e_203201
plain-language theorem explainer
Pointwise identity: the folded M2 numerator at multi-index (2,0,3,2,0,1) equals eight times the closed-form kernel value there. Gravity analysts cite it as one of 256 decide-chunks assembling the full 4D Regge midpoint M2–TT numerator identity. Proof is a single kernel decision on concrete integers.
Claim. At multi-index $(2,0,3,2,0,1)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit closed form: $m_2^{\mathrm{num}}(2,0,3,2,0,1)=8\,Z_{\mathrm{expl}}(2,0,3,2,0,1)$.
background
This module is chunk 8 of a 256-way split proving $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on the full $(\mathbb{F}_4)^6$ index set for the 4D Regge exact-midpoint M2–TT kernel.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib kernel, summed from zero. The closed form $Z_{\mathrm{expl}}$ is a sparse pattern-matched integer table on the same six $\mathrm{Fin},4$ indices (typical nonzero entries $\pm 2,,4$).
Both objects live in the kernel-cert module imported here; the chunk modules only discharge individual index tuples so the assembler can recombine them by exhaustive fin_cases.
proof idea
One-line computational proof: by decide. After reducing the six concrete Fin 4 arguments, both sides are closed integer expressions (fold of contrib versus the explicitZ match), and the kernel decision procedure checks equality.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, invoking one chunk identity per cell. That global identity is the algebraic core of the 4D Regge exact-midpoint M2–TT certification in the Gravity analysis stack. Without the pointwise chunks, the assembler has nothing to case-split onto; each e_* entry is a pure decide certificate for one lattice point.
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