e_330223
plain-language theorem explainer
For the six-index slot (3,3,0,2,2,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2 TT identity. The proof is a pure kernel decide on concrete integers.
Claim. For indices $a=b=3$, $c=0$, $d=2$, $i=2$, $j=3$ 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 $Z(a,b,c,d,i,j)$.
background
This module is one of the 256-case decide chunks that certify the algebraic identity $m_2^{\mathrm{num}}=8\cdot Z$ on the 4D Regge midpoint TT kernel. The ambient setting is discrete gravity analysis: six indices in $\mathrm{Fin},4$ label the tensor slots of a midpoint contribution.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison object $Z$ is an explicit integer-valued pattern match on the six indices (typical values $\pm 2,,4$, and zero off the listed patterns).
The full identity is assembled downstream by exhaustive fin_cases over all six indices; each chunk such as this one discharges a single concrete sextuple.
proof idea
One-line decide proof. Both sides reduce to concrete Int values for the fixed indices $(3,3,0,2,2,3)$: the left by evaluating the fold that defines the numerator, the right by evaluating the pattern-match table for $Z$ and multiplying by 8. No lemmas beyond kernel reduction are required.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by casing through all 4096 combinations and invoking the matching chunk. That global equality is the certified bridge between the folded coupling definition of the M2 numerator and the closed-form kernel used in the Regge exact midpoint TT analysis.
Within Recognition gravity work, such kernel identities keep the discrete curvature bookkeeping exact rather than approximate, so later continuum or phenomenological claims rest on machine-checked integer arithmetic rather than hand tables. This declaration is pure scaffolding glue: one of many identical decide atoms, not a conceptual step by itself.
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