e_201103
plain-language theorem explainer
Single kernel case: the folded coupling numerator at multi-index (2,0,1,1,0,3) equals eight times the tabulated explicit integer. Gravity analysts assembling the 4D Regge midpoint M2TT numerator identity cite it as one of 256 exhaustive Fin-4 cells. Proof is a pure `decide` on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,1,0,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 8 of a 256-cell kernel certifying that the 4D Regge-exact midpoint numerator equals eight times a closed-form integer table. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a local contribution at each coupling triple for the six indices. The table $Z$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (typical values $\pm 2,,4$, etc.).
The local claim is the equality at one fixed sextuple. The parent assembly theorem states the same identity for every sextuple.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern match on the explicit table), and the kernel decides equality.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting all $\mathrm{Fin},4$ cases. That global identity is the algebraic backbone of the Regge-exact midpoint M2TT numerator certification in the Gravity analysis stack.
Within Recognition Science gravity work, such exact discrete identities pin the curvature/coupling bookkeeping before continuum or phenomenological limits are taken. This cell is pure scaffolding closure: one of 256 decides that together discharge the chunked equality.
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