e_010200
plain-language theorem explainer
For the six-index slot (0,1,0,2,0,0) on Fin 4, the folded Regge coupling numerator equals eight times the explicit integer kernel. Gravity analysts cite it as one of 256 atomic equalities that certify the 4D midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices $a=0$, $b=1$, $c=0$, $d=2$, $i=0$, $j=0$ 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
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $N$ is obtained by folding a fixed coupling list: each term contributes an integer via a local contrib rule, and the fold starts at 0. The explicit kernel $Z$ is a sparse lookup table of small integers (entries such as $4$, $-2$, and so on) that encodes the closed-form side of the identity.
The module is chunk 1 of a 256-way case split: every sextuple in $(\mathrm{Fin},4)^6$ is checked separately so that the global statement $N=8Z$ can be assembled by exhaustive finitary case analysis. Local setting: certify $N=8Z$ pointwise by kernel computation.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices $(0,1,0,2,0,0)$. The left side runs the fold that defines the numerator; the right side multiplies the table lookup for $Z$ by 8. Both reduce to the same integer, so the equality is discharged by the decision procedure with no manual algebra.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and proves it by six nested fin_cases sweeps. Each atomic theorem such as this one closes one leaf of that tree. In the broader gravity stack, the identity is the algebraic core of the exact midpoint M2TT relation used in the Regge analysis; without the pointwise $N=8Z$ facts the 4D kernel certificate does not close. No Recognition forcing-chain landmark (T5–T8, RCL, phi) is directly at stake here: this is pure discrete-gravity bookkeeping inside the Gravity.Analysis layer.
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