e_003013
plain-language theorem explainer
For the six-index slot (0,0,3,0,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. 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. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,0,3,0,1,3)=8\,Z_{\mathrm{explicit}}(0,0,3,0,1,3)$.
background
In the 4D Regge midpoint analysis, the M2–TT identity is checked by comparing a folded numerator against an explicit integer table. The numerator m2Num is defined by folding contrib over couplingZList at a six-tuple of Fin-4 indices; explicitZ is the matching closed-form Int-valued table on those same indices (sparse nonzero pattern, e.g. $\pm 2,4$ on selected pairs).
This module is chunk 0 of the 256-cell kernel: each cell asserts m2Num = 8 · explicitZ at one fixed multi-index. The local setting is pure integer arithmetic on Fin 4, not continuum GR.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern-match table), and the kernel checks equality.
why it matters
Parent theorem m2Num_eq_eight_explicitZ quantifies over all $a,b,c,d,i,j : \mathrm{Fin},4$ and discharges the goal by nested fin_cases, invoking one cell theorem per tuple. This declaration is the cell for $(0,0,3,0,1,3)$. Together the cells certify that the folded Regge coupling numerator is exactly eight times the explicit table, the algebraic core of the midpoint M2–TT identity in the gravity analysis stack. No continuum limit or RS forcing step (T0–T8) is claimed here; it is discrete certificate scaffolding for that identity.
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