e_201332
plain-language theorem explainer
For the six-index tuple (2,0,1,3,3,2) on Fin 4, the folded midpoint mass-squared numerator equals eight times the explicit integer Z-coupling. Gravity analysts building the exact Regge midpoint M2TT identity cite this kernel cell. The proof is a single decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,3,3,2)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint analysis, two integer-valued kernels on $(\mathrm{Fin},4)^6$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each list entry at the six indices. The comparison target $\mathrm{explicitZ}$ is a piecewise integer table on the same six indices (sample values include $4$, $-2$, and other small integers on the listed patterns).
The local module is chunk 8 of a 256-cell kernel certification: each cell checks $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ at one concrete multi-index. The ambient goal is an exact algebraic identity for the midpoint mass-squared / TT sector in 4D Regge gravity, reduced to finite integer arithmetic on $\mathrm{Fin},4$.
proof idea
One-line computational proof: decide. Both sides are closed integer terms once the six Fin-4 indices are substituted, so the kernel evaluates $m_2^{\mathrm{num}}$ by folding the coupling list and compares the result to $8$ times the matching explicitZ clause. No lemmas beyond the two definitions are required.
why it matters
This cell is one of the 256 concrete witnesses assembled by m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ by exhaustive fin_cases. That universal equality is the certified bridge from the folded coupling definition of the midpoint $m_2$ numerator to the compact explicit $Z$ table used in the 4D Regge midpoint M2TT identity.
In the broader Recognition gravity stack, such kernel certificates keep the continuum-facing curvature/mass-squared identities on a fully discrete, machine-checked integer base rather than on symbolic simplification alone. The chunk split exists only for compile-time granularity; mathematically the content is the single pointwise factor-of-eight relation at this multi-index.
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