e_203013
plain-language theorem explainer
Pointwise identity: the folded midpoint numerator at multi-index (2,0,3,0,1,3) equals eight times the closed-form integer kernel at those same indices. Gravity analysts cite it as one cell of the 256-case kernel table that underwrites the exact Regge midpoint M2TT identity in 4D. The proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,0,1,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,0,3,0,1,3)=8\,Z(2,0,3,0,1,3)$.
background
This module is chunk 8 of a 256-cell kernel certification for the exact Regge midpoint identity in 4D. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list and summing a local contribution at each six-index slot in $(\mathbb{F}_4)^6$. The comparison object $Z=\mathrm{explicitZ}$ is a closed-form integer table on the same index domain (sample values include $4$, $-2$, and other small integers on the listed patterns).
The local claim is the scalar relation $N=8Z$ at one concrete multi-index. The module header states the global target: $N=8\cdot Z$ certified by 256 kernel decides. Upstream, both $N$ and $Z$ live in the kernel-cert module; this chunk only evaluates one slot.
proof idea
One-line computational proof: decide evaluates both sides at the fixed Fin-4 indices $(2,0,3,0,1,3)$ and checks integer equality. No algebraic lemmas are invoked; the fold defining the numerator and the pattern-match defining the explicit kernel are reduced to concrete Int values and compared.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ by exhausting all Fin-4 cases. That universal identity is the algebraic backbone of the exact midpoint M2TT kernel in 4D Regge analysis inside the Gravity domain. In the broader Recognition stack this is bookkeeping infrastructure for discrete curvature/coupling identities, not a forcing-chain step (T5–T8) or an RCL claim; it closes one cell of the certified kernel table the continuum or continuum-limit arguments rely on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.