e_110321
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at indices (1,1,0,3,2,1) equals eight times the explicit integer table entry there. Gravity analysts cite it when assembling the full six-index Regge midpoint mass-squared identity over Fin 4. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=1$, $b=1$, $c=0$, $d=3$, $i=2$, $j=1$ in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$ at those same indices.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing each term's contribution at the six indices. The companion table $Z$ is an explicit pattern-matched integer function on the same domain (typical values $\pm 2,,4$, and sparse support).
The module is chunk 5 of a 256-way split of the kernel: each chunk discharges a block of concrete index tuples so the global identity can be reassembled without a single monolithic decide. Local setting: prove $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on that block.
Upstream, only the two kernel definitions are needed; no analytic lemma intervenes. The equality is pure integer arithmetic once the indices are ground.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are literals, so the kernel decision procedure closes the equality with no lemmas, rewrites, or case splits inside this declaration.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhausting indices. That global equality is the certified algebraic core of the Regge exact-midpoint mass-squared numerator in 4D, tying the folded coupling sum to the sparse explicit $Z$ table used downstream in the gravity analysis stack.
Within Recognition Science gravity work, such kernel certificates keep the continuum-limit bookkeeping exact rather than numeric. This chunk does not itself invoke the forcing chain (T0–T8) or the J-cost; it is infrastructure under the Regge side of the gravity module. Closing all chunks discharges the pointwise obligation the assembler relies on.
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