e_010021
plain-language theorem explainer
Pointwise identity: the midpoint Regge $m_2$ numerator at multi-index $(0,1,0,0,2,1)$ equals eight times the explicit integer $Z$ table at that index. Gravity analysts assembling the 4D TT midpoint kernel cite these 256 chunk lemmas. The proof is a single kernel `decide` on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,0,2,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8\,Z(a,b,c,d,i,j)$, where $Z$ is the explicit integer table for the 4D midpoint TT kernel.
background
This module is chunk 1 of a 256-way split proving $m_2^{\mathrm{num}}=8\cdot Z$ on every 6-tuple in $(\mathrm{Fin},4)^6$. The setting is the exact midpoint Regge analysis of the 4D transverse-traceless (TT) identity used in the gravity stack.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling entry at the six indices. The companion table $Z$ is an explicit pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on at listed patterns).
The full quantified statement is assembled downstream by exhaustive fin_cases over all six indices; each chunk such as this one discharges one concrete cell.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern table and the factor $8$), so the equality is a decidable closed integer computation with no lemmas beyond the two definitions.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and proves it by six nested fin_cases invocations, each landing on a chunk like this one. That global identity is the certified bridge between the folded coupling definition and the closed-form $Z$ table in the 4D midpoint TT Regge kernel. Within Recognition gravity analysis it is bookkeeping infrastructure, not a forcing-chain step (T0–T8), but it is required for any later claim that quotes the exact midpoint $m_2$ numerator in closed form.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.