e_212322
plain-language theorem explainer
For the six-index tuple (2,1,2,3,2,2) on Fin 4, the summed coupling numerator equals eight times the tabulated explicit kernel integer. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one concrete cell of the 4^6 case grid. The proof is a single kernel decide on fully concrete integers.
Claim. With indices $a=2$, $b=1$, $c=2$, $d=3$, $i=2$, $j=2$ in $\{0,1,2,3\}$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of a 256-way split that certifies, cell by cell, the identity between two integer-valued kernels on $(\mathrm{Fin},4)^6$ arising in the 4D Regge exact-midpoint M2–TT analysis.
The numerator $N$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at 0. The explicit kernel $Z$ is a closed pattern-match table on the same six indices (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$). The local claim is that $N=8Z$ at one fixed tuple.
Upstream, both $N$ and $Z$ live in the kernel-cert module; downstream assembly will quantify over all six indices by exhaustive fin_cases.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are literals, so the kernel equality checker discharges $N(2,1,2,3,2,2)=8\cdot Z(2,1,2,3,2,2)$ with no lemmas and no case split inside this declaration.
why it matters
Parent theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by six nested fin_cases branches; each branch lands on a chunk theorem of this form. This cell is the $(2,1,2,3,2,2)$ leaf of that tree.
In the Recognition gravity stack the identity underwrites the exact midpoint evaluation of the M2–TT coupling in 4D Regge calculus, so the discrete curvature bookkeeping matches the closed kernel used in continuum limits. It does not itself touch T0–T8 or the J-cost forcing chain; it is pure integer certification inside the gravity analysis layer.
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