e_220233
plain-language theorem explainer
Single kernel case: the midpoint M2 numerator at multi-index (2,2,0,2,3,3) equals eight times the explicit Z-table entry. Gravity analysts cite it only as one tile in the full 4D identity m2Num = 8·explicitZ. Proof is a pure `decide` on a closed integer equality.
Claim. For indices $(a,b,c,d,i,j)=(2,2,0,2,3,3)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
This module is chunk 10 of a 256-case kernel that certifies the exact midpoint M2–TT identity in 4D Regge analysis. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each triple’s contribution at the six $\mathbb{F}_4$ slots. The comparison target is explicitZ, a sparse integer table on $(\mathbb{F}_4)^6$ (typical nonzero values $\pm 2,4$).
The local claim is only the equality at one concrete sextuple. The parent assembly theorem then exhausts all $4^6$ index combinations by fin_cases and glues the chunks into $\forall$-form $m_2^{\mathrm{num}}=8,Z$.
proof idea
One-line computational proof: by decide. Lean reduces both sides of the integer equality (the fold that defines the numerator versus eight times the table lookup) and closes the goal by the decidable equality instance on Int. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathbb{F}_4)^6$ by casing on each coordinate and dispatching to these chunk lemmas. That global identity is the algebraic core of the exact midpoint M2–TT certificate in the 4D Regge gravity analysis stack. It is bookkeeping, not a new physical law: once every kernel cell matches, the continuum-facing midpoint formula inherits a fully expanded, machine-checked numerator. No Recognition forcing step (T5–T8) is touched directly; the result sits downstream in the gravity numerics layer.
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