e_213113
plain-language theorem explainer
For indices (2,1,3,1,1,3) the accumulated numerator m2Num equals eight times the tabulated kernel explicitZ. Analysts certifying the 4D Regge midpoint M2TT identity cite these atomic equalities. Proof is a single decide on the concrete integer values.
Claim. The integer value of the six-index numerator $m_2^{\mathrm{num}}(2,1,3,1,1,3)$ equals $8$ times the explicit kernel integer $Z(2,1,3,1,1,3)$.
background
In the 4D Regge-calculus midpoint analysis the numerator m2Num folds a fixed coupling list, adding a local contribution at every six-tuple of Fin-4 indices. The companion explicitZ is a sparse pattern-matched table that returns small integers (chiefly 4 or -2) on the support of that kernel and zero elsewhere.
The present module is chunk 9 of the exhaustive verification that m2Num = 8·explicitZ on all 4^6 tuples; each chunk discharges a block of cases by kernel decide. The two upstream definitions supply the exact left- and right-hand sides being compared.
proof idea
One-line decide. Both sides reduce to concrete Int literals: m2Num by folding the coupling list at the fixed indices, explicitZ by matching the same indices in its table. Integer equality is then decided by the kernel.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which exhausts all six Fin-4 arguments by fin_cases and invokes the chunk lemmas. That global identity is the algebraic core of the ReggeExactMidpointM2TTIdentity4D certificate. It sits inside the broader gravity-analysis layer that checks discrete curvature identities before continuum limits are taken; no Recognition-forcing landmark (T5–T8, RCL, phi) is directly invoked here.
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