e_032003
plain-language theorem explainer
For the six-index slot (0,3,2,0,0,3) 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 M2=8Z relation in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,0,0,3)$ in $\{0,1,2,3\}^6$, the folded coupling numerator $N(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint certification that the discrete numerator $N$ equals eight times an explicit integer table $Z$. The ambient setting is gravity analysis on Fin-4 multi-indices: each argument runs over the four discrete directions of a 4-simplex edge labeling.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple. The table $Z=\mathrm{explicitZ}$ is a pattern-matched integer function on six Fin-4 indices (typical nonzero values $\pm 2,,4$ on diagonal and off-diagonal pairings).
Chunk 3 packages 256 such pointwise equalities so the global identity can be assembled by exhaustive case split rather than a single huge decide.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side evaluates the fold of contrib over couplingZList at the fixed indices $(0,3,2,0,0,3)$; the right-hand side is $8$ times the matching clause of explicitZ. Lean’s kernel decision procedure checks integer equality with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by nested fin_cases over all six Fin-4 arguments. Each chunk identity such as this one discharges one concrete cell so the universal statement does not rely on a monolithic decide.
In the Recognition gravity stack this is bookkeeping for the Regge midpoint M2/TT identity in 4D: once $N=8Z$ is certified pointwise, the continuum-facing curvature and mass-ladder comparisons can quote a fully reduced integer kernel rather than an opaque fold. It does not itself invoke the forcing chain (T5–T8) or the RCL; it is infrastructure under the discrete gravity side.
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