e_030100
plain-language theorem explainer
Pointwise check that the Regge midpoint numerator m2Num at multi-index (0,3,0,1,0,0) equals eight times the explicit kernel value explicitZ there. Gravity analysts cite it as one of the 256 Fin-4 kernel cells in chunk 3 of the m2Num = 8·explicitZ identity. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,3,0,1,0,0)$ with each coordinate in $\mathbb{F}_4$, the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one cell of the 4D Regge exact-midpoint analysis: it certifies the numerator identity $m_2^{\mathrm{num}}=8\cdot Z$ on a fixed Fin-4 multi-index, as part of a 256-way kernel split (chunk 3).
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a coupling list and summing integer contributions at those six indices. The comparison target $Z$ is an explicit case-table on $(\mathrm{Fin},4)^6\to\mathbb{Z}$ (sample clauses include values $4$, $-2$, and so on for distinguished index patterns).
The local claim is only the equality at $(0,3,0,1,0,0)$; neighboring siblings cover the other index tuples in the same chunk.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold defining the numerator versus eight times the explicit table entry) and closes the propositional equality. No lemmas are invoked beyond the kernel definitions of the two sides.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ by exhausting Fin cases. That global equality is the certified numerator half of the Regge exact-midpoint $M_2$ TT identity in 4D gravity analysis inside the monolith.
In the broader Recognition stack this is bookkeeping infrastructure for discrete curvature/edge couplings, not a forcing-chain landmark (T5–T8). It closes one kernel cell so the assembled identity can be cited without residual decides at use sites.
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