e_030230
plain-language theorem explainer
For the six-index slot (0,3,0,2,3,0) on the 4D midpoint Regge kernel, the summed numerator m2Num equals eight times the closed-form table entry explicitZ. Gravity analysts cite these micro-identities when assembling the global m2Num = 8·explicitZ certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,2,3,0)$ in $\mathrm{Fin}\,4$, 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, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the fold starts from zero. The companion table $Z$ is an explicit case-split function on the same six indices, returning small integers such as $4$, $-2$, and so on for the nonzero patterns.
The module is one chunk of a 256-way case split that checks $m_2^{\mathrm{num}}=8Z$ pointwise. Chunk 3 covers a block of those index sextuples; the present declaration is the single sextuple $(0,3,0,2,3,0)$. Downstream assembly will recombine all chunks into a universal statement over $\mathrm{Fin},4^6$.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side evaluates the fold of contrib over couplingZList at the fixed indices, the right-hand side multiplies the matching explicitZ clause by eight. The kernel closes the equality with no lemmas beyond definitional reduction.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over the six indices and dispatch into these chunk lemmas. That global identity is the certified bridge between the summed coupling definition and the closed-form table used in the Regge midpoint $M_2$ TT analysis.
Within Recognition gravity, such kernel certificates keep the discrete curvature bookkeeping exact rather than numeric. They do not themselves invoke the forcing chain (T5–T8) or the RCL, but they underwrite the 4D geometric side of the gravity stack that those landmarks eventually constrain.
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