e_120203
plain-language theorem explainer
For the six-index tuple (1,2,0,2,0,3) on Fin 4, the midpoint Regge numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite this as one atomic decide-cell in the 4D M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}1$, $b{=}2$, $c{=}0$, $d{=}2$, $i{=}0$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The comparison target is an explicit piecewise integer table $Z$ (explicitZ) that records the closed-form values needed for the M2TT identity.
The module is chunk 6 of a 256-cell decide grid that discharges $m_2^{\mathrm{num}}=8Z$ pointwise. Each cell fixes one concrete six-tuple; the present cell is $(1,2,0,2,0,3)$. Upstream, both kernels live in the KernelCert module: $m_2^{\mathrm{num}}$ as a fold over couplingZList, and $Z$ as a pattern-matched Int table.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides at the concrete Fin-4 indices $(1,2,0,2,0,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
This cell feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over Fin 4. That global identity is the algebraic backbone of the exact midpoint M2TT relation in the 4D Regge gravity analysis.
Within Recognition Science gravity work, such exact discrete kernel identities underwrite continuum limits and curvature bookkeeping on the recognition lattice. The chunking into 256 decide theorems keeps each certificate tiny and machine-checkable while the assembler rebuilds the full quantified claim.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.