e_021203
plain-language theorem explainer
For the six-index slot (0,2,1,2,0,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D midpoint Regge kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0$, $b=2$, $c=1$, $d=2$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-case kernel certification that the 4D midpoint Regge numerator equals eight times a tabulated integer kernel. The ambient setting is discrete gravity analysis: couplings on Fin 4 index sextuples are summed into an integer numerator, then matched against a closed-form table.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution function over a fixed coupling list: start at 0 and add each term's contribution at the six indices. The explicit kernel $Z$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by pattern-matched integer values (e.g. $4$, $-2$, and further cases in the full table).
Chunk 2 packages many pointwise equalities of the form $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ so a later assembler can discharge the universal statement by finite case split.
proof idea
One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold that defines the numerator at the fixed indices $(0,2,1,2,0,3)$; the right-hand side multiplies the pattern-matched kernel entry by 8. Lean’s kernel decision procedure checks integer equality, with no lemmas or rewriting beyond definitional reduction.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z$ and proves it by exhaustive fin_cases on all six Fin 4 arguments. Each chunk theorem such as this one supplies one concrete cell so the case tree closes without sorry.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge analysis in 4D: once the numerator is identified with eight times the explicit kernel, downstream curvature and mass-response identities can quote a closed integer table rather than a fold. It is pure finite certification, not a continuum GR derivation, and sits downstream of the discrete coupling definitions rather than the T0–T8 forcing chain.
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