e_223012
plain-language theorem explainer
For index tuple (2,2,3,0,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on the concrete integers.
Claim. For the six Fin-4 indices $a{=}2,b{=}2,c{=}3,d{=}0,i{=}1,j{=}2$, the integer $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding the coupling list equals $8$ times the explicit kernel entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator side m2Num folds a fixed coupling list, accumulating a contribution at each triple; the closed form explicitZ is a sparse case table of small integers (entries such as 4, -2, and so on).
The module is chunk 10 of a 256-cell partition of the Fin-4^6 domain. Each cell asserts the same scalar identity m2Num = 8 · explicitZ at one concrete index tuple. The factor eight is the global normalization that makes the folded sum match the hand-written kernel.
Upstream, both sides are pure definitions in the KernelCert module: m2Num is the fold, explicitZ is the pattern-matched table. No analytic lemma is required at this layer; only evaluation of those definitions at fixed indices.
proof idea
One-line proof by decide. Lean reduces both sides of the equality at the concrete Fin-4 sextuple (2,2,3,0,1,2) to bare integers and checks equality in the kernel. No lemmas are invoked beyond definitional unfolding of m2Num and explicitZ.
why it matters
This cell is one of the 256 atomic facts consumed by m2Num_eq_eight_explicitZ in the Assemble module. That parent theorem states the identity for every Fin-4 sextuple by exhaustive fin_cases, discharging each branch with a chunk theorem of this form.
In the broader Gravity track, the equality certifies that the folded Regge midpoint numerator agrees with the explicit 4D kernel used in the M2TT identity. Without the cell-by-cell match, the global assembly cannot close. The result is pure discrete bookkeeping; it does not itself invoke the Recognition forcing chain (T0–T8), but it sits inside the gravity-analysis stack that those foundations eventually feed.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.