e_020320
plain-language theorem explainer
Single kernel identity: the Regge midpoint mass-squared numerator at multi-index (0,2,0,3,2,0) equals eight times the explicit integer coupling Z at those indices. Gravity analysts assembling the 4D midpoint M2–TT identity cite it as one exhaustive Fin-4 case. Proof is a pure kernel decide on the concrete integers from the folded coupling list versus the explicit Z table.
Claim. For indices $a=0$, $b=2$, $c=0$, $d=3$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the mass-squared numerator obtained by folding the coupling contribution list equals $8$ times the explicit integer kernel value $Z(0,2,0,3,2,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ folds a fixed coupling list, accumulating each term's contribution at the given multi-index. The comparison target is an explicit piecewise integer table $Z$ on the same six indices (for example $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
The local module is chunk 2 of a 256-way case split that discharges $m_2^{\mathrm{num}}=8Z$ pointwise by kernel decision. The ambient goal is the exact midpoint mass-squared / TT identity in four dimensions, reduced to these finite combinatorial checks on the discrete index cube.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete sextuple $(0,2,0,3,2,0)$. The left-hand side runs the fold that defines the numerator over the coupling list; the right-hand side looks up the explicit $Z$ table and multiplies by eight. Equality of the resulting integers is decided by the kernel with no further lemmas or rewrites.
why it matters
Feeds the universal assembly theorem that states $m_2^{\mathrm{num}}=8Z$ for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive case split on the six indices. That quantified equality is the algebraic content of the Regge exact midpoint $M_2$–TT identity in 4D: the folded coupling numerator is identically eight times the explicit $Z$ kernel. Within Recognition Science gravity this closes a finite combinatorial certificate rather than an analytic continuum argument. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is infrastructure on the discrete gravity side.
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