e_010322
plain-language theorem explainer
For the six-index slot (0,1,0,3,2,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,3,2,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8\,Z_{\mathrm{explicit}}(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-case kernel certification that the 4D Regge exact-midpoint numerator equals eight times an explicit integer kernel. Indices run over $\mathrm{Fin},4$, matching spacetime coordinate slots in the discrete curvature bookkeeping.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the six indices. The comparison target $Z_{\mathrm{explicit}}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample values include $\pm 2,4$ on diagonal and off-diagonal patterns).
Local goal: discharge one concrete equality so the assembler can rebuild the universal statement by exhaustive fin_cases.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the fixed indices $(0,1,0,3,2,2)$ to concrete integers (the fold of contrib versus $8$ times the matched explicitZ clause) and checks equality in the kernel. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts the identity for every six-tuple in $\mathrm{Fin},4$ by nested fin_cases. That universal equality is the certified algebraic core of the 4D Regge exact-midpoint M2TT identity in the Gravity analysis stack.
In Recognition Science gravity work, such kernel identities underwrite discrete curvature and coupling normalizations on the lattice side before continuum or phenomenological limits. This chunk does not itself touch T0–T8 or the J-cost forcing chain; it is pure integer certification supporting the Regge midpoint bookkeeping.
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