e_312230
plain-language theorem explainer
For the six-index tuple (3,1,2,2,3,0) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge exact-midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}2$, $d{=}2$, $i{=}3$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the numerator m2Num is defined by folding a fixed coupling list: it sums contrib terms over couplingZList for six Fin 4 indices. The companion explicitZ is a pattern-matched integer table on the same six indices (sample cells include 4, -2, and so on).
The local module is chunk 13 of a 256-cell kernel certification that m2Num = 8 · explicitZ pointwise. The parent assembly theorem ranges over all Fin 4 sextuples by fin_cases and dispatches each cell to a decide lemma of this form.
Upstream, only the two definitions are needed: the fold that builds m2Num and the closed-form table explicitZ. No analytic identity is invoked at this layer; the equality is pure integer arithmetic on one concrete sextuple.
proof idea
One-line tactic proof: by decide. Lean evaluates both sides as concrete Int values (the fold of contrib over couplingZList versus eight times the matching explicitZ clause) and closes by kernel computation. No lemmas beyond the two definitions are applied.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every Fin 4 sextuple and is proved by exhaustive fin_cases dispatching to cells such as this one. That global equality is the certified numerator half of the Regge exact-midpoint M2TT identity in 4D gravity analysis inside the Recognition Science stack.
Within the broader framework this sits in the gravity domain rather than the T0–T8 forcing chain; it is infrastructure that makes the discrete curvature/midpoint algebra machine-checkable. Chunk 13 is one of many parallel decide modules that together close the 256-cell kernel.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.