e_201133
plain-language theorem explainer
For the discrete index sextuple (2,0,1,1,3,3), the folded M2 coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of 256 kernel decides. The proof is a single native decide on the concrete integers.
Claim. For indices $a=2$, $b=0$, $c=1$, $d=1$, $i=3$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals $8$ times the explicit integer kernel value at those indices.
background
In the 4D Regge midpoint M2–TT analysis, six indices in $\mathrm{Fin},4$ label discrete edge/face data. The folded coupling numerator is defined by folding a fixed coupling list and summing integer contributions at those indices. The explicit integer kernel is a piecewise table on the same six indices (sample values $4$, $-2$, and so on).
This module is chunk 8 of the 256 kernel decides that certify the pointwise identity numerator $= 8\cdot$ kernel. The local goal is exhaustive, machine-checked equality on each concrete sextuple rather than a symbolic closed form.
proof idea
One-line wrapper: decide. Both sides reduce to concrete integers (the fold over the coupling list on the left; the table lookup, scaled by 8, on the right), and the decision procedure checks equality.
why it matters
Parent theorem is the assembled pointwise identity: for every sextuple in $(\mathrm{Fin},4)^6$, the folded numerator equals eight times the explicit kernel. That proof introduces all six indices and runs fin_cases on each, so each chunk equality such as this one is a leaf of the case tree.
The identity is part of the certified 4D Regge exact-midpoint M2–TT kernel used in the gravity analysis stack. It does not itself touch the Recognition forcing chain (T5–T8) or the RCL; it is infrastructure for discrete curvature bookkeeping.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.