e_323101
plain-language theorem explainer
For the six Fin-4 indices (3,2,3,1,0,1), the folded Regge coupling numerator equals eight times the closed-form integer table. Gravity analysts cite it when assembling the full midpoint M2 TT identity over the 4D kernel. The proof is a single kernel decide on two concrete integers.
Claim. For indices $a{=}3,b{=}2,c{=}3,d{=}1,i{=}0,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint analysis, the 4D TT kernel is certified by comparing two integer-valued maps on six indices in $\mathrm{Fin},4$. 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 zero. The companion map $Z$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
The local module is chunk 14 of a 256-way case split: each chunk discharges one concrete multi-index equality $N=8Z$ by kernel computation. The ambient claim is that this scalar identity holds for every six-tuple, which is the algebraic content of the midpoint M2 TT identity in four dimensions.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted: the left-hand side evaluates the fold that defines the numerator, the right-hand side multiplies the table entry by eight. The kernel checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
This case is one atom of the universal identity assembled downstream as m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over $\mathrm{Fin},4$. That global equality is the certified numerator form of the Regge exact-midpoint M2 TT identity in 4D, sitting in the Gravity analysis layer that supports continuum limits and discrete curvature bookkeeping in Recognition Science. It does not itself touch the T0–T8 forcing chain or the J-cost; it is pure kernel arithmetic feeding the gravity side.
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