e_203203
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (2,0,3,2,0,3) equals eight times the explicit integer table at that same index. Gravity analysts assembling the 4D Regge midpoint M2TT numerator identity cite this chunk. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(2,0,3,2,0,3)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer table entry: $m_2^{\mathrm{num}}(2,0,3,2,0,3)=8\,Z_{\mathrm{expl}}(2,0,3,2,0,3)$.
background
This module is chunk 8 of a 256-case kernel certification that the 4D Regge midpoint M2TT numerator equals eight times a closed-form integer table. Indices run over $\mathbb{F}_4$ (four discrete directions).
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The table $Z_{\mathrm{expl}}$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on).
The local claim is one concrete instance of $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$. Downstream assembly will quantify over all six indices by exhaustive fin_cases.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are fixed: the fold that defines the numerator evaluates fully, and the explicit table matches on that pattern. Lean’s decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and discharges the universal claim by six nested fin_cases over $\mathrm{Fin},4$ (256 points). This declaration is one of those 256 kernel facts (chunk 8).
In the Recognition gravity stack, the identity certifies that the discrete Regge midpoint M2TT numerator collapses to a sparse explicit integer kernel, a prerequisite for exact 4D midpoint identities used in the discrete gravity analysis. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure algebraic bookkeeping inside the gravity analysis layer.
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