e_022011
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (0,2,2,0,1,1) equals eight times the explicit integer table at those same indices. Gravity analysts cite it as one kernel cell in the 4D Regge midpoint M2TT numerator certification. The proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,2,0,1,1)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator of the M2TT identity is assembled by folding a fixed coupling list: each term contributes an integer depending on six Fin 4 indices, and the fold sum is the numerator at that multi-index. Parallel to that fold sits an explicit integer table on the same six indices, recording the closed-form value expected after all cancellations.
This module is chunk 2 of a 256-cell kernel certification that the fold equals eight times the table entry at every multi-index. The local claim is only the single cell $(0,2,2,0,1,1)$. Upstream, the fold and the table are pure definitions; no analytic lemma is required beyond evaluating both sides at concrete indices.
proof idea
One-line kernel proof: decide evaluates both the fold-sum definition and the explicit table at the six concrete Fin 4 values and checks integer equality. No lemmas are invoked; the decision procedure discharges the ground equality directly.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive case split. That global equality is the certified numerator half of the 4D Regge exact-midpoint M2TT identity used in the gravity analysis stack. The chunking (256 kernel decides) keeps each cell small enough for the kernel while the assembler glues them into the universal statement. No Recognition forcing-chain landmark (T5–T8, RCL, phi) is touched here; the result is pure discrete tensor arithmetic supporting the continuum gravity side.
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