e_203301
plain-language theorem explainer
For the index sextuple (2,0,3,3,0,1) on Fin 4, the folded M2 numerator equals eight times the explicit Z table entry. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on two concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}3$, $d{=}3$, $i{=}0$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded M2 numerator equals eight times the explicit integer Z-value at those indices: $\mathrm{m2Num}(2,0,3,3,0,1)=8\cdot\mathrm{explicitZ}(2,0,3,3,0,1)$.
background
In the 4D Regge midpoint analysis, the M2 numerator at six Fin-4 indices is defined by folding a fixed coupling list: each term contributes an integer, and the fold accumulates their sum. The companion table explicitZ assigns a closed-form integer to each sextuple (with sparse nonzero patterns such as $\pm 2$ and $4$ on selected diagonal and off-diagonal slots).
The module is chunk 8 of a 256-cell kernel certification that every sextuple satisfies numerator $= 8\cdot$ explicitZ. Upstream, both the fold definition and the explicit table live in the KernelCert module; this chunk only evaluates one concrete cell.
The local setting is pure integer arithmetic on Fin 4 indices: no continuum limit, no metric signature choice beyond the discrete coupling data already baked into the list and table.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed indices $(2,0,3,3,0,1)$ and checks integer equality. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the table lookup by eight. No lemmas are invoked beyond decidable equality on $\mathbb{Z}$.
why it matters
The parent theorem m2Num_eq_eight_explicitZ quantifies over all six Fin-4 indices and discharges the identity by exhaustive fin_cases, each branch landing on a chunk cell such as this one. Without the cell-level facts, the assemble step cannot close.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2–TT kernel in 4D Regge calculus: it certifies that the folded numerator is exactly eight copies of the explicit Z pattern, so later continuum or continuum-limit arguments can quote a single closed form rather than a fold. It does not itself touch T5–T8 or the J-cost; it is infrastructure under the discrete gravity side of the monolith.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.