e_300022
plain-language theorem explainer
Kernel equality: the Regge midpoint mass-squared numerator at multi-index (3,0,0,0,2,2) equals eight times the explicit Z-table value there. Gravity analysts building the 4D TT midpoint identity cite the assembled universal form; this chunk is one of 256 finite-case checks. Proof is a single kernel `decide`.
Claim. For indices $(a,b,c,d,i,j)=(3,0,0,0,2,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at a six-index slot $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, and the fold starts from zero. The companion table $Z$ is an explicit piecewise integer function on the same index set (sample values include $4$, $-2$, and so on at distinguished patterns).
The local module is chunk 12 of a 256-way case split whose sole job is to certify $m_2^{\mathrm{num}}=8Z$ pointwise. The ambient goal is the exact midpoint TT identity in four dimensions: once every index tuple matches, the numerator is identically eight times the closed-form table, so algebraic identities that mention only $Z$ lift to the folded coupling sum.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(3,0,0,0,2,2)$ and checks integer equality in the kernel. No lemmas are invoked beyond the reducible definitions of the fold m2Num and the piecewise table explicitZ.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhausting all Fin-4 cases. That universal equality is the bridge from the explicit Z-table (used in closed-form Regge/TT algebra) to the actual folded coupling numerator that appears in the 4D midpoint mass-squared identity. Within Recognition gravity, this is bookkeeping infrastructure for the exact discrete curvature/TT sector, not a forcing-chain step (T0–T8), but it is required before continuum or phenomenological claims can quote the simplified $8Z$ form.
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