e_033203
plain-language theorem explainer
One kernel cell of the 4D Regge midpoint identity: the numerator mass-squared coupling at multi-index (0,3,3,2,0,3) equals eight times the explicit closed-form integer table at that cell. Gravity analysts assembling the full m2Num = 8·explicitZ identity cite these decide-chunks. The proof is a single kernel decide on concrete Fin-4 indices.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,3,3,2,0,3)$ in $(\mathrm{Fin}\,4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ 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 maps on six Fin-4 indices are compared. The numerator map $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each multi-index. The comparison target is an explicit piecewise integer table $Z$ (named explicitZ upstream), with hard-coded values such as $4$, $-2$, and so on on selected index patterns.
The module is chunk 3 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell is an independent equality on concrete indices; the present declaration is the cell at $(0,3,3,2,0,3)$. The local setting is pure finite enumeration over $\mathrm{Fin},4$, not continuum GR.
proof idea
One-line kernel proof: decide evaluates both sides at the fixed Fin-4 sextuple and checks integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contrib over couplingZList) and the explicit $Z$ table are required; the checker closes the ground term.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over all six indices. That global identity is the algebraic backbone of the Regge-exact midpoint $M_2$ TT identity in 4D within the Gravity analysis stack. In Recognition Science terms this is infrastructure for the discrete curvature/mass-squared side of the gravity sector, not a T0–T8 forcing step; it certifies that the folded coupling numerator matches the closed-form table used downstream. Without the chunk equalities, the assemble proof cannot discharge every case.
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