e_311002
plain-language theorem explainer
Concrete kernel check: the folded M2 numerator at index sextuple (3,1,1,0,0,2) equals eight times the explicit Z-table entry. Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite it as one of 256 decided cases. The proof is a single `decide` on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(3,1,1,0,0,2)$ 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
In the 4D Regge exact-midpoint analysis, two integer-valued maps on sextuples of $\mathbb{F}_4$ indices appear. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local kernel, and the fold starts at $0$. The companion table $Z$ is an explicit piecewise function on the same six indices, returning small integers such as $4$, $-2$, and so on.
The local module is chunk 13 of a 256-case kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. The ambient goal is the exact midpoint M2–TT identity in four dimensions, reduced to finite arithmetic on $\mathrm{Fin},4$ labels rather than symbolic continuum tensors.
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions in the kernel certificate module; no analytic hypotheses remain once the indices are fixed.
proof idea
One-line computational proof: decide. With all six indices concrete, both sides reduce to closed Int expressions (a finite fold versus a table lookup), and Lean’s decision procedure checks equality by evaluation. No lemmas are invoked beyond the definitions of the numerator fold and the explicit $Z$ table.
why it matters
This case is one tile in the exhaustive identity m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and discharges the claim by fin_cases over all six $\mathrm{Fin},4$ arguments. That global equality is the algebraic backbone of the 4D Regge exact-midpoint M2–TT certificate: once every kernel cell matches, the continuum-facing TT identity inherits an exact discrete prefactor of eight.
Within Recognition gravity, the certificate sits in the discrete curvature bookkeeping that must line up before continuum limits and ladder normalizations are trusted. It does not itself invoke $\phi$, the eight-tick octave, or $D=3$; those enter at higher assembly layers that consume the certified identity.
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