e_023213
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at index sextuple (0,2,3,2,1,3) equals eight times the explicit Z-table entry. Gravity analysts cite it as one cell of the 4D Regge midpoint M2TT kernel certification. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,3,2,1,3)$ with each coordinate in $\mathbb{F}_4$, 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 2 of a 256-way kernel split certifying $m_2^{\mathrm{num}}=8\cdot Z$ on all sextuples in $(\mathbb{F}_4)^6$. The setting is the 4D Regge exact-midpoint analysis of the M2TT identity in the Gravity domain.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums a contribution functional over that list and returns an integer. The companion table $Z$ is an explicit case-split function $\mathbb{F}_4^6\to\mathbb{Z}$ listing the closed-form integer values expected after that fold (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).
The full universal statement is assembled downstream by exhausting all Fin-4 coordinates; each chunk theorem such as this one discharges one concrete cell.
proof idea
One-line computational proof: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed, so the kernel decision procedure compares the folded sum against $8$ times the matching $Z$ clause and closes the equality with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and proves it by nested fin_cases over the six coordinates, invoking one chunk identity per cell. That universal equality is the numerical backbone of the 4D Regge exact-midpoint M2TT kernel certificate: it confirms the folded coupling numerator matches the closed-form Z table everywhere on the discrete index space. Within Recognition gravity analysis this is bookkeeping infrastructure rather than a forcing-chain landmark (T0–T8), but without the pointwise cells the assembly cannot close.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.