e_120101
plain-language theorem explainer
For the six-index tuple (1,2,0,1,0,1) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite this as one atomic cell in the 4D Regge exact-midpoint M2TT identity. The proof is a single kernel decide on concrete integer arithmetic.
Claim. For indices $a=1$, $b=2$, $c=0$, $d=1$, $i=0$, $j=1$ 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
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts from zero. The comparison target is an explicit piecewise integer function $Z$ on the same index domain, with hard-coded values on selected patterns (for example $4$ or $-2$ on particular matchings of coordinate pairs).
This module is chunk 6 of a 256-cell decide grid that checks the pointwise identity $m_2^{\mathrm{num}}=8Z$ one multi-index at a time. The local setting is pure finite enumeration over $\mathrm{Fin},4^6$, with no continuum limit or curvature hypothesis in play at this layer.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern-match definition of $Z$; the kernel closes the equality by evaluation.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles every cell of the grid by nested fin_cases on the six indices and invokes each chunk equality such as this one. That global identity is the algebraic backbone of the Regge exact-midpoint M2TT certification in four dimensions: it converts the folded coupling expression into an eightfold multiple of the closed-form kernel, which is the form needed for later gravity-analysis identities. Within Recognition Science this sits in the gravity analysis layer that supports discrete curvature bookkeeping consistent with the forced $D=3$ spatial skeleton and the eight-tick octave structure, though the present lemma itself is purely combinatorial.
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