e_312211
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at multi-index (3,1,2,2,1,1) equals eight times the tabulated explicit kernel value. Gravity analysts assembling the full 4D Regge midpoint M2TT identity cite this as one of 256 kernel cells. Proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,2,2,1,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table entry: $N(3,1,2,2,1,1)=8\,Z(3,1,2,2,1,1)$.
background
This module is chunk 13 of a 256-cell kernel certifying that the Regge exact-midpoint mass-squared numerator equals eight times a closed-form table on every 4D multi-index. The ambient setting is discrete gravity analysis: couplings on $\mathrm{Fin},4$ edges are summed into an integer numerator, then matched against an explicit integer table.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list and adding each contribution at the six indices. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (sample clauses give values such as $4$, $-2$, etc.). Both live in the kernel-cert module imported here.
The local claim is only the single cell $(3,1,2,2,1,1)$; sibling theorems cover the other cells in the same chunk.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (the fold of contributions versus eight times the matched table entry), so the equality is a decidable integer comparison with no further lemmas.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and discharges the goal by exhaustive fin_cases on all six indices, invoking one cell theorem per case. Without the full 256-cell cover, the global midpoint M2TT identity in 4D cannot close.
In the Recognition gravity stack this is bookkeeping infrastructure for the discrete curvature/mass-squared side of the Regge analysis, not a forcing-chain landmark (T5–T8). It closes a scaffolding gap: every multi-index cell must hold before the assembled identity is available downstream.
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