e_010033
plain-language theorem explainer
For the discrete index tuple (0,1,0,0,3,3), the midpoint Regge numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0$, $b{=}1$, $c{=}0$, $d{=}0$, $i{=}3$, $j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-case kernel certification that the discrete midpoint Regge numerator equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, matching a 4D combinatorial skeleton for the Regge exact-midpoint identity.
The numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing local contributions at those six indices. The companion table $\mathrm{explicitZ}$ 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 local theoretical setting is purely algebraic: both sides are concrete integers once the indices are fixed, so each chunk reduces to a decidable equality.
proof idea
One-line kernel proof: by decide. With all six arguments ground terms in $\mathrm{Fin},4$, both $\mathrm{m2Num}$ (a fold of integer contributions) and $\mathrm{explicitZ}$ (a pattern match to an integer) evaluate to concrete Int values; Lean’s decision procedure checks equality to $8$ times the table entry.
why it matters
Feeds the assembly theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ by exhaustive fin_cases and discharge of each ground instance. That global equality is the certified algebraic core of the 4D Regge exact-midpoint $M_2$ TT identity in the Gravity analysis stack.
In the Recognition Science gravity line, such kernel identities lock the discrete curvature/coupling bookkeeping before continuum or phenomenological claims. This chunk does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure under the Regge midpoint analysis.
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