e_133010
plain-language theorem explainer
Single kernel identity: the midpoint Regge numerator m2Num at multi-index (1,3,3,0,1,0) equals eight times the closed-form explicitZ value at the same index. Gravity analysts assembling the 4D TT midpoint identity cite these pointwise facts. Proof is a pure `decide` on concrete integer arithmetic after unfolding the fold and the case table.
Claim. For the six indices $(a,b,c,d,i,j)=(1,3,3,0,1,0)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-way kernel split that discharges the pointwise claim $m_2^{\mathrm{num}}=8\cdot Z$ on $\mathrm{Fin},4^6$ for the 4D Regge exact-midpoint TT identity.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined as a left fold of contrib over couplingZList, yielding an integer. The companion explicitZ is a total case table $\mathrm{Fin},4^6\to\mathbb{Z}$ listing the closed-form values (e.g. $4$, $-2$, and further patterns on the remaining tuples).
The local goal is mechanical: evaluate both sides at one fixed six-tuple and certify equality in $\mathbb{Z}$, so the assembler can recombine all chunks into a single universal statement.
proof idea
One-line kernel proof: by decide. Lean unfolds m2Num (the fold of contrib over couplingZList) and explicitZ (the matching case of the integer table) at the concrete indices $1,3,3,0,1,0$, reduces both sides to numerals, and checks $n=8\cdot z$ by decidable integer equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the parent assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices, each case landing on a chunk theorem of this form.
In the Recognition gravity stack this identity is the algebraic core of the 4D Regge exact-midpoint TT kernel certificate: the folded coupling numerator is forced to match eight times the explicit $Z$-table, so the continuum midpoint identity can be read off without residual combinatorial slack. It is pure discrete linear algebra on the coupling list; it does not itself invoke the forcing chain (T5–T8) or the RCL, but it is infrastructure those continuum limits sit on once the kernel is certified.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.