e_020232
plain-language theorem explainer
Pointwise identity: the folded m2 numerator at multi-index (0,2,0,2,3,2) equals eight times the explicit Z table entry. Gravity analysts cite it as one cell of the 4D Regge midpoint kernel certification. The proof is a single kernel decide on closed integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(0,2,0,2,3,2)$ 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
This module is chunk 2 of a 256-way split certifying $m_2^{\mathrm{num}}=8\cdot Z$ on every sextuple of $\mathrm{Fin},4$ indices in the 4D Regge exact-midpoint analysis.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The comparison target $\mathrm{explicitZ}$ is a closed integer table on $(\mathrm{Fin},4)^6$, with sparse nonzero pattern (e.g. $4$ on diagonal-like slots, $-2$ on selected off-diagonal pairs).
The local goal is purely algebraic: evaluate both sides at one concrete multi-index and check equality in $\mathbb{Z}$.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed: the fold that defines $m_2^{\mathrm{num}}$ becomes a finite sum of integer contributions, and $\mathrm{explicitZ}$ is a pattern-match table. Lean’s kernel decision procedure discharges the resulting closed $\mathbb{Z}$ equality with no lemmas beyond the two definitions.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ by exhausting all $\mathrm{Fin},4$ cases and invoking the chunk identities. That universal identity is the certified algebraic core of the 4D Regge exact-midpoint $M_2$ TT kernel used in the gravity analysis stack.
In the broader RS gravity program this is bookkeeping, not a forcing-chain step: it locks the discrete curvature/mass-squared numerator against the explicit $Z$ table so later continuum or continuum-limit arguments can quote a proved pointwise match rather than a schematic factor of eight.
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