e_013202
plain-language theorem explainer
Pointwise kernel identity: the midpoint numerator m2Num at indices (0,1,3,2,0,2) equals eight times the explicit integer table explicitZ at the same indices. Gravity analysts cite it as one of 256 decide-certified cells feeding the universal m2Num = 8·explicitZ assembly. Proof is a single native decide on closed integer arithmetic.
Claim. For the index tuple $(a,b,c,d,i,j)=(0,1,3,2,0,2)$ with each index in $\mathbb{F}_4$, the folded midpoint 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
This module is chunk 1 of a 256-cell kernel certification that the midpoint numerator equals eight times an explicit integer table on all of $(\mathbb{F}_4)^6$. The local setting is exact midpoint identities for 4D Regge-type gravity analysis in the Recognition Science stack.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from $0$ and add each contribution term at the six indices. The table $Z$ is an explicit pattern-matched map $\mathbb{F}_4^6\to\mathbb{Z}$ with sparse nonzero entries (e.g. $4$, $-2$ on selected diagonal and off-diagonal patterns).
The assembly theorem downstream cases on every index and invokes one decide cell per tuple; this declaration is the cell for $(0,1,3,2,0,2)$.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed literals, so the kernel decision procedure discharges equality with no lemmas beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the parent assembly m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases over $(\mathbb{F}_4)^6$ and one decide cell per case. Without the full 256-cell cover, the universal midpoint identity does not close.
In the gravity analysis layer this identity is bookkeeping infrastructure for exact midpoint/Regge-style 4D kernel relations, not a forcing-chain landmark (T5–T8) by itself. It closes a pure computational obligation so higher geometric claims can quote a single quantified equality rather than a table of raw folds.
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