e_311333
plain-language theorem explainer
For the six-index tuple (3,1,1,3,3,3) in Fin 4, the folded numerator m2Num equals eight times the closed-form table value explicitZ. Gravity analysts assembling the exact midpoint M2TT identity cite this as one of the 256 kernel-decided cells in chunk 13. The proof is a single decide, discharging both sides by computation.
Claim. For indices $a{=}3,\,b{=}1,\,c{=}1,\,d{=}3,\,i{=}3,\,j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table entry: $m_2^{\mathrm{num}}(3,1,1,3,3,3)=8\,Z_{\mathrm{expl}}(3,1,1,3,3,3)$.
background
In the Regge exact-midpoint M2TT analysis, the numerator m2Num is the fold of a fixed coupling list: it sums integer contributions over that list at a six-tuple of Fin-4 indices. The companion table explicitZ is a pattern-matched integer function on the same six-tuples (sample clauses give values such as 4, -2, etc.).
The local module certifies the pointwise identity m2Num = 8·explicitZ in a 256-cell chunk of the 4^6 index space. Chunk 13 is one such block of kernel-decidable equalities; the present declaration is the cell at (3,1,1,3,3,3).
Upstream, both sides are pure definitions in the kernel-cert module: no analytic hypotheses, only the fold and the table.
proof idea
One-line computational proof: by decide. Lean evaluates both the fold that defines m2Num at (3,1,1,3,3,3) and the matching clause of explicitZ, then checks the integer equality against the factor 8. No lemmas are invoked beyond the kernel decision procedure.
why it matters
This cell is consumed by the assembler m2Num_eq_eight_explicitZ, which states the full pointwise identity for every six-tuple in Fin 4 and discharges it by exhaustive fin_cases. The chunk theorems supply the concrete equalities that make that sweep succeed.
In the gravity stack the identity is the algebraic backbone of the exact midpoint M2TT kernel: once numerator and explicit table agree up to the universal factor 8, later curvature and continuum-limit arguments can quote a closed form instead of a fold. It is bookkeeping inside the Regge analysis, not a forcing-chain landmark (T0–T8), but it is required scaffolding for any claim that the discrete M2 numerator matches the explicit Z table.
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