e_030301
plain-language theorem explainer
For the multi-index (0,3,0,3,0,1) on Fin 4, the folded M2 numerator equals eight times the explicit integer kernel entry. Gravity analysts cite it when discharging one cell of the 4D Regge midpoint TT identity. The proof is a single kernel decide on concrete integers.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. For the sextuple $(0,3,0,3,0,1)$, the summed coupling numerator $\mathrm{m2Num}(0,3,0,3,0,1)$ equals $8$ times the explicit kernel value at those indices.
background
In the 4D Regge midpoint analysis, the M2 numerator at six Fin-4 indices is the fold of a fixed coupling list: each term contributes an integer via a local contrib, and the total is an Int. The companion table explicitZ is a sparse case-split on those same six indices, returning small integers such as $4$, $-2$, and so on.
The module is chunk 3 of a 256-cell kernel certification that $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ pointwise. Each cell fixes one concrete sextuple so the equality is a closed integer computation rather than a symbolic identity over all indices.
Upstream, m2Num is pure accumulation over couplingZList; explicitZ is the hand-written target table the fold is checked against.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values for the fixed indices $(0,3,0,3,0,1)$: the left by folding the coupling list, the right by evaluating the explicitZ match and multiplying by 8. The kernel decision procedure checks integer equality; no lemmas beyond the two definitions are required.
why it matters
This cell is one brick in the exhaustive identity m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$. The assembler case-splits all six Fin-4 coordinates; chunk theorems such as this one discharge individual branches (or certify the same equalities the decide tactic would rediscover).
In the gravity stack, that global identity is the algebraic certificate that the midpoint M2 TT kernel matches its closed-form table, a step toward exact Regge curvature bookkeeping in four dimensions. It does not itself invoke the RS forcing chain (T5–T8) or the J-cost; it is pure discrete kernel arithmetic supporting the continuum gravity side of the monolith.
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