e_001200
plain-language theorem explainer
Pointwise identity: the folded M2 numerator coupling at multi-index (0,0,1,2,0,0) equals eight times the explicit integer kernel Z at the same index. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite it as one of 256 kernel cells. Proof is a single kernel decide on concrete Fin-4 data.
Claim. For indices $(a,b,c,d,i,j)=(0,0,1,2,0,0)$ in $(\mathrm{Fin}\,4)^6$, 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
This module sits in the Gravity analysis stack for the exact midpoint Regge identity in 4D. The local setting is the chunk-0 batch of 256 kernel cells that certify $m_2^{\mathrm{num}}=8\cdot Z$ pointwise by decision procedure.
The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: it sums contribution terms contrib over couplingZList at a six-tuple of $\mathrm{Fin},4$ indices. The explicit kernel $Z$ is a closed-form integer table on the same six-tuple (pattern-matched constants such as $4$, $-2$, and defaults).
Both objects live in the KernelCert module. The present cell fixes the multi-index $(0,0,1,2,0,0)$ and compares the two integers.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are closed, so the kernel decision procedure discharges equality with no lemmas and no case split in this file.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices. Each cell such as this one is a named decide fact the assembler (or the generated goals) can hit.
In the broader Recognition gravity stack, the identity is bookkeeping for the 4D Regge midpoint M2 TT kernel: it converts a folded coupling sum into an explicit sparse integer table, scaled by eight. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel arithmetic supporting the continuum gravity claims built on top.
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