e_020223
plain-language theorem explainer
Pointwise kernel identity: the midpoint M2 numerator at discrete indices (0,2,0,2,2,3) equals eight times the explicit Z-table entry. Gravity analysts cite it as one cell of the 4D Regge midpoint TT identity. The proof is a single decide on concrete integers.
Claim. For the index sextuple $(a,b,c,d,i,j)=(0,2,0,2,2,3)$ 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 table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on $(\mathbb{F}_4)^6$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple. The comparison table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ with sparse nonzero entries (e.g. $4$ or $-2$ on selected patterns).
The local module is chunk 2 of a 256-decide kernel certifying $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The ambient goal is the exact midpoint M2 TT identity in four dimensions, reduced to finite arithmetic on face and edge labels in $\mathrm{Fin},4$.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(0,2,0,2,2,3)$. The left side runs the fold that defines the numerator; the right side looks up (or defaults) the explicit $Z$ entry and multiplies by $8$. Equality of the resulting integers is discharged by the kernel decision procedure.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j\in\mathbb{F}_4$, $m_2^{\mathrm{num}}=8\cdot Z$, by exhausting all sextuples (via fin_cases or chunked decides). That global identity is the algebraic core of the Regge exact-midpoint M2 TT certificate in 4D gravity analysis.
Within Recognition Science this sits in the gravity sector that checks discrete curvature/action identities against the forced $D=3$ spatial skeleton and eight-tick structure; it is bookkeeping arithmetic, not a new forcing step, but it closes a finite case obligating the continuum-facing midpoint formula.
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