e_203033
plain-language theorem explainer
For the Fin-4 multi-index (2,0,3,0,3,3), the folded coupling numerator equals eight times the explicit integer table entry. Gravity analysts building the 4D Regge midpoint M2TT identity cite this as one kernel-decided cell among the chunk-8 cases. The proof is a single decide on concrete integers.
Claim. For indices $a=2$, $b=0$, $c=3$, $d=0$, $i=3$, $j=3$ in $\{0,1,2,3\}$, the coupling-fold numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module sits in the Gravity analysis stack that certifies an exact midpoint identity for the 4D Regge M2TT structure. The local goal, per the module header, is to check $m_2^{\mathrm{num}}=8\cdot Z$ on a 256-cell chunk by kernel decision.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six Fin-4 indices. The comparison value $Z$ is an explicit six-index integer table (pattern cases such as $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$). Both sides are pure Int once the indices are concrete.
Upstream, those two definitions live in the kernel-cert module; this chunk only instantiates one ordered sextuple.
proof idea
One-line proof by decide. With all six indices closed Fin-4 literals, both the fold that defines the numerator and the table lookup that defines $Z$ reduce to concrete integers; the kernel checks equality to $8Z$ by computation. No lemmas are invoked beyond the two definitions.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full pointwise identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting Fin-4 cases; this declaration is the leaf for the sextuple $(2,0,3,0,3,3)$ inside chunk 8. That global algebraic identity is the certified numerator form used in the Regge exact-midpoint M2TT 4D analysis. It is bookkeeping inside the gravity stack, not a forcing-chain step (T0–T8), but it hardens the discrete curvature side that later couples to RS units and ladder masses.
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