e_013012
plain-language theorem explainer
For the six Fin-4 indices (0,1,3,0,1,2), the integer numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts cite it when assembling the full midpoint M2 TT identity over the 4D index cube. The proof is a single kernel decide on both integer sides.
Claim. For indices $a=0$, $b=1$, $c=3$, $d=0$, $i=1$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,1,3,0,1,2)=8\,Z_{\mathrm{explicit}}(0,1,3,0,1,2)$.
background
In the Regge exact-midpoint analysis, the 4D TT mass-squared numerator is the integer-valued map m2Num on six Fin-4 indices. It is defined by folding a fixed coupling list and summing each term's contribution at those indices.
The companion map explicitZ is a sparse closed-form table on the same six indices, returning small integers (typically $\pm 2$ or $4$ on the nonzero support). The local module is chunk 1 of a 256-way kernel split whose sole job is to certify m2Num = 8 · explicitZ pointwise.
Upstream, both maps live in the kernel certificate module; this lemma only evaluates one concrete sextuple.
proof idea
One-line proof by decide. Lean reduces both the fold that defines m2Num at (0,1,3,0,1,2) and the matching explicitZ clause to concrete integers, then checks equality with 8 times the right-hand side. No lemmas beyond the two definitions are invoked.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every sextuple in (Fin 4)^6. Its proof case-splits on all six indices; each atomic goal is discharged by a chunk lemma of this form. Without the pointwise certificates the global assembly cannot close.
In the Recognition gravity stack this identity is the algebraic core of the exact midpoint M2 TT reduction in 4D. It sits downstream of the kernel certificate definitions and upstream of any continuum or continuum-limit statements that quote the closed form 8·explicitZ instead of the folded sum.
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