e_033223
plain-language theorem explainer
For the six-index tuple (0,3,3,2,2,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one cell of the 4D Regge midpoint M2TT identity certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}3$, $c{=}3$, $d{=}2$, $i{=}2$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(0,3,3,2,2,3)=8\,Z(0,3,3,2,2,3)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num is defined by folding a fixed coupling list and summing a local contribution at each six-tuple of face indices in Fin 4. The companion explicitZ is a sparse integer table on the same six indices (typical nonzero values are $\pm 2,\pm 4$).
The module is chunk 3 of a 256-cell kernel certificate that the folded numerator is identically eight times that table. The ambient claim is the pointwise identity $N=8Z$ on all of $(\mathrm{Fin},4)^6$, assembled later by exhaustive fin_cases.
This cell fixes the concrete arguments $(0,3,3,2,2,3)$. No continuous geometry enters: both sides are pure integers.
proof idea
One-line kernel proof: decide evaluates both sides of the integer equality after unfolding the fold that defines the numerator and the pattern-match table that defines the explicit kernel. No lemmas are invoked beyond decidable equality on Int.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ by casing on all six indices and dispatching each cell. That global identity is the algebraic core of the Regge-exact midpoint M2TT certificate in the gravity analysis stack.
Within Recognition Science this sits in the discrete gravity / Regge calculus layer that underwrites curvature bookkeeping on the eight-tick, $D=3$ scaffold (T7–T8). It does not itself touch the J-cost or the forcing chain; it is pure integer kernel hygiene needed before continuum limits or mass-ladder comparisons are trustworthy.
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