e_100200
plain-language theorem explainer
For the six-index slot (1,0,0,2,0,0) on Fin 4, the folded midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel point checks. The proof is a single decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,0,0,2,0,0)$ 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 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: start at 0 and add each contribution term at the given indices. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (sample values include $4$, $-2$, and so on for listed patterns).
This module is chunk 4 of a 256-way kernel decide campaign whose local goal is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$. Each chunk theorem fixes one concrete six-tuple and discharges equality by computation rather than by a structural argument over the fold.
proof idea
One-line kernel decide: both sides are closed integer expressions at the fixed indices $(1,0,0,2,0,0)$, so decide evaluates $m_2^{\mathrm{num}}$ (the fold of contributions) and $8\cdot Z$ and checks equality in $\mathbb{Z}$. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ and $Z$ are required.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8\cdot Z$ for all Fin-4 indices, which proceeds by exhaustive fin_cases on the six arguments and dispatches each cell to a chunk equality such as this one. That global identity is the certified algebraic core of the Regge exact midpoint M2TT 4D kernel comparison in the Gravity analysis stack.
Within Recognition Science gravity work, these kernel certificates lock the discrete curvature/coupling bookkeeping before continuum or phenomenological claims are attached. The result is pure finite enumeration support, not a derivation of $G$, $\phi$-ladder masses, or the T0–T8 forcing chain.
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