e_203231
plain-language theorem explainer
For the six-index tuple (2,0,3,2,3,1) in (Fin 4)^6, the folded M2 numerator equals eight times the explicit integer Z-table entry. Gravity analysts assembling the Regge exact midpoint TT identity in 4D cite this as one of 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. Let $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ be the fold of coupling contributions over the fixed coupling list, and let $Z(a,b,c,d,i,j)$ be the explicit integer table on $(\mathrm{Fin}\,4)^6$. Then $m_2^{\mathrm{num}}(2,0,3,2,3,1)=8\,Z(2,0,3,2,3,1)$.
background
This module is chunk 8 of a 256-cell kernel certification that the midpoint M2 numerator equals eight times an explicit integer table on six Fin-4 indices. The local setting is discrete Regge-style gravity analysis in 4D: one checks a closed-form TT midpoint identity by exhausting the finite index cube.
Upstream, m2Num is defined by folding a fixed coupling list and summing a contribution functional at each tuple; explicitZ is a pattern-matched Int table on the same six indices (sample cells include values 4, -2, and so on). The identity claimed here is the cell at (2,0,3,2,3,1).
The full universal statement is assembled downstream by nested fin_cases over all six coordinates, each leaf discharging one such cell theorem.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed, so the kernel decision procedure closes the equality with no further lemmas.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states that for every $(a,b,c,d,i,j)$ in $(\mathrm{Fin},4)^6$ one has $m_2^{\mathrm{num}}=8,Z$. That assembly is the bridge from the explicit Z-table to the Regge exact midpoint M2/TT identity certification in 4D.
Within Recognition gravity analysis this is pure finite bookkeeping: it locks one cell of the discrete curvature/coupling kernel so the midpoint identity can be quoted without residual case splits. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi-ladder; those enter only if the certified identity is later tied to continuum or RS-native constants.
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