e_201302
plain-language theorem explainer
For the six Fin-4 indices (2,0,1,3,0,2), the Regge midpoint numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts certifying the 4D M2TT midpoint identity cite this as one kernel cell in chunk 8 of the 256-case table. The proof is a single `decide` that evaluates both integer sides.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,3,0,2)$ with each index in $\{0,1,2,3\}$, the midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module sits in the 4D Regge exact-midpoint analysis for the M2TT identity. The numerator m2Num is the integer obtained by folding a fixed coupling list: each term contributes via a local contrib at six Fin-4 indices, then summing. The companion explicitZ is a piecewise integer table on the same six indices (typical values 4, -2, and so on), intended as the closed form of that sum divided by 8.
The local setting is chunk 8 of a 256-cell kernel: every ordered 6-tuple in $(\mathrm{Fin},4)^6$ is a separate equality m2Num = 8 · explicitZ. Upstream, both sides are pure definitions in the KernelCert module; no analytic lemma is required beyond evaluating the fold and the match.
proof idea
One-line kernel proof: by decide. Lean reduces m2Num 2 0 1 3 0 2 by running the fold over couplingZList and reduces explicitZ 2 0 1 3 0 2 by matching the six-index pattern, then checks the integer equality $n = 8m$ by computation. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$. That forall is the algebraic backbone of the Regge exact-midpoint M2TT certification in 4D gravity: once every cell (including this one) matches, the numerator is identically eight times the explicit kernel, so continuum or continuum-limit arguments can replace the fold by the closed table.
Within Recognition Science gravity work this is bookkeeping, not a forcing-chain step (T0–T8), but it is required infrastructure for any claim that the discrete midpoint curvature weights match the predicted integer pattern. Downstream assembly simply case-splits all Fin-4 indices and discharges each goal by the corresponding chunk theorem.
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