e_301033
plain-language theorem explainer
Pointwise identity: at multi-index (3,0,1,0,3,3) the folded M2 numerator equals eight times the explicit Z-table entry. One of 256 kernel cases in chunk 12 of the 4D Regge midpoint M2TT certification. Downstream assembly cites it to obtain the universal equality. Proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,0,1,0,3,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module sits in the 4D Regge midpoint analysis for the M2TT identity. The local goal, stated in the module doc, is to certify $m_2^{\mathrm{num}}=8\cdot Z$ on a 256-point kernel by chunked decide lemmas.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from $0$ and add each contribution contrib t a b c d i j. The comparison target explicitZ is a total function $\mathbb{F}_4^6\to\mathbb{Z}$ given by an explicit pattern-match table (sample entries include $4$, $-2$, and so on).
Both objects live in the kernel certificate module imported here. The present declaration fixes one concrete six-tuple of Fin-4 indices and asserts numerical equality after the factor of eight.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers (the fold over the coupling list on the left; the table lookup, scaled by 8, on the right), so the kernel closes the equality with no further lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j., m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal claim by exhaustive fin_cases on all six Fin-4 indices. Each case lands on a chunk lemma of this form.
In the gravity analysis stack, that universal identity is the certified link between the folded coupling numerator and the closed-form Z table used in the 4D Regge midpoint M2TT identity. Without the full 256-case cover, the assembly step cannot finish. This entry is the $(3,0,1,0,3,3)$ cell of chunk 12; it carries no independent physics content beyond completing that cover.
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