e_110322
plain-language theorem explainer
For the multi-index (1,1,0,3,2,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts certifying the 4D Regge midpoint M2 TT identity cite this as one kernel cell in the 256-point enumeration. The proof is a single decide on concrete integer arithmetic.
Claim. For indices $a=1$, $b=1$, $c=0$, $d=3$, $i=2$, $j=2$ in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(1,1,0,3,2,2)=8\,Z_{\mathrm{ex}}(1,1,0,3,2,2)$.
background
This module is chunk 5 of the pointwise certification that the 4D Regge midpoint M2 numerator equals eight times an explicit integer kernel on every sextuple of Fin-4 indices (256 cells). The local slogan is $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{ex}}$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those indices. The companion $Z_{\mathrm{ex}}$ is a total function Fin 4$^6\to\mathbb{Z}$ given by an exhaustive pattern match (e.g. $(0,0,1,1,2,2)\mapsto 4$, off-diagonal pairs $\mapsto -2$, and so on).
The surrounding analysis sits in the Gravity domain of the Recognition Science mirror: discrete Regge-calculus identities that underwrite continuum TT structure in $D=3$ spatial dimensions after the eight-tick forcing chain.
proof idea
One-line computational discharge: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted, so the kernel evaluates the fold defining the numerator against the matched clause of explicitZ and checks equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every index cell into the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$. That global equality is the algebraic backbone of the Regge exact-midpoint M2 TT identity in 4D. This declaration closes one concrete cell (the 1,1,0,3,2,2 slot) inside chunk 5 of the 256-kernel partition. Without the full pointwise match, the continuum TT reduction and the discrete-to-continuum gravity bridge remain uncertified. It does not itself touch T5–T8 or the J-cost; it is pure integer kernel bookkeeping supporting those later geometric claims.
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