e_322012
plain-language theorem explainer
One kernel cell of the 4D Regge midpoint identity: the folded numerator m2Num at index sextuple (3,2,2,0,1,2) equals eight times the explicit integer table explicitZ at the same point. Gravity analysts assembling the global m2Num = 8·explicitZ identity cite this cell. Proof is a single decide on concrete Fin 4 integers.
Claim. For the discrete index sextuple $(a,b,c,d,i,j)=(3,2,2,0,1,2)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a 256-cell kernel certification that the 4D Regge midpoint numerator equals eight times an explicit integer table. Indices run over $\mathrm{Fin},4$, labeling discrete edge/face slots in the midpoint triangulation scheme.
Upstream, $\mathrm{m2Num}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums contribution terms $\mathrm{contrib},t,a,b,c,d,i,j$ over that list, yielding an integer. The companion $\mathrm{explicitZ}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample clauses include values $4$, $-2$, and so on).
The local goal is purely algebraic bookkeeping: verify the scalar identity cell-by-cell so a later assembler can recombine all 256 decides into a single quantified theorem.
proof idea
One-line kernel discharge: decide evaluates both sides at the concrete sextuple $(3,2,2,0,1,2)$. The left side reduces by unfolding the fold over couplingZList and summing the integer contributions; the right side multiplies the looked-up explicitZ entry by 8. No lemmas beyond decidable equality on Int are required.
why it matters
Feeds the parent assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one closes one concrete branch of that case split.
In the Recognition gravity stack, the identity certifies that the midpoint Regge mass/coupling numerator collapses to a sparse explicit integer kernel, a prerequisite for exact 4D discrete curvature bookkeeping. It is infrastructure rather than a forcing-chain landmark (T0–T8), but without the cell-wise decides the global algebraic simplification does not typecheck.
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