e_312322
plain-language theorem explainer
For the six Fin-4 indices (3,1,2,3,2,2), the folded midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite it as one of the 256 kernel cells in the Regge midpoint M2TT identity. The proof is a single kernel decide on integer arithmetic.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}2$, $d{=}3$, $i{=}2$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the tabulated kernel entry $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-cell kernel certification that the Regge-exact midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The ambient setting is 4D discrete gravity analysis: index sextuples label midpoint contributions to an M2TT identity.
The numerator $\mathrm{m2Num}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple. The comparison table $\mathrm{explicitZ}$ is a total function on six $\mathrm{Fin},4$ arguments with hard-coded integer values on the nonzero support (e.g. $4$, $-2$, and symmetric permutations).
The global claim is assembled elsewhere by exhaustive case split; each chunk theorem pins one concrete sextuple so the kernel can decide the integer equality without carrying the full fold in the assembler.
proof idea
One-line proof by decide. Lean reduces both sides on the concrete indices $(3,1,2,3,2,2)$: the left-hand fold of coupling contributions evaluates to an integer, the right-hand side is $8$ times the matching explicitZ clause (or zero if outside the tabulated support), and the kernel checks equality.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ and proves it by six nested fin_cases over $\mathrm{Fin},4$, dispatching each cell to a chunk identity of this form.
In the Recognition gravity stack this closes the algebraic midpoint identity used when matching discrete Regge curvature bookkeeping to the continuum TT sector. It is pure integer certification, not a dynamical derivation of $G$ or the $\phi$-ladder masses, but it is a required kernel step before those continuum limits can quote an exact discrete numerator.
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