e_021011
plain-language theorem explainer
For the six-index slot (0,2,1,0,1,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2TT equality. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,1,0,1,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib and the fold starts at 0. The companion map explicitZ is a closed-form pattern match that returns small integers (typically $\pm 2,\pm 4$, or 0) on the same index sextuples.
The module is one of several chunks that discharge the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ by kernel decision. Chunk 2 covers a block of the $4^6=4096$ index combinations; each local theorem pins one sextuple. The factor of eight is the combinatorial normalisation that links the summed coupling contributions to the explicit kernel table used downstream in the M2TT identity.
proof idea
One-line decide proof. Both sides are closed integer expressions once the six concrete Fin-4 indices are substituted: the left-hand side evaluates the fold of contrib over couplingZList, the right-hand side multiplies the pattern-matched explicitZ value by 8. Lean’s kernel decision procedure checks the resulting integer equality with no tactics or lemmas beyond evaluation.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices by exhaustive fin_cases and invokes each chunk theorem such as this one. That global equality is the numerical backbone of the Regge-exact midpoint M2TT identity in 4D gravity analysis inside the monolith.
Within Recognition Science gravity work, these kernel certificates keep the discrete curvature bookkeeping exact rather than approximate. They do not themselves invoke the forcing chain (T5–T8) or the J-cost, but they underwrite the discrete geometric layer that later couples to continuum limits and mass-ladder phenomenology.
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