e_030332
plain-language theorem explainer
For the six-index tuple (0,3,0,3,3,2) in Fin 4, the folded m2 numerator equals eight times the explicit kernel value Z. Gravity analysts cite it when assembling the full 4D Regge midpoint identity m2Num = 8·explicitZ. The proof is a single kernel decide on two closed integer expressions.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,3,3,2)$ with each index in $\{0,1,2,3\}$, the integer $m_2^{\mathrm{num}}(0,3,0,3,3,2)$ obtained by folding coupling contributions equals $8\,Z_{\mathrm{explicit}}(0,3,0,3,3,2)$.
background
In the 4D Regge midpoint analysis, two integer kernels on six Fin-4 indices are compared. The numerator side m2Num folds a fixed coupling list, accumulating a contribution at each triple for the given multi-index. The closed form explicitZ is a sparse case table of small integers (typically $\pm 2,\pm 4$, or zero off the listed patterns).
This module is chunk 3 of a 256-way split of the finite kernel: each chunk discharges a block of concrete index tuples so the global identity can be reassembled by exhaustive fin_cases. The local claim is only the single tuple $(0,3,0,3,3,2)$.
The surrounding development certifies an exact algebraic identity used in the discrete gravity / Regge curvature bookkeeping, not a continuum limit statement.
proof idea
One-line proof: by decide. Both sides reduce to concrete Int values (the fold for m2Num and the matching clause or default of explicitZ), and the kernel checks the equality $n = 8\cdot z$ by computation. No lemmas are invoked beyond the definitions of the two kernels.
why it matters
Parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in $(\mathrm{Fin},4)^6$. Its proof runs nested fin_cases on all six indices; this chunk supplies the discharged case $(0,3,0,3,3,2)$ inside that enumeration.
In the Recognition gravity stack the identity is bookkeeping infrastructure: it equates a summed coupling numerator to an explicit sparse kernel, scaled by eight, so later curvature or mass-ladder arguments can quote a closed form rather than a fold. It does not itself touch T5–T8, the RCL, or the phi-ladder; it is a finite combinatorial certificate inside the 4D Regge midpoint layer.
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