e_002302
plain-language theorem explainer
For the six-index slot (0,0,2,3,0,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2 TT identity cite this as one of the 256 finite-case checks. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator $m_2^{\mathrm{num}}(0,0,2,3,0,2)$ equals $8\,Z_{\mathrm{expl}}(0,0,2,3,0,2)$, where $Z_{\mathrm{expl}}$ is the explicit integer kernel table and $m_2^{\mathrm{num}}$ is the fold of coupling contributions over the fixed coupling list.
background
This module is one chunk of the 256-case kernel certification that the Regge midpoint $M_2$ numerator equals eight times an explicit integer table on all six-tuples of $\mathrm{Fin},4$ indices. The local slogan is "$m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$, chunk 0 (256 kernel decides)."
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a contribution functional at those indices. The companion table $Z_{\mathrm{expl}}$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on for distinguished index patterns).
The ambient setting is 4D discrete gravity analysis: exact algebraic identities for the midpoint $M_2$ TT sector, reduced to finite integer arithmetic on a $4^6$ index grid.
proof idea
One-line kernel proof: by decide. Both sides are closed integer expressions once the six concrete Fin 4 indices are substituted, so the decision procedure evaluates the fold defining $m_2^{\mathrm{num}}$ and the pattern match defining $Z_{\mathrm{expl}}$ and checks equality with the factor eight. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for all six-tuples, which is proved by exhaustive fin_cases on each index and discharge of every leaf by a chunk theorem of this form. Without the full 256-case cover, the closed-form replacement of the folded numerator by the explicit kernel table is not certified.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint $M_2$ TT sector in 4D Regge-type analysis: it turns a sum-over-couplings definition into a sparse integer table, which is what later curvature and continuum-limit arguments actually consume. It is not itself a forcing-chain step (T0–T8); it is infrastructure under the discrete gravity identities those continuum claims rest on.
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