e_033133
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (0,3,3,1,3,3) equals eight times the explicit integer table at that same index. Gravity analysts cite it when discharging one cell of the 4D midpoint M2TT kernel. The proof is a pure kernel decision on concrete integers.
Claim. For indices $a{=}0,\,b{=}3,\,c{=}3,\,d{=}1,\,i{=}3,\,j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(0,3,3,1,3,3)=8\,Z(0,3,3,1,3,3)$.
background
In the 4D Regge midpoint analysis the numerator $N(a,b,c,d,i,j)$ is obtained by folding a fixed coupling list: start at $0$ and add each term's contribution at the six $\mathrm{Fin},4$ indices. The companion table $Z$ is an explicit case-split integer function on the same six indices (sample values include $4$, $-2$, and so on).
This module is chunk 3 of the 256-cell kernel that asserts $N=8Z$ at every multi-index. The local setting is purely combinatorial: no continuum limit, no metric ansatz beyond the discrete index set. Upstream, $N$ and $Z$ are the two definitions being compared cell by cell.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(0,3,3,1,3,3)$ and checks integer equality. No lemmas are invoked beyond the reducibility of the fold defining $N$ and the case table defining $Z$.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhausting all $\mathrm{Fin},4$ cases. That global identity is the algebraic backbone of the exact midpoint M2TT certificate in the gravity analysis stack. Within Recognition Science it sits in the discrete-geometry layer that underwrites continuum gravity limits; it does not itself touch the T0–T8 forcing chain, $\varphi$, or the eight-tick octave, but it clears a necessary kernel obligation before those continuum claims can be certified.
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