e_312323
plain-language theorem explainer
At multi-index (3,1,2,3,2,3) the Regge midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic case in the 4D M2TT identity. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,2,3,2,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(3,1,2,3,2,3)=8\,Z_{\mathrm{explicit}}(3,1,2,3,2,3)$.
background
This module is chunk 13 of a 256-case kernel certification that the 4D Regge-exact midpoint numerator agrees with a closed form: $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ on every tuple in $(\mathbb{F}_4)^6$.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating integer contributions at those six indices. The comparison value $Z_{\mathrm{explicit}}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (typical entries $\pm 2,,4$, and so on).
Both definitions live in the KernelCert import; the present file only discharges one concrete point of the table.
proof idea
One-line computational proof: decide. After the six indices are substituted, both sides reduce to concrete integers (the fold for the numerator versus the matched clause for the explicit kernel), and the kernel decision procedure checks equality.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles the full identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ by exhausting all $\mathrm{Fin},4$ cases; this declaration is the witness for the single tuple $(3,1,2,3,2,3)$.
In the gravity analysis stack, that global equality is the algebraic certificate that the midpoint M2TT numerator matches its closed kernel form in 4D. The chunk split (256 decides) keeps each goal tiny and machine-checkable. No Recognition forcing-chain landmark (T5–T8, RCL, $\varphi$) is invoked here; the result is pure discrete tensor arithmetic supporting the Regge gravity side.
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