e_230321
plain-language theorem explainer
For the six-index slot (2,3,0,3,2,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these 256 pointwise identities when assembling the global Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(a,b,c,d,i,j)=(2,3,0,3,2,1)$ equals $8$ times the explicit integer kernel at the same indices: $N(2,3,0,3,2,1)=8\,Z(2,3,0,3,2,1)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2–TT identity certification. The local goal, stated in the module header, is to prove pointwise that the folded numerator equals eight times an explicit integer table, across a 256-case kernel on six Fin-4 indices.
The numerator $N=m2Num$ is defined by folding a fixed coupling list: start from $0$ and add a contribution term for each coupling triple at the six indices. The comparison table $Z=explicitZ$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of integer values (e.g. $4$, $-2$, and related entries on matched index patterns).
Chunk 11 isolates one concrete sextuple so the global assembly can case-split without re-evaluating the whole fold in one tactic.
proof idea
One-line proof by decide. Both sides are closed integer expressions once the six Fin-4 arguments are literals: the left side reduces by unfolding the fold over couplingZList and summing contributions; the right side multiplies the looked-up explicitZ entry by $8$. Kernel decision discharges the resulting numeral equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m2Num,a,b,c,d,i,j=8\cdot explicitZ,a,b,c,d,i,j$ by exhaustive fin_cases on all six indices. Each chunk such as this one supplies one leaf of that case tree.
In the gravity analysis stack this identity is the algebraic backbone of the Regge exact-midpoint M2–TT relation in 4D: matching the folded coupling numerator to a simple multiple of an explicit kernel lets later certificates treat the midpoint identity as a finite integer check rather than a symbolic tensor expansion. It is infrastructure inside the RS gravity lane, not a forcing-chain (T0–T8) step, but it is required for trustworthy discrete curvature bookkeeping on the eight-tick / $D=3$ spacetime skeleton.
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