e_032102
plain-language theorem explainer
For the six-index slot (0,3,2,1,0,2) on Fin 4, the folded Regge midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D midpoint M2–TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,1,0,2)$ in $\mathrm{Fin}\,4$, the midpoint numerator $m_2(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator m2Num is the fold of a coupling list: it sums a local contribution over every term in couplingZList. The comparison target explicitZ is a closed-form pattern-match table of small integers (entries such as 4, -2, and so on).
The module is chunk 3 of a 256-cell partition of the Fin-4^6 index space. Each cell is a concrete equality m2Num a b c d i j = 8 * explicitZ a b c d i j, discharged by the kernel so the full universal statement can be assembled by exhaustive case split.
Local setting (module doc): prove m2Num = 8·explicitZ cellwise via 256 kernel decides.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values once the six Fin-4 indices are fixed to 0,3,2,1,0,2, so the decision procedure closes the equality with no lemmas beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices and proves it by nested fin_cases over the 256 cells. Each cell theorem such as this one discharges one branch of that split.
In the gravity stack this is bookkeeping for the Regge midpoint M2–TT identity in 4D: the folded numerator must match eight times the explicit kernel before continuum or curvature comparisons proceed. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete-kernel certification inside the gravity analysis layer.
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