e_310231
plain-language theorem explainer
For the six Fin-4 indices (3,1,0,2,3,1), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel point checks. The proof is a single decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}0$, $d{=}2$, $i{=}3$, $j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $\mathrm{m2Num}(3,1,0,2,3,1)=8\cdot\mathrm{explicitZ}(3,1,0,2,3,1)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num is defined by folding a fixed coupling list: it sums contrib terms over couplingZList for six Fin-4 indices. The companion map explicitZ is a closed-form integer table on the same six indices (pattern-matched cases such as (0,0,1,1,2,2)mapsto 4 and (0,0,1,2,1,2)mapsto -2).
The local module is chunk 13 of the exhaustive verification that m2Num = 8·explicitZ at every kernel point. The ambient goal is the exact midpoint M2TT identity in four dimensions; the kernel certificate supplies the two maps being compared, and the chunk files discharge the 256 concrete equalities by decision procedures.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are fixed at (3,1,0,2,3,1): m2Num unfolds the foldl over couplingZList, explicitZ hits its matching clause (or the default), and decide checks the integer equality against the factor 8.
why it matters
Feeds the universal statement m2Num_eq_eight_explicitZ in the assemble module, which introduces all six Fin-4 variables and runs fin_cases on each, routing every concrete cell to a chunk theorem of this form. That universal equality is the algebraic spine of the Regge exact-midpoint M2TT identity in 4D: it replaces the folded coupling sum by eight times the explicit kernel table, clearing the numerator side of the gravity identity. Within Recognition Science gravity work this is pure certificate scaffolding, not a forcing-chain step (T0–T8), but it is required before continuum or continuum-limit claims that quote the midpoint identity can be trusted.
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