e_101000
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at multi-index (1,0,1,0,0,0) equals eight times the explicit integer table entry. Gravity analysts assembling the 4D midpoint M2 TT identity cite this among the 256 kernel decides. Proof is a single computational decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(1,0,1,0,0,0)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(1,0,1,0,0,0)=8\,Z(1,0,1,0,0,0)$.
background
This module is chunk 4 of a 256-way kernel certification that the folded midpoint numerator equals eight times an explicit integer table on all of $(\mathbb{F}_4)^6$. The setting is the exact midpoint analysis of the 4D Regge M2 TT identity in the Gravity.Analysis stack.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: it sums a local contribution over that list at the six Fin-4 indices. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on the same six indices (sample values include $4$, $-2$, and further sparse entries). The claim is the scalar equality $N=8Z$ at one concrete multi-index.
Upstream, both $N$ and $Z$ live in the KernelCert module; this chunk only evaluates them at fixed arguments.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted into the fold definition of the numerator and the pattern match of the explicit table, so the equality is discharged by Lean's decision procedure on integer arithmetic. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8Z(a,b,c,d,i,j)$ by exhausting all Fin-4 cases. Each chunk lemma such as this one closes one case cell in that $4^6=4096$-free but kernel-structured grid (module doc: 256 kernel decides).
In the broader RS gravity stack this underwrites the exact midpoint form of the 4D Regge M2 TT identity, a discrete curvature/mass-side algebraic identity rather than a continuum GR derivation. It does not itself touch the forcing chain (T0–T8), RCL, or the $\varphi$-ladder mass formula; it is pure kernel arithmetic supporting the discrete gravity layer.
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