e_030202
plain-language theorem explainer
For the six-index tuple (0,3,0,2,0,2) on Fin 4, the folded Regge midpoint numerator equals eight times the explicit kernel value Z. Gravity analysts cite it as one cell of the 256-case cover that rebuilds the global m2Num = 8·explicitZ identity. The proof is a single kernel decide on closed integer arithmetic.
Claim. For indices $a=0,b=3,c=0,d=2,i=0,j=2$ in $\mathrm{Fin}\,4$, the midpoint 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 3 of a 256-cell case split proving that the Regge exact-midpoint numerator coincides with eight times a closed-form integer table on $(\mathrm{Fin},4)^6$. The ambient setting is 4D discrete gravity analysis: couplings are summed, then compared to an explicit sparse kernel.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and accumulating a contribution at each multi-index. The comparison target $\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,,4$ on diagonal and off-diagonal patterns).
Chunk theorems such as this one discharge one concrete hexuple so the assembler can recombine them by exhaustive fin_cases.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values (the fold for the numerator versus the pattern match for the explicit kernel), and the kernel checks equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every hexuple in $(\mathrm{Fin},4)^6$ by nested fin_cases over the chunk lemmas. That global equality is the certified algebraic core of the Regge exact-midpoint $M_2$ TT identity in 4D inside the Gravity analysis stack.
In the broader Recognition framework this is infrastructure, not a forcing-chain landmark: it underwrites discrete curvature bookkeeping used when matching continuum limits and coupling normalizations, rather than T5–T8 or the RCL themselves. Closing the 256 decides removes a scaffolding surface on the midpoint identity.
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