e_311331
plain-language theorem explainer
For the six-index tuple (3,1,1,3,3,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts certifying the 4D Regge exact-midpoint M2TT kernel cite this as one atomic decide cell in chunk 13. The proof is a single kernel decision on concrete integers.
Claim. For indices $a=3,b=1,c=1,d=3,i=3,j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table entry: $m_2^{\mathrm{num}}(3,1,1,3,3,1)=8\,Z_{\mathrm{explicit}}(3,1,1,3,3,1)$.
background
The module certifies, cell by cell, that the 4D Regge exact-midpoint M2TT numerator matches eight times a sparse explicit integer table. Local setting (module doc): m2Num = 8·explicitZ, chunk 13, among 256 kernel decides.
m2Num folds couplingZList, accumulating contrib at a fixed six-index in Fin 4. explicitZ is the matching closed form: a pattern match on Fin 4^6 returning small integers (e.g. 4, -2, and defaults). Upstream, both live in the KernelCert module as the two sides of the identity being checked.
Chunk theorems such as this one pin one concrete multi-index so the assembler can discharge the universal statement by fin_cases without re-evaluating the fold.
proof idea
One-line proof by decide. Lean reduces both sides at the concrete Fin 4 values (3,1,1,3,3,1): the left-hand fold of contrib over couplingZList and the right-hand 8 * explicitZ table lookup become ground integers, and the kernel checks equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states forall a b c d i j, m2Num = 8 * explicitZ, by exhaustive fin_cases on Fin 4; each case lands on a chunk cell such as this one. That universal identity is the algebraic core of the 4D Regge exact-midpoint M2TT kernel certificate in the Gravity analysis stack.
In the broader Recognition framework this is scaffolding for discrete curvature bookkeeping (Regge-type edge and deficit data) rather than a T0–T8 forcing step; it keeps the midpoint M2 numerator on a fully explicit integer footing so later continuum or continuum-limit claims can cite a machine-checked table identity instead of an opaque sum.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.