e_200320
plain-language theorem explainer
For the discrete index tuple (2,0,0,3,2,0) on Fin 4, the summed midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge exact midpoint M2–TT identity cite this as one of the 256 kernel point checks. The proof is a pure kernel decision: both sides evaluate to concrete integers and Lean closes equality by decide.
Claim. Let $m_2^{\mathrm{num}}$ be the fold of coupling contributions over the fixed coupling list, and let $Z$ be the explicit integer kernel on six $\mathrm{Fin}\,4$ indices. Then $m_2^{\mathrm{num}}(2,0,0,3,2,0)=8\,Z(2,0,0,3,2,0)$.
background
This module sits in the Gravity analysis stack that certifies an exact midpoint identity for the M2–TT sector of 4D Regge calculus. The local setting (module doc) is the chunked verification that the summed numerator equals eight times an explicit kernel on every six-tuple of $\mathrm{Fin},4$ indices: "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)."
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a contribution function over a fixed coupling list, starting from zero. The companion $Z(a,b,c,d,i,j)$ is a pattern-matched integer table on the same six indices (sample clauses return $4$, $-2$, and so on). Both are pure combinatorial objects: no continuum limit or metric ansatz is involved at this layer.
The full identity is $\forall$ six indices, $m_2^{\mathrm{num}}=8Z$. Because the domain is finite ($4^6=4096$ would be heavy in one go; here the work is chunked to 256 decides), each concrete tuple is discharged separately and later reassembled by nested fin_cases.
proof idea
One-line computational proof: by decide. Both sides of the equality are closed integer terms once the six concrete Fin 4 arguments are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern table for $Z$. The kernel evaluates the fold and the table lookup and checks integer equality; no rewriting lemmas or induction are required.
why it matters
This declaration is one atomic cell in the 256-point chunk that feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple and proves it by exhaustive fin_cases on each index. That global equality is the algebraic backbone of the Regge exact midpoint M2–TT identity certificate in 4D: it replaces a summed coupling expression by a sparse explicit kernel, making later curvature and deficit identities tractable.
In the broader Recognition Science gravity line, such exact discrete identities underwrite continuum limits and comparison with the forced $D=3$ spatial sector (T8) and the eight-tick structure (T7) without introducing free fitting constants. The chunk does not itself touch $\phi$-ladder masses or $\alpha$; it is pure combinatorial support for the Regge side of the ledger.
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