e_301331
plain-language theorem explainer
Pointwise identity: the midpoint M2 numerator at multi-index (3,0,1,3,3,1) equals eight times the explicit Z-table entry. Gravity analysts cite it as one of 256 kernel cells in the Regge exact midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,0,1,3,3,1)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer table on six $\mathbb{F}_4$ arguments.
background
In the 4D Regge exact-midpoint analysis, the M2 numerator is defined by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums contribution terms over that list. The companion table $Z$ is an explicit pattern-matched integer function on six $\mathbb{F}_4$ indices (sample values include $4$, $-2$, and so on for distinguished index patterns).
The local module is chunk 12 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every multi-index. Each chunk theorem pins one concrete sextuple so the assembly proof can discharge the universal statement by exhaustive fin_cases.
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel certificate module; this declaration only evaluates them at $(3,0,1,3,3,1)$.
proof idea
One-line computational proof: decide evaluates both sides at the fixed Fin-4 sextuple and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit Z table; the kernel reduces the equality to a closed arithmetic fact.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by casing on all six Fin-4 indices and invoking the matching chunk cell at each leaf. That universal identity is the algebraic core of the Regge exact-midpoint M2–TT certification in 4D gravity analysis inside the monolith.
Within Recognition Science gravity work, such kernel equalities underwrite discrete curvature and coupling bookkeeping on the eight-tick, $D=3$ side of the forcing chain; they are infrastructure rather than a new physical law. Closing all 256 cells removes scaffolding from the midpoint identity path.
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