e_220311
plain-language theorem explainer
For the six Fin-4 indices (2,2,0,3,1,1), the folded coupling numerator equals eight times the explicit integer table entry. Gravity analysts cite it when discharging one cell of the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=2$, $c=0$, $d=3$, $i=1$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 10 of a 256-cell kernel certification that the 4D Regge midpoint M2TT numerator matches an explicit integer table. Indices run over $\mathrm{Fin},4$, labeling discrete edge and face slots in the midpoint stencil.
The numerator $N$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term evaluated at the six indices. The table $Z$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by pattern-matched constants (examples in the kernel cert include $4$, $-2$, and other small integers on the matched patterns).
The local claim is the scalar identity $N=8Z$ at one concrete 6-tuple. The full quantified statement is assembled downstream by exhaustive fin_cases over all six coordinates.
proof idea
One-line proof: by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are literals, so the kernel evaluates the fold defining the numerator and the pattern match defining the table, then checks equality with the factor eight. No lemmas are invoked beyond decidable arithmetic on Int.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and discharges the goal by six nested fin_cases, each leaf a chunk identity of this form. That global equality is the algebraic core of the Regge-exact midpoint M2TT identity in 4D discrete gravity inside the Recognition Science gravity stack.
Within RS, such exact discrete curvature identities sit under the geometric forcing chain (T8 forces $D=3$ spatial dimensions in the continuum limit; the 4D simplex calculus here is the relativistic packaging). Closing every kernel cell removes scaffolding from the midpoint gravity analysis and locks the numerator against the explicit table used in later continuum and phenomenological comparisons.
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