e_033221
plain-language theorem explainer
At multi-index (0,3,3,2,2,1), the Regge numerator coupling equals eight times the explicit kernel table entry. Gravity analysts certifying the 4D midpoint M₂TT identity cite this as one atomic kernel decide among the chunk-3 block. Proof is a single computational `decide` on the two integer definitions.
Claim. The integer numerator coupling at indices $(0,3,3,2,2,1)$ equals eight times the explicit kernel value at those indices: $m_2(0,3,3,2,2,1)=8\,Z(0,3,3,2,2,1)$.
background
In the 4D Regge exact-midpoint stack, the numerator coupling is an integer obtained by folding a contribution map over a fixed coupling list, one value per 6-tuple of indices in $\mathrm{Fin},4$. The companion table explicitZ is a pattern-matched closed form on the same index domain (entries such as $4$, $-2$, etc. at listed patterns).
This module is chunk 3 of the pointwise certificate that the numerator equals eight times the explicit table. The ambient goal is the full identity on all $4^6$ index tuples; the chunk discharges a 256-case block by kernel decides, as stated in the module doc.
proof idea
One-line wrapper: decide. Both sides reduce by unfolding the fold definition of the numerator and the pattern match of the explicit table to concrete integers, which the kernel compares for equality. No intermediate lemmas are invoked.
why it matters
Supplies one case to the assembly theorem that states the numerator equals eight times the explicit kernel for every 6-tuple of $\mathrm{Fin},4$ indices (proved there by exhaustive fin_cases). That global equality is the algebraic backbone of the 4D Regge midpoint $M_2TT$ identity certificate in the Gravity analysis layer. It secures discrete curvature couplings on the Regge side; it does not itself touch the forcing chain (T0–T8), the $J$-cost, or continuum limits.
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