e_020023
plain-language theorem explainer
Pointwise identity: the folded Regge M2 numerator at multi-index (0,2,0,0,2,3) equals eight times the explicit integer table value. Gravity analysts cite it as one of the 256 kernel cases that assemble the global m2Num = 8·explicitZ identity. Proof is a single decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,0,2,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit table entry: $m_2^{\mathrm{num}}(0,2,0,0,2,3)=8\,Z_{\mathrm{expl}}(0,2,0,0,2,3)$.
background
In the 4D Regge exact-midpoint M2 TT analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib kernel, and the fold starts at 0. The comparison target $Z_{\mathrm{expl}}$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on on the diagonal and off-diagonal slots).
The module is chunk 2 of a 256-case kernel certification: each declaration fixes one sextuple and asserts $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at that point. The factor 8 is the structural constant relating the folded sum to the closed table across the whole kernel.
proof idea
One-line kernel decision. Both sides reduce to concrete integers once the six indices are substituted into the fold definition of $m_2^{\mathrm{num}}$ and the pattern table for $Z_{\mathrm{expl}}$. The tactic decide discharges the resulting numeral equality; no lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for all $\mathrm{Fin},4$ sextuples, which exhausts the index space by fin_cases and consumes these chunk lemmas. That global identity is the algebraic backbone of the Regge exact-midpoint M2 TT certification in 4D gravity: it replaces a folded coupling sum by an explicit sparse integer table, making downstream curvature and deficit identities computable by pure case analysis. Within Recognition Science gravity work this is bookkeeping infrastructure, not a forcing-chain step (T0–T8), but it is required before continuum or continuum-limit claims can quote a closed M2 numerator.
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