e_013111
plain-language theorem explainer
For the Fin-4 index sextuple (0,1,3,1,1,1), the folded Regge midpoint mass-squared numerator equals eight times the explicit integer kernel entry. Analysts certifying the 4D Regge exact midpoint M2TT identity cite this as one decided cell in chunk 1 of the 256-case kernel block. The proof is a single kernel decide on a closed integer equality.
Claim. For $a=0$, $b=1$, $c=3$, $d=1$, $i=1$, $j=1$ in $\{0,1,2,3\}$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module sits in the Gravity analysis stack that certifies an exact midpoint identity for the 4D Regge M2TT numerator. The local setting (module doc) is the first 256-cell chunk of the exhaustive check that the folded numerator equals eight times an explicit kernel table.
The numerator $N(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list and summing integer terms at the six Fin-4 indices. The explicit kernel $Z$ is a piecewise integer function on $(\mathrm{Fin},4)^6$, tabulated by pattern (for example $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
Both objects live in the kernel-certificate module imported here. The present declaration fixes one concrete sextuple and asserts the scalar identity at that point.
proof idea
One-line kernel proof: decide evaluates both sides of the integer equality at the concrete indices $(0,1,3,1,1,1)$ and accepts the closed true proposition. No lemmas are invoked beyond the elaborator reducing $N$ (the fold over the coupling list) and $Z$ (the piecewise table) to numerals.
why it matters
The parent theorem is the universal assembly statement: for every $a,b,c,d,i,j:\mathrm{Fin},4$, the numerator equals eight times the explicit kernel. That assembly proceeds by exhaustive fin_cases on the six indices; this declaration discharges the single cell $(0,1,3,1,1,1)$ inside chunk 1.
In the Recognition gravity stack, the identity $N=8Z$ is the numerical backbone of the 4D Regge exact midpoint M2TT certificate. It does not itself invoke the T0–T8 forcing chain, the Recognition Composition Law, or the phi-ladder mass formula; it is pure discrete linear-algebra bookkeeping that later continuum or phenomenological layers can quote without rechecking 4096 cells.
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