e_310020
plain-language theorem explainer
For the multi-index (3,1,0,0,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite this as one of the 256 kernel cells. The proof is a single decide on concrete integer arithmetic.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=0$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The goal across chunks is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on all six $\mathrm{Fin},4$ indices (256 cells).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $Z$ is an explicit integer-valued kernel on the same six indices, tabulated by pattern (e.g. diagonal-type slots map to $4$, certain off-diagonal slots to $-2$).
Chunk 13 holds a block of these cells; the present declaration is the single cell with indices $(3,1,0,0,2,0)$. Downstream assembly will reassemble every cell into the universal statement.
proof idea
One-line kernel decision: decide evaluates both sides as concrete Int expressions (the fold of m2Num at the fixed indices versus 8 * explicitZ at those indices) and closes the equality by computation. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices, discharging each cell by a chunk theorem of this form.
In the Recognition gravity stack this identity is part of the exact midpoint $M_2$ TT analysis in 4D Regge calculus: matching the folded coupling numerator to eight times the closed-form kernel is the algebraic certificate that the discrete curvature/mass-squared contribution is under exact control. It does not itself invoke the forcing chain (T5–T8) or the RCL, but sits in the gravity-analysis layer that those continuum limits ultimately constrain.
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