e_311202
plain-language theorem explainer
For the six-index slot (3,1,1,2,0,2) in the 4D Regge midpoint kernel, the folded numerator equals eight times the closed-form integer table entry. Gravity analysts cite it as one atomic case in the exhaustive m2Num = 8·explicitZ identity. The proof is a single kernel decide on two concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the midpoint numerator at $(a,b,c,d,i,j)=(3,1,1,2,0,2)$ satisfies $m_2^{\mathrm{num}}(3,1,1,2,0,2)=8\,Z_{\mathrm{expl}}(3,1,1,2,0,2)$.
background
This module is chunk 13 of the 256-case kernel that certifies the exact midpoint identity $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on all six-tuples of $\mathrm{Fin},4$. The setting is 4D Regge calculus at the midpoint: discrete curvature couplings are reduced to integer arithmetic so the continuum TT-sector identity can be checked by finite casework.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating each term's contribution at those indices. The companion table $Z_{\mathrm{expl}}$ is an explicit pattern-matched integer function on the same six indices (sample values include $4$, $-2$, and so on). The claim is pure equality of two integers at one fixed multi-index.
proof idea
One-line proof: decide. Both sides reduce to concrete Int values (the fold for the numerator and the pattern match for the table), so the kernel discharges the equality with no lemmas and no case splits inside this declaration.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every six-tuple by nested fin_cases and invokes the chunk lemmas; this declaration is the witness for indices $(3,1,1,2,0,2)$. Closing all 256 decides yields the global algebraic identity used in the Regge-exact midpoint analysis of the 4D TT sector. It is bookkeeping infrastructure for the gravity side of the Recognition stack, not a forcing-chain step (T0–T8), but it underwrites the discrete curvature identities that later feed continuum limits.
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