e_300133
plain-language theorem explainer
For the six Fin-4 indices (3,0,0,1,3,3), the midpoint mass-matrix numerator equals eight times the explicit integer coupling kernel. Gravity analysts building the 4D Regge exact-midpoint M2TT identity cite this as one decided cell in chunk 12 of the 256-kernel block. The proof is a single computational decide on concrete integer arithmetic.
Claim. With indices in $\{0,1,2,3\}$, the mass-matrix numerator satisfies $m_2^{\mathrm{num}}(3,0,0,1,3,3)=8\,Z(3,0,0,1,3,3)$, where $Z$ is the explicit sparse coupling kernel and $m_2^{\mathrm{num}}$ is the folded sum of coupling contributions.
background
In the 4D Regge exact-midpoint gravity analysis, the mass-matrix numerator is the integer $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ obtained by folding a fixed list of coupling contributions over six indices in $\mathrm{Fin},4$. The companion map $Z$ is an explicit sparse lookup: it returns a handful of nonzero integers on patterned index sextuples (for example $4$ on $(0,0,1,1,2,2)$ and $-2$ on several crossed pairs) and zero elsewhere.
The present module certifies the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ one cell at a time. Module documentation labels this file chunk 12 of a 256-kernel decide block covering part of the $4^6$ index space. Both $m_2^{\mathrm{num}}$ and $Z$ are defined upstream in the KernelCert module; this theorem only evaluates them at one concrete point.
proof idea
One-line computational discharge. The tactic decide fully evaluates both sides at the literal indices $(3,0,0,1,3,3)$: the fold that defines the numerator and the pattern-match that defines the explicit kernel. Integer equality is then checked by the kernel. No named lemmas are applied beyond the two definitions.
why it matters
The cell is consumed by the universal assembly theorem m2Num_eq_eight_explicitZ, which introduces six Fin-4 variables and runs fin_cases on each, routing every branch to a matching decide lemma of this family. That universal identity is the algebraic core of the 4D Regge exact-midpoint M2TT certificate in the gravity analysis stack: it locks the folded numerator to eight times a sparse closed-form kernel, so later geometric identities can quote a single explicit table instead of a fold.
Within the broader Recognition gravity layer this is scaffolding for discrete curvature bookkeeping (Regge-type midpoint mass matrices), not a direct step of the T0–T8 forcing chain.
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