e_021330
plain-language theorem explainer
For the six-index slot (0,2,1,3,3,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form explicitZ value at that slot. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. At multi-index $(a,b,c,d,i,j)=(0,2,1,3,3,0)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit integer table entry: $m_2^{\mathrm{num}}(0,2,1,3,3,0)=8\,Z_{\mathrm{ex}}(0,2,1,3,3,0)$.
background
This module sits in the 4D Regge midpoint analysis for the M2–TT identity. 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 Fin-4 indices. The companion table $Z_{\mathrm{ex}}$ is an explicit pattern-match on those same indices, returning small integers (e.g. $4$, $-2$, and defaults).
The local goal, stated in the module header, is to certify $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{ex}}$ cell by cell across a 256-way kernel split (chunk 2). Each cell is an equality of concrete integers once the indices are fixed, so the kernel can discharge it without symbolic algebra.
proof idea
One-line kernel proof: by decide. Both sides reduce to closed integers for the fixed indices $(0,2,1,3,3,0)$: the left via the fold definition of $m_2^{\mathrm{num}}$, the right via the pattern table $Z_{\mathrm{ex}}$ scaled by 8. No lemmas are invoked beyond definitional reduction and integer decision.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 cell by nested fin_cases and consumes this chunk equality as one branch. That universal identity is the algebraic backbone of the Regge exact-midpoint M2–TT certification in 4D gravity analysis inside the monolith. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure discrete tensor bookkeeping needed before continuum or phenomenological gravity claims.
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