e_310022
plain-language theorem explainer
Pointwise kernel cell: the folded Regge numerator at multi-index (3,1,0,0,2,2) equals eight times the explicit integer table at that same point. Gravity analysts certifying the 4D midpoint M2 TT identity cite it as one decided cell among the 256 Fin-4 cases. The proof is a single kernel decide on concrete integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,0,2,2)$ with each index in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on $(\mathbb{F}_4)^6$ are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: it sums a local contribution over every term in couplingZList. The comparison target $Z$ is an explicit piecewise table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
The module is chunk 13 of a 256-cell kernel certification whose global claim is $m_2^{\mathrm{num}}=8\cdot Z$ at every multi-index. Each chunk discharges a block of those cells by direct evaluation rather than by a symbolic identity on the fold.
proof idea
One-line computational proof: decide. Both sides are closed integer terms once the six Fin 4 arguments are concrete literals, so the kernel evaluates the fold defining the numerator and the matching clause of the explicit table and checks numerical equality.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathbb{F}_4$ and discharges the universal quantifier by exhaustive fin_cases. That global equality is the algebraic core of the Regge exact midpoint M2 TT identity certificate in 4D. The factor of eight is the bookkeeping bridge between the folded coupling sum and the closed-form table used downstream in the gravity analysis stack. No Recognition forcing-chain landmark (T5–T8, RCL, $\varphi$) is invoked here; the result is pure discrete tensor arithmetic inside the Regge sector.
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