e_001022
plain-language theorem explainer
For the six-index slot (0,0,1,0,2,2) on Fin 4, the folded Regge midpoint numerator m2Num equals eight times the explicit kernel table explicitZ. Gravity analysts cite this as one atomic case in the 256-way kernel certification that m2Num = 8·explicitZ. The proof is a single decide, discharging the integer equality by computation.
Claim. At multi-index $(a,b,c,d,i,j)=(0,0,1,0,2,2)$ with each index in $\{0,1,2,3\}$, 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 layer that certifies an exact midpoint identity for a 4D Regge-type coupling. The local setting is the chunk-0 slice of a 256-case kernel: every tuple in $(\mathrm{Fin},4)^6$ must satisfy $m_2^{\mathrm{num}}=8\cdot Z$ by direct decision.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each list entry at the six indices. The explicit kernel $Z$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by a pattern-match table (e.g. $Z(0,0,1,1,2,2)=4$, $Z(0,0,1,2,1,2)=-2$, and similarly for the remaining nonzero slots).
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel-cert module; no analytic hypothesis is left open. The present declaration pins one concrete six-tuple in that table.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold for $m_2^{\mathrm{num}}$ at $(0,0,1,0,2,2)$ and eight times the matched $Z$ entry) and closes the equality in the integer decidable instance. No lemmas are invoked beyond the elaborator’s reduction of the two definitions.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ by exhausting $\mathrm{Fin},4$ on each coordinate. This chunk theorem is one of the 256 atomic decides that make that exhaustion honest rather than axiomatic.
In the broader Recognition gravity stack, the identity underwrites exact midpoint evaluations of the Regge-type coupling that feeds continuum limits and effective Newtonian kernels. It is bookkeeping infrastructure, not a forcing-chain landmark (T5–T8), but without the certified numerator–kernel match the later continuum and phenomenology layers would rest on an unchecked finite table.
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