e_210310
plain-language theorem explainer
For the six Fin-4 indices (2,1,0,3,1,0), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on integer arithmetic.
Claim. With indices in $\{0,1,2,3\}$, the summed coupling numerator $m_2^{\mathrm{num}}(2,1,0,3,1,0)$ equals $8$ times the explicit integer kernel $Z(2,1,0,3,1,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices appear. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes via a local contrib and the fold starts at 0. The companion map explicitZ is a closed-form pattern match returning small integers (e.g. $\pm 2,,4$) on selected index patterns and (implicitly) zero elsewhere.
The module is chunk 9 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The local setting is pure finite enumeration over $(\mathrm{Fin},4)^6$, not continuum GR. Upstream, both maps live in the KernelCert module; this chunk only evaluates one concrete sextuple.
proof idea
One-line computational proof: by decide. Lean reduces both sides of the integer equality for the fixed indices $(2,1,0,3,1,0)$ and checks equality in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement by fin_cases on all six indices; each chunk cell such as this one discharges one branch. That identity is the algebraic core of the exact midpoint M2–TT relation in the 4D Regge analysis under Recognition Science gravity. It does not itself touch the forcing chain (T0–T8) or the J-cost, but it underwrites a discrete curvature identity used downstream in the gravity stack.
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