e_210122
plain-language theorem explainer
At multi-index (2,1,0,1,2,2) the Regge midpoint mass-squared numerator equals eight times the explicit integer Z-kernel. Gravity analysts cite it as one of the 256 finite-case checks that assemble the global m2Num = 8·Z identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,0,1,2,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of a 256-way case split proving that the midpoint Regge mass-squared numerator coincides with eight times an explicit integer kernel on every 4D multi-index. The ambient setting is exact algebraic identities for discrete gravity couplings, not continuum GR.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the six Fin-4 slots. The comparison target $Z$ is an explicitly tabulated integer function on $(\mathbb{F}_4)^6$ (sample values include $4$, $-2$, and other small integers on the listed patterns).
The parent assembly theorem states the identity for all six indices and discharges it by exhaustive fin_cases, so each chunk only needs one concrete equality.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(2,1,0,1,2,2)$. The left side runs the fold that defines the numerator; the right side multiplies the tabulated kernel entry by 8. No lemmas beyond the two definitions are invoked.
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 through all of $(\mathbb{F}_4)^6$. Without each point check, the global midpoint mass-squared identity in the 4D Regge analysis does not close.
In the Recognition gravity stack this is pure discrete algebra supporting the exact midpoint TT identity, not a continuum curvature claim. It does not touch the T0–T8 forcing chain, RCL, or $\phi$-ladder masses; it is infrastructure for the certified 4D kernel comparison.
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