e_203313
plain-language theorem explainer
Pointwise kernel check: the folded M2 numerator coupling at multi-index (2,0,3,3,1,3) equals eight times the explicit integer table Z at those same indices. Gravity analysts cite it as one of 256 decide-cells that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decision on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,3,1,3)$ in $(\mathbb{F}_4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, the TT-sector mass-squared numerator is built by folding a fixed coupling list: each term contributes an integer depending on six $\mathbb{F}_4$ indices, and $m_2^{\mathrm{num}}$ is their sum. Parallel to that fold sits an explicit closed-form table $Z$ on the same six indices, with sparse nonzero entries such as $\pm 2$ and $4$ on selected diagonal and off-diagonal patterns.
The local module is chunk 8 of a 256-cell kernel: every concrete six-tuple is checked separately so that a later assembly theorem can recombine them by exhaustive fin_cases. The claimed relation is the scalar identity $m_2^{\mathrm{num}}=8Z$ at one fixed tuple.
proof idea
One-line computational proof: decide evaluates both sides on the concrete Fin-4 indices. The left side runs the fold that defines $m_2^{\mathrm{num}}$; the right side looks up $8\cdot Z(2,0,3,3,1,3)$. No lemmas are invoked beyond the kernel decision procedure on integers.
why it matters
Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and discharges the goal by six nested fin_cases over $\mathbb{F}_4$, each leaf a chunk decide such as this one. Closing the pointwise cells is what turns the explicit table $Z$ into a certified closed form for the folded numerator in the 4D Regge midpoint TT identity. That identity is infrastructure for the discrete gravity side of the Recognition stack (curvature and mass-squared bookkeeping on the lattice), not a forcing-chain landmark itself.
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