$ table.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles every cell by nested fin_cases on the six $\mathrm{Fin},4$ indices, quoting each chunk lemma such as this one. That universal identity is the algebraic backbone of the Regge exact midpoint M2-TT certification in the gravity stack: once numerator and explicit kernel agree up to the constant factor 8, downstream continuum and curvature arguments can treat $Z$ as a closed-form stand-in for the folded coupling sum.
Within Recognition Science this is infrastructure for the discrete gravity side (Regge calculus on the recognition complex), not a forcing-chain step. It closes one of the 256 kernel obligations in chunk 15 and has no remaining sorry.
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