e_023202
plain-language theorem explainer
For the six-index slot (0,2,3,2,0,2) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts cite these micro-identities when assembling the global Regge midpoint M2–TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,2,3,2,0,2)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-case kernel certification that the Regge-exact midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$ on $(\mathrm{Fin},4)^6$. The local setting is pure finite arithmetic: no continuum limit, only integer contributions from a fixed coupling list.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing a contribution functional at those six indices. The companion table explicitZ hard-codes the expected integers on the nonzero support (e.g. $4$, $-2$, and zeros elsewhere). The identity $m_2^{\mathrm{num}}=8Z$ is the algebraic content being certified case by case.
Chunk 2 holds one block of those decides; the present declaration is the single point $(0,2,3,2,0,2)$ inside that block.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values: the left-hand side by evaluating the fold that defines $m_2^{\mathrm{num}}$ at the six fixed Fin 4 arguments, the right-hand side by looking up explicitZ and multiplying by 8. Lean’s kernel decision procedure discharges the resulting integer equality with no lemmas or rewriting.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over $(\mathrm{Fin},4)^6$ and routing each tuple to its chunk theorem. That global identity is the certified algebraic core of the 4D Regge-exact midpoint M2–TT analysis in the Gravity domain.
Within Recognition Science gravity work, such kernel equalities underwrite discrete curvature bookkeeping before continuum or phenomenological claims. This declaration does not itself touch the forcing chain (T0–T8), RCL, or the $\varphi$-ladder; it is pure finite support arithmetic supporting the Regge midpoint identity stack.
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