e_313222
plain-language theorem explainer
Pointwise identity: the midpoint Regge m₂ numerator at Fin-4 indices (3,1,3,2,2,2) equals eight times the explicit Z kernel at those indices. Gravity analysts cite it as one cell of the 256-case kernel that assembles the global m2Num = 8·explicitZ theorem. The proof is a single decide on concrete integers.
Claim. For the index sextuple $(3,1,3,2,2,2)\in(\mathbb{F}_4)^6$, the midpoint $m_2$ numerator equals eight times the explicit $Z$ coupling value: $m_2^{\mathrm{num}}(3,1,3,2,2,2)=8\,Z_{\mathrm{expl}}(3,1,3,2,2,2)$.
background
This module is chunk 13 of a 256-cell kernel certification that the midpoint Regge $m_2$ numerator coincides with eight times an explicit integer kernel $Z$ on every sextuple of indices in $\mathbb{F}_4$. The ambient setting is 4D discrete gravity analysis: couplings are tabulated, then folded into a numerator.
Upstream, m2Num a b c d i j is the fold of a contribution function over couplingZList, returning an Int. The companion explicitZ is a total pattern-matched table $\mathbb{F}_4^6\to\mathbb{Z}$ (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$). The claim here is only the single cell with indices $(3,1,3,2,2,2)$.
Sibling theorems in the same chunk discharge neighboring cells; the assembly layer will case-split on all six Fin 4 arguments and invoke each cell.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once m2Num is unfolded to its fold over the coupling list and explicitZ is evaluated on the fixed sextuple $(3,1,3,2,2,2)$. Lean’s decision procedure checks integer equality; no algebraic lemma is required beyond definitional reduction.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}4,; m_2^{\mathrm{num}}=8,Z{\mathrm{expl}}$ and proves it by six nested fin_cases, discharging each cell by the corresponding chunk theorem. Without this cell the universal identity has a hole at $(3,1,3,2,2,2)$.
In the Recognition gravity stack the identity is bookkeeping infrastructure: it certifies that the tabulated midpoint numerator matches the closed-form kernel used downstream in Regge/TT analyses. It does not itself invoke the forcing chain (T5–T8), RCL, or the $\varphi$-ladder; it is pure discrete-kernel hygiene supporting those layers.
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