e_222020
plain-language theorem explainer
For the Fin-4 multi-index (2,2,2,0,2,0), the folded coupling numerator equals eight times the explicit integer table entry. Gravity analysts certifying the 4D Regge midpoint M2 TT identity cite this as one concrete kernel case among 256. The proof is a single native decide on both integer sides.
Claim. For indices $a=b=c=2$, $d=0$, $i=2$, $j=0$ in $\{0,1,2,3\}$, the coupling-fold numerator equals eight times the explicit table value: $N(2,2,2,0,2,0)=8\,Z(2,2,2,0,2,0)$.
background
In the 4D Regge exact-midpoint analysis, the numerator m2Num accumulates integer contributions by folding a fixed coupling list through a local contrib map on six Fin-4 indices. The companion table explicitZ hard-codes the closed-form integers that this fold is expected to match, up to a universal factor of eight (snippet: pattern cases such as (0,0,1,1,2,2) ↦ 4 and (0,0,1,2,1,2) ↦ −2).
This module is chunk 10 of a 256-case kernel certification that the fold equals eight times the table pointwise. Each chunk discharges a block of concrete index sextuples by decidable integer arithmetic, feeding a later exhaustive assembly over all Fin-4 sextuples.
proof idea
One-line wrapper: by decide. Both sides are closed integer terms (the fold of contrib over couplingZList versus eight times the pattern-matched table entry at (2,2,2,0,2,0)). The kernel evaluates them and checks equality; no lemmas are invoked beyond decidable Int arithmetic.
why it matters
This case is consumed by the assembly theorem m2Num_eq_eight_explicitZ, which states ∀ a b c d i j, m2Num = 8 · explicitZ and proves it by nested fin_cases on all six Fin-4 indices, dispatching each concrete sextuple to a chunk lemma of this form. That pointwise identity is infrastructure for the Gravity.Analysis certification of the Regge exact-midpoint M2 TT relation in 4D. It does not itself touch the T0–T8 forcing chain, RCL, or phi-ladder mass formula; it is discrete-gravity bookkeeping that keeps the midpoint kernel exact.
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