e_023323
plain-language theorem explainer
For the six Fin-4 indices (0,2,3,3,2,3), the folded numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts cite it as one kernel cell in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. With $m_2^{\mathrm{num}}$ the fold of coupling contributions over the 4D index sextuple and $Z_{\mathrm{ex}}$ the explicit integer table, $m_2^{\mathrm{num}}(0,2,3,3,2,3)=8\,Z_{\mathrm{ex}}(0,2,3,3,2,3)$.
background
This module is chunk 2 of a 256-cell kernel certification that the 4D Regge midpoint numerator equals eight times an explicit integer table. Indices run in $\mathrm{Fin},4$, labeling discrete directions in the midpoint TT identity for Regge gravity.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing each contribution at those six indices. The companion table $Z_{\mathrm{ex}}$ is a pattern-matched integer function on the same sextuples (sample values include $4$, $-2$, and so on).
The local claim is one concrete cell of the identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$, proved by computation rather than by a symbolic closed form.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values once the six indices are fixed, so the kernel equality checker discharges the goal with no lemmas or rewriting.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 sextuple into the global statement $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$. This cell is one of the chunk-2 decides that fill that case split.
In the Recognition gravity stack, the identity is bookkeeping for the exact midpoint M2–TT kernel in 4D Regge analysis: it certifies that the folded coupling numerator matches the explicit table used downstream. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete linear-algebra certification inside the gravity analysis layer.
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