e_123103
plain-language theorem explainer
At Fin-4 indices (1,2,3,1,0,3), the folded coupling numerator equals eight times the explicit integer Z-kernel. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one pointwise certificate in a 256-case chunk. Proof is a single kernel decide on the two integer sides.
Claim. For $a=1$, $b=2$, $c=3$, $d=1$, $i=0$, $j=3$ 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 value $Z(a,b,c,d,i,j)$.
background
This module is chunk 6 of the Gravity analysis stack that certifies the Regge exact-midpoint M2TT identity in 4D. The local claim family is pointwise equality of a folded numerator against an explicit integer table, scaled by eight.
The numerator $m_2^{\mathrm{num}}$ is defined by folding couplingZList and summing a contribution at each six-index slot in $\mathrm{Fin},4$. The comparison target is explicitZ, a closed-form $\mathrm{Int}$-valued pattern match on the same six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, and similar small integers).
The module doc states the program directly: establish $m_2^{\mathrm{num}} = 8\cdot Z$ by 256 kernel decides, of which this declaration is one concrete sextuple.
proof idea
One-line wrapper proved by decide. Both sides are closed integer expressions at fixed Fin 4 indices, so the kernel evaluates $m_2^{\mathrm{num}}(1,2,3,1,0,3)$ and $8\cdot Z(1,2,3,1,0,3)$ and checks definitional equality. No lemmas are invoked beyond the two upstream defs.
why it matters
Feeds the assemble theorem m2Num_eq_eight_explicitZ, which asserts the same equality for every sextuple in $(\mathrm{Fin},4)^6$ and discharges the universal claim by exhaustive fin_cases. Without the pointwise chunk certificates, the assemble step has nothing to quote at each concrete index.
In the broader Recognition gravity stack this is bookkeeping for the 4D Regge midpoint mass-squared numerator identity: once numerator equals eight times the explicit kernel everywhere, later layers can treat $Z$ as the closed form. It does not itself touch the forcing chain (T0–T8), RCL, or the $\varphi$-ladder; it is pure discrete certificate infrastructure under the Gravity domain.
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