e_210332
plain-language theorem explainer
At multi-index (2,1,0,3,3,2) the folded 4D Regge midpoint numerator m2Num equals eight times the explicit integer table explicitZ. Gravity analysts cite it as one of the 256 kernel point checks that assemble the universal identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,0,3,3,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the TT-sector mass-squared numerator is built by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums integer contributions over that list. The companion table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ that records the closed-form value at each multi-index (examples: $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$).
The local module is chunk 9 of a 256-way kernel partition whose sole job is to certify $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. Each chunk theorem fixes one concrete 6-tuple in $\mathbb{F}_4$ and discharges the integer equality by computation. The present declaration treats the tuple $(2,1,0,3,3,2)$.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 indices and checks integer equality. No lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contrib over couplingZList) and the case table $Z$.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j., m_2^{\mathrm{num}}=8\cdot Z$ by exhaustive fin_cases on all six indices; each case reduces to one of these chunk equalities. That universal identity is the algebraic core of the 4D Regge midpoint M2-TT certificate: it replaces a folded sum by an explicit integer table, enabling later closed-form gravity identities in the Recognition stack. The module doc frames the whole file as “m2Num = 8·explicitZ, chunk 9 (256 kernel decides).”
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.