e_230333
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at multi-index (2,3,0,3,3,3) equals eight times the explicit integer table entry. Gravity analysts assembling the exact 4D midpoint M2TT identity cite this kernel cell. Proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(2,3,0,3,3,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $N(2,3,0,3,3,3)=8\,Z(2,3,0,3,3,3)$.
background
This module is chunk 11 of a 256-cell kernel certifying that the folded midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The setting is the exact 4D Regge midpoint M2TT identity in the gravity analysis stack.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The table $Z$ is an explicit pattern-matched integer function on six $\mathrm{Fin},4$ arguments (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$). The claim is the pointwise factor-of-eight match at one concrete multi-index.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices $(2,3,0,3,3,3)$ and checks integer equality. No algebraic lemmas are invoked; the kernel reduces the folded sum and the table lookup to numerals and compares them.
why it matters
Feeds the universal assembly theorem stating $\forall a,b,c,d,i,j,, N=8Z$, proved by exhaustive fin_cases over $(\mathrm{Fin},4)^6$. That assembly is the algebraic backbone of the Regge exact midpoint M2TT identity in 4D: the folded coupling numerator is replaced by a closed integer table times eight, clearing a large case-bash from later gravity derivations. Within Recognition gravity analysis this is pure kernel bookkeeping, not a forcing-chain step (T0–T8), but it is required scaffolding for exact midpoint identities used downstream in the discrete curvature/mass bookkeeping.
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