e_312333
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at discrete indices (3,1,2,3,3,3) equals eight times the explicit kernel value Z at those indices. Gravity analysts cite it when assembling the full 4D m2Num = 8·explicitZ certificate. The proof is a single kernel decide on two closed integer expressions.
Claim. For indices $(a,b,c,d,i,j)=(3,1,2,3,3,3)$ in $(\mathbb{F}_4)^6$, the summed 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 kernel on six $\mathbb{F}_4$ arguments.
background
In the 4D Regge midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer depending on six $\mathbb{F}_4$ indices $(a,b,c,d,i,j)$. The companion map $Z$ is an explicit case table on the same six indices, returning small integers (e.g. $4$, $-2$) on the support patterns that appear in the kernel.
The local module is chunk 13 of a 256-case decide grid that checks $m_2^{\mathrm{num}}=8Z$ at every index tuple. The module doc states the goal directly: "$m_2^{\mathrm{num}}=8\cdot\mathrm{explicit}Z$, chunk 13 (256 kernel decides)." Upstream, $m_2^{\mathrm{num}}$ is the foldl of contrib over couplingZList, and $Z$ is the pattern-matched integer table.
proof idea
One-line computational proof: decide evaluates both sides as concrete Int values at the fixed tuple $(3,1,2,3,3,3)$ and checks equality. No algebraic lemmas are invoked; the kernel reduces the fold defining $m_2^{\mathrm{num}}$ and the case table for $Z$, then compares $8Z$ to the sum.
why it matters
This is one cell of the exhaustive grid that feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for all six $\mathbb{F}_4$ indices by case split. That global equality is the algebraic certificate that the midpoint Regge $m_2$ numerator is exactly eight times the closed-form kernel, a step in the 4D gravity analysis chain inside Recognition Science.
The chunk layout (256 decides) keeps each obligation tiny and machine-checkable. Downstream assembly glues the cells into the universal statement used by later Regge identity work. No forcing-chain landmark (T5–T8) is touched directly; the result is pure discrete kernel bookkeeping for the gravity side.
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