e_313002
plain-language theorem explainer
Pointwise identity: the Regge midpoint $m_2$ numerator at multi-index $(3,1,3,0,0,2)$ equals eight times the explicit $Z$ kernel value there. Gravity analysts cite it as one of the 256 kernel cells assembled into the global $m_2=8Z$ theorem. Proof is a single `decide` on the concrete integers.
Claim. For indices $a=3,b=1,c=3,d=0,i=0,j=2$ in $\mathbb{F}_4$, the integer $m_2$-numerator (fold of coupling contributions) equals $8$ times the explicit integer kernel $Z$ at those same six indices.
background
This module is chunk 13 of a 256-cell kernel certification that the midpoint Regge $m_2$ numerator coincides with eight times an explicit integer table $Z$ on $(\mathbb{F}_4)^6$.
The numerator $m_2(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at $0$. The companion table $\mathrm{explicit}Z$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (e.g. $(0,0,1,1,2,2)\mapsto 4$, several off-diagonal patterns $\mapsto -2$).
The local goal is purely algebraic bookkeeping: check one concrete six-tuple so the assembler can discharge the universal statement by exhaustive fin_cases.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(3,1,3,0,0,2)$. The left side runs the fold that defines the numerator; the right side multiplies the pattern-matched explicit $Z$ entry by $8$. Equality of the resulting integers is decided in the kernel.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2=8,Z$ and proves it by six nested fin_cases over $\mathbb{F}_4$, invoking one chunk theorem per cell. That global identity is the certified algebraic core of the 4D Regge exact-midpoint $m_2$/$TT$ analysis in the Gravity lane. It does not itself touch the T0–T8 forcing chain or the $\varphi$-ladder mass formula; it is infrastructure for the discrete curvature side of Recognition gravity.
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