e_233100
plain-language theorem explainer
Pointwise kernel identity: at multi-index (2,3,3,1,0,0) the folded coupling numerator equals eight times the explicit integer table. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT numerator identity over (F_4)^6. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,3,3,1,0,0)$ 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
This module is chunk 11 of a 256-way split of the 4D Regge exact-midpoint M2–TT numerator identity. The claim under audit is the pointwise relation $m_2^{\mathrm{num}}=8Z$ on one concrete sextuple of indices in $\mathbb{F}_4$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib kernel, and the fold starts at 0. The companion table explicitZ is a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by exhaustive pattern match (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).
The local setting is pure integer arithmetic on a finite index set: no continuum limit, no metric signature choice beyond the discrete 4D skeleton already fixed by the kernel certificate module.
proof idea
One-line computational proof: decide. Lean reduces both sides at the concrete indices $(2,3,3,1,0,0)$ to closed integers (the fold for the numerator versus eight times the matched table clause) and checks equality by the kernel decision procedure. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the universal statement $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by six nested fin_cases sweeps over $\mathbb{F}_4$. Each leaf is one of these chunk identities; this declaration discharges the leaf $(2,3,3,1,0,0)$.
In the Recognition gravity stack the identity certifies that the discrete midpoint M2–TT numerator matches the closed-form eightfold table, a necessary algebraic step before continuum or continuum-adjacent gravity claims. It does not itself touch T5–T8 or the RCL; it is infrastructure inside the 4D Regge analysis layer.
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