e_030133
plain-language theorem explainer
For the six-index slot (0,3,0,1,3,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,0,1,3,3)$ in $\mathrm{Fin}\,4$, the folded coupling 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 midpoint analysis, the numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes an integer weight depending on six Fin-4 indices, and the fold starts from zero. The companion map $Z$ is an explicit piecewise integer kernel on the same six indices (sample values include $4$, $-2$, and so on for listed patterns).
The local module is chunk 3 of a 256-cell decide grid that checks $m_2^{\mathrm{num}}=8Z$ pointwise. The ambient goal is the exact midpoint M2–TT identity in four dimensions, certified by exhausting all index sextuples rather than by a symbolic closed-form argument at this layer.
proof idea
One-line decide proof. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed: the left-hand side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right-hand side multiplies the matching clause of the explicit kernel $Z$ by eight. No lemmas are invoked beyond kernel evaluation.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple by fin_cases on each index and dispatches each cell to a chunk lemma of this form. That universal equality is the algebraic backbone of the Regge exact-midpoint M2–TT identity in 4D gravity analysis inside the monolith.
Within Recognition Science gravity work, these kernel identities pin the discrete curvature bookkeeping that later couples to the forced $D=3$ spatial sector and the eight-tick structure; the present cell is pure integer bookkeeping, not a dynamical claim.
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