e_210322
plain-language theorem explainer
For the six-index slot (2,1,0,3,2,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these 256 pointwise identities when assembling the global Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2,b=1,c=0,d=3,i=2,j=2$ in $\mathrm{Fin}\,4$, the folded coupling 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
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the given indices. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample values include $4$, $-2$, and so on for the listed patterns).
The module is chunk 9 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every multi-index. Each chunk theorem pins one concrete sextuple so the assembler can discharge the universal statement by exhaustive fin_cases.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contributions at $(2,1,0,3,2,2)$, the right by looking up explicitZ at those indices and multiplying by 8. Kernel decision closes the equality with no further lemmas.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over Fin 4, invoking one pointwise identity per cell. Without the full grid of chunk equalities, the global midpoint M2TT numerator identity in the 4D Regge analysis does not close. This is pure discrete kernel bookkeeping inside the gravity analysis stack, not a forcing-chain (T0–T8) step; it supports exact algebraic control of the midpoint curvature coupling before continuum or phenomenological claims.
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