e_011003
plain-language theorem explainer
For the six-index slot (0,1,1,0,0,3) on the 4D midpoint kernel, the folded numerator m2Num equals eight times the closed-form table entry explicitZ. Gravity analysts assembling the exact Regge midpoint M2 identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(a,b,c,d,i,j)=(0,1,1,0,0,3)$ equals $8$ times the explicit integer table value at those same indices: $N(0,1,1,0,0,3)=8\,Z(0,1,1,0,0,3)$.
background
This module is chunk 1 of a 256-cell kernel certification that the folded numerator of the 4D Regge exact-midpoint $M_2$ tensor equals eight times a sparse closed-form integer table. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels $0..3$.
The numerator $N=\texttt{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\texttt{contrib},t,a,b,c,d,i,j$. The table $Z=\texttt{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (typical nonzero values $\pm 2,,4$). The local claim is the pointwise identity $N=8Z$ at one concrete multi-index.
The parent assembly theorem will range over all $4^6$ slots by nested finite case splits and invoke one such cell theorem per slot.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values (the fold of m2Num at fixed indices versus 8 * explicitZ at those indices), so the decidable equality checker discharges the goal 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 exhaustive fin_cases on the six indices and dispatch to the matching cell. That global identity is the algebraic core of the Regge exact-midpoint $M_2$ TT-identity certification in 4D gravity analysis inside the monolith.
In the Recognition Science gravity stack this is bookkeeping infrastructure, not a forcing-chain landmark (T5–T8). It closes one of 256 discrete obligations so the continuum/Regge side can quote a fully checked numerator–table match rather than an unexpanded fold. No open physics question is settled here; the value is machine-checked completeness of the kernel.
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