e_013322
plain-language theorem explainer
Pointwise kernel identity: the folded numerator coupling at index sextuple (0,1,3,3,2,2) equals eight times the explicit integer table entry at those indices. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite it as one of 256 concrete decides. Proof is a single kernel decision on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(0,1,3,3,2,2)$ in $(\mathrm{Fin}\,4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator coupling $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list and summing local contributions at the six indices. The companion map $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold, up to a universal factor of eight.
This module is chunk 1 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8Z$ holds at every sextuple. Each chunk theorem pins one concrete point so the global assembly can discharge the identity by exhaustive fin_cases rather than by a symbolic rewrite of the fold.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,1,3,3,2,2)$. The left side reduces the fold that defines the numerator coupling; the right side looks up the explicit table entry and multiplies by eight. Equality of the resulting integers is decided in the kernel.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, invoking one pointwise decide per cell. That global identity is the algebraic backbone of the 4D Regge midpoint M2–TT certification in the gravity analysis stack. Within Recognition Science it sits in the discrete geometric bookkeeping that supports continuum gravity limits, not in the T0–T8 forcing chain itself; it closes a finite kernel obligation rather than an open physical hypothesis.
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