e_323331
plain-language theorem explainer
For the six Fin-4 indices (3,2,3,3,3,1), the folded coupling numerator m2Num equals eight times the tabulated explicitZ value. Gravity analysts assembling the full Regge midpoint M2 identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the coupling numerator at $(3,2,3,3,3,1)$ satisfies $m_2^{\mathrm{num}}(3,2,3,3,3,1)=8\,Z_{\mathrm{ex}}(3,2,3,3,3,1)$, where $m_2^{\mathrm{num}}$ is the fold of local contributions and $Z_{\mathrm{ex}}$ is the closed-form integer table.
background
This module is chunk 14 of a 256-cell kernel certifying that the Regge-exact midpoint numerator equals eight times an explicit integer table on every 4D multi-index. The ambient setting is discrete gravity analysis: six indices in $\mathrm{Fin},4$ label edge/face data for a 4-simplex midpoint identity.
Upstream, m2Num is defined by folding couplingZList and summing a local contrib at the six indices; it is an integer accumulator, not a closed form. explicitZ is the matching closed table: a pattern-matched function $\mathrm{Fin},4^6\to\mathbb{Z}$ with sparse nonzero entries (e.g. $4$, $-2$ on selected pairs). The claim is the pointwise equality of those two integers at one fixed tuple.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values (the fold on the left, the table lookup on the right), so the equality is a decidable integer comparison discharged by the kernel with no lemmas or case splits in the source.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$ by fin_cases on all six indices; each cell such as this one supplies one branch. Without the full grid of decides, the midpoint M2/TT identity in 4D cannot be closed as an exact integer relation.
In the Recognition gravity stack this is bookkeeping infrastructure for Regge-calculus identities, not a forcing-chain landmark (T5–T8). It removes a sorry-shaped gap in the exact discrete curvature algebra that later continuum or phenomenological gravity claims depend on.
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