e_113111
plain-language theorem explainer
For the six-index slot (1,1,3,1,1,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these pointwise identities when assembling the global Regge midpoint M2=8Z certificate. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=b=i=j=1$, $c=3$, $d=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 5 of a 256-case kernel certification that the Regge-exact midpoint numerator equals eight times a sparse explicit integer table on six Fin-4 indices. The ambient setting is 4D discrete gravity analysis: couplings are summed into an integer numerator, then matched against a hand-written closed form.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and accumulating a contribution at each six-index argument. The comparison target explicitZ is a pattern-matched Int-valued table on Fin 4^6 (nonzero only on a thin set of index patterns such as (0,0,1,1,2,2)mapsto 4 and several -2 entries).
The local claim is one concrete six-tuple evaluation inside that table identity, not the quantified statement.
proof idea
One-line decide proof. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed: the fold defining the numerator evaluates fully, and explicitZ matches its pattern (or defaults), so equality of Ints is decided by the kernel with no lemmas invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the full quantified identity $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over Fin 4. That global certificate is the algebraic backbone of the Regge-exact midpoint M2–TT identity in the 4D gravity analysis stack.
Within Recognition Science gravity work, these chunk decides close the discrete curvature/coupling bookkeeping that sits under continuum limits and effective Newtonian reconstructions. They do not themselves invoke the forcing chain (T5–T8) or the J-cost; they are pure integer kernel facts supporting the geometric side.
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