e_113121
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at multi-index (1,1,3,1,2,1) equals eight times the explicit integer Z-table entry. Gravity analysts cite it as one of 256 kernel cells in the 4D midpoint TT identity. The proof is a single kernel decision on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,1,3,1,2,1)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator is assembled by folding a fixed coupling list: each term contributes an integer depending on six $\mathbb{F}_4$ indices, and $m_2^{\mathrm{num}}$ is their sum. Parallel to that fold sits an explicit integer table $Z$ on the same six-index domain, given by a finite pattern match (nonzero only on a sparse set of index patterns such as $(0,0,1,1,2,2)\mapsto 4$).
This module is chunk 5 of the 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every multi-index. The local setting is pure finite enumeration over $\mathrm{Fin},4$, not continuum GR: each cell is an integer equality that the kernel can decide.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete six-tuple $(1,1,3,1,2,1)$ and checks integer equality. The left side runs the fold that defines the numerator; the right side looks up (or defaults) the explicit $Z$ table and multiplies by 8. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and discharges the universal quantifier by fin_cases on all six indices, routing each cell to its chunk theorem. That global identity is the certified bridge between the folded coupling definition and the closed-form $Z$ table in the Regge exact-midpoint TT analysis. Within Recognition gravity work it is bookkeeping infrastructure: it locks the discrete 4D kernel so later continuum or phenomenological claims can quote a proved numerator identity rather than an unchecked table.
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