e_203012
plain-language theorem explainer
Pointwise identity: the summed coupling numerator at multi-index (2,0,3,0,1,2) equals eight times the explicit integer kernel at that same index. One of 256 kernel cells in chunk 8 of the 4D Regge midpoint M2TT numerator certification. Downstream assembly cites it when discharging the universal equality by exhaustive fin_cases. Proof is a single kernel decide.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,0,1,2)$ 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 exact-midpoint analysis, the numerator of the M2TT identity is assembled by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and $m_2^{\mathrm{num}}$ is that total. The companion map $Z$ is an explicit piecewise-integer kernel on the same six indices (sample values include $4$, $-2$, and so on for distinguished patterns).
The local module is chunk 8 of a 256-cell decide grid whose sole job is to certify $m_2^{\mathrm{num}}=8Z$ at every multi-index. The present cell fixes the concrete tuple $(2,0,3,0,1,2)$. Upstream definitions supply both the fold and the closed-form table; no analytic expansion is needed beyond evaluating those defs.
proof idea
One-line kernel proof: decide evaluates both sides of the integer equality after unfolding the fold that defines the numerator and the pattern-match that defines the explicit kernel at $(2,0,3,0,1,2)$. No lemmas beyond the two upstream defs are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, each leaf a chunk decide such as this one. That universal equality is the certified numerator half of the 4D Regge exact-midpoint M2TT identity in the Gravity analysis stack. Within Recognition Science it is bookkeeping infrastructure for the discrete curvature/coupling side of the gravity sector, not a forcing-chain (T0–T8) step; it closes a finite computational obligation so later geometric identities can quote a clean closed form.
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