e_123013
plain-language theorem explainer
Pointwise kernel identity: for the Fin-4 sextuple (1,2,3,0,1,3), the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it as one cell of the 4D Regge midpoint M2 TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,3,0,1,3)$ in $(\mathrm{Fin}\,4)^6$, the integer numerator $m_2^{\mathrm{num}}(1,2,3,0,1,3)$ equals $8\,Z(1,2,3,0,1,3)$, where $Z$ is the explicit six-index kernel table.
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 and summing a local contribution at each term. The comparison target is an explicit piecewise table $Z$ on the same six indices, with values such as $\pm 2,\pm 4$ on the listed patterns and (implicitly) zero elsewhere.
This module is chunk 6 of a 256-cell decide grid that discharges $m_2^{\mathrm{num}}=8Z$ one sextuple at a time. The local setting is pure integer arithmetic on a finite index set: no continuum limit and no metric reconstruction yet, only the algebraic identity needed for the midpoint M2 TT certificate.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(1,2,3,0,1,3)$. The left side runs the fold that defines $m_2^{\mathrm{num}}$; the right side looks up $8Z$ from the explicit table. Equality of the resulting integers is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all Fin-4 sextuples, which exhausts the index space by fin_cases and stitches the chunk identities into a single universal statement. That universal identity is the algebraic core of the Regge exact midpoint M2 TT certificate in 4D gravity analysis inside the monolith.
Within Recognition Science this sits on the gravity side of the forcing chain rather than on T5–T8 themselves: it certifies a discrete curvature/kernel identity used when matching Regge midpoint data to the continuum TT sector. Closing the full grid removes a scaffolding burden on the 4D midpoint certificate.
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