e_323232
plain-language theorem explainer
At multi-index (3,2,3,2,3,2) the folded Regge coupling numerator equals eight times the explicit integer kernel. Gravity analysts cite it as one of the 256 atomic cases behind the global m2Num = 8·explicitZ identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3,b=2,c=3,d=2,i=3,j=2$ in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 14 of a 256-way case split establishing $m_2^{\mathrm{num}}=8\cdot Z$ on the 4D Regge midpoint kernel. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels ${0,1,2,3}$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each local contribution at the six indices. The comparison target $Z$ (explicitZ) is a closed-form integer table on the same six indices, with sparse nonzero entries such as $4$ on matched pairs and $-2$ on crossed pairs.
The local setting is pure integer arithmetic certification of a discrete gravity identity, not continuum GR. Upstream supplies only the two definitions being compared; no analytic lemma is required beyond decidable equality of Int.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete hex-tuple $(3,2,3,2,3,2)$ to closed integers (the fold of contrib versus $8$ times the matching explicitZ clause) and checks equality in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles all $4^6=4096$ index tuples by nested fin_cases; each leaf is one of these e_* decide facts (here chunk 14). That global identity is the algebraic certificate that the midpoint Regge $M_2$ TT coupling matches eight times the explicit kernel table used downstream in 4D gravity analysis.
Within Recognition Science this sits in the discrete-gravity layer that supports continuum limits and effective Newtonian structure; it does not itself invoke the T0–T8 forcing chain, $\varphi$, or the eight-tick octave. It closes a pure bookkeeping obligation so later curvature and mass-ladder arguments can treat the kernel as a single closed form rather than a fold.
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