e_202101
plain-language theorem explainer
Single kernel certificate: at multi-index (2,0,2,1,0,1) the folded Regge numerator m2Num equals eight times the explicit integer table explicitZ. Gravity analysts cite it as one of 256 concrete decides that assemble the full 4D midpoint identity. Proof is a one-line native decide on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(2,0,2,1,0,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit table value: $N(2,0,2,1,0,1)=8\,Z(2,0,2,1,0,1)$.
background
Module setting is the 4D Regge exact-midpoint M2TT identity: prove the folded numerator equals eight times a sparse explicit integer kernel on all sextuples in $(\mathrm{Fin},4)^6$. The module is chunk 8 of 256 kernel decides.
Upstream, $N=\mathrm{m2Num}$ is the fold of contrib over couplingZList at six $\mathrm{Fin},4$ indices, returning an Int. $Z=\mathrm{explicitZ}$ is the matching closed-form table (pattern-matched integers such as $4$, $-2$, and defaults). The claim is the pointwise equality $N=8Z$ at one fixed sextuple.
These point certificates are the raw fuel for the exhaustive assembly theorem that quantifies over every $a,b,c,d,i,j$.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are literals, so the kernel decision procedure discharges $N(2,0,2,1,0,1)=8Z(2,0,2,1,0,1)$ with no lemmas or rewriting.
why it matters
Feeds m2Num_eq_eight_explicitZ in the assemble module, which states $\forall a,b,c,d,i,j,,N=8Z$ and proves it by nested fin_cases over the six indices. Each chunk theorem such as this one closes one cell of that $4^6=4096$-free but kernel-tabulated grid (here framed as 256 decides).
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint discrete curvature / TT sector in 4D Regge analysis. It does not itself invoke the forcing chain (T5–T8) or the J-cost RCL; it is pure integer kernel hygiene supporting the continuum-limit and continuum-matching arguments built on top of the certified discrete operator.
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