e_023012
plain-language theorem explainer
For the six-index slot (0,2,3,0,1,2), the Regge midpoint m2 numerator equals eight times the explicit integer kernel Z. Gravity analysts cite it as one concrete cell of the 4D midpoint TT identity. The proof is a pure kernel decide on fixed Fin-4 arguments.
Claim. For indices $(a,b,c,d,i,j)=(0,2,3,0,1,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 2 of a 256-cell kernel certification that the midpoint m2 numerator is identically eight times an explicit integer table on all six Fin-4 indices. The ambient setting is 4D Regge calculus at the exact midpoint, where TT-sector identities are reduced to finite integer arithmetic.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple $(a,b,c,d,i,j)$. The comparison object $Z$ is an explicit pattern-matched integer table on the same six indices (typical entries $\pm 2,,4$, and zero off the matched patterns).
The parent assembly theorem states the identity for every index sextuple by exhaustive fin_cases; each chunk theorem such as this one discharges one concrete cell.
proof idea
One-line computational proof: both sides are closed integer terms once the six Fin-4 arguments are literals, so decide evaluates $m_2^{\mathrm{num}}(0,2,3,0,1,2)$ against $8\cdot Z(0,2,3,0,1,2)$ and closes the equality. No algebraic lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by casing on all six Fin-4 coordinates. That universal identity is the certified algebraic core of the 4D Regge exact-midpoint TT kernel used in the gravity analysis stack.
Within Recognition Science gravity work, these kernel cells turn a continuum midpoint identity into a finite, machine-checked integer statement, so downstream curvature and mass-ladder arguments can quote a proved numerator rather than a schematic coupling. This declaration is one cell in chunk 2 of that 256-decide grid; it does not itself touch T5–T8 or the RCL, but it hardens the discrete geometric substrate those continuum claims sit on.
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