e_202011
plain-language theorem explainer
One kernel cell of the 4D Regge midpoint identity: the numerator m2Num at multi-index (2,0,2,0,1,1) equals eight times the explicit table value explicitZ at the same indices. Gravity analysts assembling the global m2Num = 8·explicitZ identity cite this cell. The proof is a single decide on concrete integer arithmetic.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,0,2,0,1,1)$ with each coordinate in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six indices in $\mathrm{Fin},4$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts from zero. The companion table $Z$ is an explicit pattern-matched function on the same six indices, returning small integers such as $4$, $-2$, and so on at the listed patterns.
The module is chunk 8 of a 256-cell kernel certification: the claim under audit is one concrete multi-index equality inside the global identity $m_2^{\mathrm{num}}=8Z$. The ambient setting is discrete gravity bookkeeping for the exact midpoint mass-squared identity in four dimensions, not continuum GR.
proof idea
One-line kernel decide. Both sides reduce to closed integer expressions once the six $\mathrm{Fin},4$ arguments are substituted: the left-hand side evaluates the fold that defines $m_2^{\mathrm{num}}$ at $(2,0,2,0,1,1)$, the right-hand side multiplies the pattern value of $Z$ at those indices by eight. decide discharges the resulting numeral equality; no lemmas beyond the two definitions are invoked.
why it matters
This cell is consumed by the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies one of the 256 concrete equalities the case split lands on.
In the broader Recognition gravity stack, the identity certifies that the midpoint Regge numerator matches the closed-form eightfold table used downstream in mass-squared and curvature bookkeeping. It is pure discrete algebra: no continuum limit, no forcing-chain step (T0–T8), and no appeal to $\varphi$ or the eight-tick octave. Its value is mechanical completeness of the kernel certificate.
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