e_201301
plain-language theorem explainer
For the six Fin-4 indices (2,0,1,3,0,1), the folded numerator m2Num equals eight times the closed-form kernel entry explicitZ. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one atomic case. The proof is a single kernel decide on the concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,3,0,1)$ in $\mathrm{Fin}\,4$, the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding coupling contributions equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2–TT identity certification. The local claim is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on a block of index sextuples, discharged by kernel decides.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of contrib over couplingZList, an integer accumulator of all coupling terms at those six Fin-4 slots. The companion explicitZ is a pattern-matched closed form on the same six indices, returning small integers such as $4$, $-2$, and so on for the nonzero patterns.
The ambient setting is discrete gravity: verifying that the midpoint-evaluated second-moment numerator matches eight times an explicit algebraic kernel, index by index, before the cases are reassembled.
proof idea
One-line proof by decide. Both sides are concrete Int values once the six Fin-4 arguments are fixed: the left folds a finite coupling list, the right is a single pattern match in explicitZ. Lean’s kernel decision procedure evaluates the equality and closes the goal with no lemmas or rewriting.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal quantifier by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies one concrete case in that case-split tree.
In the Recognition gravity stack this identity is bookkeeping for the 4D Regge midpoint M2–TT kernel: once every index sextuple matches, the folded numerator may be replaced by the closed form everywhere downstream. It does not itself invoke the forcing chain (T0–T8) or the RCL; it is pure discrete-tensor arithmetic supporting the continuum-limit gravity side.
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