e_202031
plain-language theorem explainer
Case check that the midpoint Regge mass-squared numerator at multi-index (2,0,2,0,3,1) equals eight times the explicit Z-coupling there. Gravity analysts cite it as one cell of the 256-kernel partition of the 4D midpoint M2–TT identity. The proof is a single kernel decide on two concrete integers.
Claim. At the multi-index $(a,b,c,d,i,j)=(2,0,2,0,3,1)$ with each coordinate in $\{0,1,2,3\}$, the integer midpoint mass-squared numerator equals eight times the explicit $Z$-coupling: $N_{\mathrm{m2}}(2,0,2,0,3,1)=8\,Z(2,0,2,0,3,1)$.
background
This module is chunk 8 of a 256-way partition of the discrete kernel for the 4D midpoint Regge identity $N_{\mathrm{m2}}=8Z$. Indices run over $\mathrm{Fin},4$, i.e. the four coordinate directions of a 4-simplex edge labeling.
The numerator $N_{\mathrm{m2}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at those six indices. The explicit coupling $Z$ is a closed-form integer table on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and $0$ off the listed patterns).
The local claim is only the numerical instance of $N_{\mathrm{m2}}=8Z$ at one concrete six-tuple. Upstream, both sides are pure definitions in the kernel certificate module; no analytic continuum limit is taken here.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted into the fold definition of the numerator and the pattern-match table for explicit $Z$. Lean’s kernel decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
The parent assembly theorem states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N_{\mathrm{m2}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$, proved by exhausting all $4^6$ index tuples. This declaration discharges the single cell $(2,0,2,0,3,1)$ inside chunk 8 of that exhaustion.
In the Recognition gravity stack the identity is bookkeeping for the exact midpoint Regge mass-squared / TT sector in 4D: once every kernel cell matches, the discrete curvature coupling is certified before any continuum or phenomenological reading. It does not itself invoke the forcing chain (T5–T8), $\phi$-ladder masses, or the RCL; it is pure integer kernel hygiene supporting those later gravity claims.
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