e_312202
plain-language theorem explainer
Pointwise identity: at multi-index (3,1,2,2,0,2) the folded M2 numerator equals eight times the explicit Z-table entry. Gravity analysts cite it only as one cell of the 4D midpoint kernel certification. The proof is a single kernel `decide` on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(3,1,2,2,0,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint M2TT analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib rule, and the fold starts at $0$. The comparison target explicitZ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) encoding the closed-form Z-coupling.
The ambient module is chunk 13 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ holds at every multi-index. The full quantified statement is assembled downstream by exhaustive case split on the six indices.
proof idea
One-line computational proof: decide. Both sides are closed integer expressions once the six concrete $\mathrm{Fin},4$ arguments are substituted into m2Num and explicitZ, so the kernel evaluates the fold and the table lookup and checks numeral equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and discharges the universal quantifier by fin_cases on all six indices. That global identity is the algebraic backbone of the exact midpoint M2TT kernel certificate in the 4D Regge gravity analysis. The factor-of-eight normalization ties the folded coupling sum to the sparse Z-table used elsewhere in the discrete curvature bookkeeping. This cell is pure scaffolding glue: it does not itself touch continuum limits, phi-ladder masses, or the T0–T8 forcing chain.
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