e_211321
plain-language theorem explainer
For the six-index tuple (2,1,1,3,2,1) on Fin 4, the Regge midpoint numerator equals eight times the explicit kernel value Z. Gravity analysts cite it as one concrete kernel check among the 4^6 cases that assemble into the global m2Num = 8·explicitZ identity. The proof is a single computational decide on two concrete integers.
Claim. For indices $a=2$, $b=1$, $c=1$, $d=3$, $i=2$, $j=1$ in $\{0,1,2,3\}$, the integer numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on six indices drawn from Fin 4 are compared pointwise. The numerator m2Num is defined by folding a fixed coupling list and summing contribution terms at the given multi-index. The companion map explicitZ is a closed-form pattern match returning small integers (e.g. 4, -2) on those same indices.
This module is chunk 9 of a split that discharges 256 concrete kernel equalities of the form m2Num = 8·explicitZ. The local setting is pure decidable integer arithmetic on the discrete index set; no continuum or curvature interpretation is invoked inside the chunk.
proof idea
One-line computational proof: decide evaluates both sides under the definitions of m2Num and explicitZ, reduces the fold and the pattern match to concrete Int numerals, and checks equality. No intermediate lemmas are applied.
why it matters
The pointwise certificate is consumed by m2Num_eq_eight_explicitZ, which asserts the same identity for every six-tuple in Fin 4 and is proved by exhaustive fin_cases. That global equality is a building block of the Regge exact-midpoint M2TT identity chain in the Gravity analysis layer. It supports discrete curvature bookkeeping used when matching continuum gravity limits in Recognition Science; it does not itself touch the T0–T8 forcing chain, the J-cost, or the phi ladder.
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