e_210302
plain-language theorem explainer
Pointwise kernel identity: the midpoint numerator at index sextuple (2,1,0,3,0,2) equals eight times the explicit integer table at those same indices. Gravity analysts cite it as one cell of the 256-case cover that builds the global m2Num = 8·Z identity. The proof is a single decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,0,3,0,2)$ in $(\mathrm{Fin}\,4)^6$, the folded midpoint numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of a 256-cell kernel that certifies, index by index, the algebraic identity $m_2^{\mathrm{num}}=8\cdot Z$ used in the exact midpoint treatment of the 4D Regge M2 TT sector.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start from $0$ and add a contribution term for each coupling entry at the six Fin-4 indices. The comparison object $Z$ is an explicit integer-valued table on $(\mathrm{Fin},4)^6$, with sparse nonzero patterns such as $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$.
The local claim is only the single sextuple $(2,1,0,3,0,2)$. Sibling chunk theorems cover the remaining cells; the assembler glues them into the universal statement.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed, so the kernel decision procedure evaluates the fold that defines the numerator and the pattern match that defines the table, then checks integer equality. No lemmas are invoked beyond the definitions of the two sides.
why it matters
Feeds the assembler theorem that states $\forall(a,b,c,d,i,j),, m_2^{\mathrm{num}}=8\cdot Z$, proved by exhaustive fin_cases over $(\mathrm{Fin},4)^6$. That global identity is the certified algebraic core of the Regge exact-midpoint M2 TT analysis in four dimensions: it replaces a folded coupling sum by a closed integer table, scaled by eight.
Within Recognition Science gravity work, such kernel certificates keep the discrete curvature bookkeeping exact rather than approximate, so downstream continuum or continuum-limit arguments inherit a fully checked combinatorial identity rather than a numerical fit. The chunk exists only to keep the 256-case cover modular and machine-checkable.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.