e_313120
plain-language theorem explainer
For the six-index tuple (3,1,3,1,2,0) in Fin 4, the folded numerator m2Num equals eight times the closed-form explicitZ value. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite this as one of 256 kernel certificates. The proof is a single decide on the concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,3,1,2,0)$ with each index in $\{0,1,2,3\}$, the folded coupling numerator equals eight times the explicit integer table: $m_2^{\mathrm{num}}(3,1,3,1,2,0)=8\,Z_{\mathrm{explicit}}(3,1,3,1,2,0)$.
background
This module is chunk 13 of a 256-way kernel certification that the 4D Regge-exact midpoint M2 TT numerator agrees with a closed integer table. The ambient setting is discrete gravity analysis: edge and face couplings on a 4-simplex lattice with midpoint evaluation.
The numerator m2Num is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The comparison target explicitZ is a pattern-matched integer table on six Fin 4 indices (sample clauses give values such as 4, -2, etc.). Both live in the KernelCert module imported here.
The local claim is only the single multi-index (3,1,3,1,2,0). Sibling theorems cover the other tuples in the same chunk.
proof idea
One-line computational discharge: by decide. Lean evaluates both sides as concrete Int values (the fold of m2Num versus 8 * explicitZ at the fixed indices) and closes the equality by kernel decision. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in Fin 4 by exhaustive fin_cases and invokes each chunk certificate such as this one. That global equality is the algebraic backbone of the Regge-exact midpoint M2 TT identity in 4D discrete gravity inside the Recognition Science gravity stack.
It does not itself touch the forcing chain (T0–T8), RCL, or the phi-ladder mass formula; it is pure integer bookkeeping that lets the continuum-facing gravity identities rest on a fully certified discrete kernel rather than an unchecked table.
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