e_112003
plain-language theorem explainer
For the six-index slot (1,1,2,0,0,3) on Fin 4, the folded numerator coupling m2Num equals eight times the explicit kernel table explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,1,2,0,0,3)$ in $\mathrm{Fin}\,4$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
The ambient module is a chunk of the 4D Regge-calculus midpoint analysis: one must check that a folded numerator coupling built from a fixed list of edge contributions equals eight times a sparse explicit integer table on six Fin-4 indices. The module header states the goal bluntly: $m_2^{\mathrm{num}}=8\cdot Z$, chunk 5, discharged by 256 kernel decides.
Upstream, $m_2^{\mathrm{num}}$ is the fold of couplingZList under a local contribution map, returning an Int. The companion table $Z$ is a pattern-matched function $\mathrm{Fin},4^6\to\mathbb{Z}$ with a handful of nonzero entries (e.g. $4$ on diagonal-like pairs, $-2$ on crossed pairs) and implicit zero elsewhere. Both live in the kernel-certificate module imported by this file.
The six indices label a discrete midpoint tensor component in the Regge M2–TT identity; each concrete sextuple is an independent integer equality.
proof idea
One-line computational discharge: by decide. Lean reduces both sides on the concrete Fin-4 literals to bare integers (the fold on the left, the pattern match on the right) and checks equality in the kernel. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
This cell is one of the 256 decides in chunk 5 that feed the universal assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for every sextuple in $\mathrm{Fin},4^6$ by exhaustive fin_cases. That assembly is the algebraic backbone of the exact midpoint M2–TT identity in the 4D Regge gravity analysis.
In the broader Recognition Science gravity stack, such exact discrete identities underwrite continuum limits and effective Newtonian/relativistic regimes without free fitting parameters. The result does not itself touch the forcing chain (T5–T8) or the RCL; it is pure discrete tensor algebra supporting the gravity side of the monolith.
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