e_311013
plain-language theorem explainer
One of 256 kernel identities equating the midpoint Regge numerator m2Num at index tuple (3,1,1,0,1,3) to eight times the closed-form explicitZ value. Gravity analysts assembling the exact 4D midpoint M2TT identity cite it as a decided case. The proof is a single kernel decide on concrete integers.
Claim. For the six $4$-valued discrete indices $(a,b,c,d,i,j)=(3,1,1,0,1,3)$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the numerator of the M2TT kernel is assembled by folding a fixed coupling list: $\mathrm{m2Num}(a,b,c,d,i,j)$ sums contribution terms over that list for six indices in $\mathrm{Fin},4$. The companion table $\mathrm{explicitZ}$ is a sparse integer function on the same six indices, with nonzero entries such as $4$ on diagonal-type pairs and $-2$ on selected off-diagonal patterns.
The local module is chunk 13 of a 256-case kernel certification that $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ pointwise. Each chunk discharges a block of concrete index sextuples so the global assembly can finish by exhaustive case split rather than symbolic summation.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed sextuple $(3,1,1,0,1,3)$ and checks integer equality. No lemmas beyond the definitions of $\mathrm{m2Num}$ (fold of contributions) and $\mathrm{explicitZ}$ (lookup table) are required; the kernel reduces the fold to a numeral and compares.
why it matters
Feeds the parent theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$ by nested fin_cases and invokes these chunk lemmas as the decided leaves. That global equality is the algebraic core of the exact midpoint M2TT identity in the 4D Regge gravity analysis: once the numerator is replaced by $8\cdot\mathrm{explicitZ}$, closed-form simplification of the kernel can proceed without residual sums.
Within Recognition Science gravity work this is pure computational scaffolding for the continuum/Regge bridge, not a forcing-chain step (T0–T8). It closes one of the 256 decide obligations that make the assembled identity fully machine-checked.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.