e_311031
plain-language theorem explainer
For the six-index tuple (3,1,1,0,3,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one of the 256 kernel cells that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=1$, $d=0$, $i=3$, $j=1$ in $\{0,1,2,3\}$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the Regge exact-midpoint 4D analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N$ (m2Num) is obtained by folding a fixed coupling list and summing local contributions at each tuple. The comparison target $Z$ (explicitZ) is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and defaults).
The module is chunk 13 of a 256-cell partition of the Fin-4^6 domain. Each cell asserts $N=8Z$ at one concrete point so that a later assembler can recombine them by exhaustive fin_cases.
Upstream, explicitZ supplies the closed form and m2Num the fold definition; both live in the kernel certificate module imported here.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted: the left-hand side evaluates the fold of coupling contributions, the right-hand side multiplies the table lookup by eight. Kernel decision discharges the equality with no further lemmas.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple by fin_cases over all indices and invokes each chunk cell. That global equality is the algebraic backbone of the Regge exact-midpoint M2 TT identity in 4D gravity analysis inside Recognition Science.
The factor of eight ties the folded coupling sum to the sparse explicit kernel, closing a computational certificate rather than an analytic derivation. It does not itself invoke the forcing chain (T0–T8) or the Recognition Composition Law; it is pure discrete bookkeeping supporting the continuum gravity side.
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