e_111130
plain-language theorem explainer
For the six-index slot (1,1,1,1,3,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=b=c=d=1$, $i=3$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-case kernel certification that the folded numerator of the midpoint M2 tensor equals eight times an explicit integer table on six Fin-4 indices. The local setting is exact 4D Regge calculus at the midpoint evaluation used in the Recognition gravity analysis.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a contribution function over that list at the six indices. The companion table $Z$ is a pattern-matched integer function on the same six indices (sample values include $\pm 2,,4$ on selected diagonal and off-diagonal patterns).
The claim is the pointwise identity at one concrete sextuple. Sibling chunks cover the other Fin-4 combinations; the assemble theorem glues them into a universal statement.
proof idea
One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold of contributions at $(1,1,1,1,3,0)$, the right-hand side multiplies the table lookup by eight. Lean’s decision procedure checks integer equality with no further lemmas.
why it matters
Feeds the assemble theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over Fin 4, dispatching each sextuple to a chunk such as this one. That identity is the algebraic core of the exact midpoint M2 TT identity in 4D Regge analysis under Recognition Science gravity.
In the broader framework it supports the discrete curvature bookkeeping that sits under the forced $D=3$ spatial sector (T8) and the eight-tick octave structure (T7), by making the 4D kernel numerator fully explicit and machine-checked rather than schematic. No open scaffold remains on this slot: the equality is closed by decide.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.