e_012103

This module is chunk 1 of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. The ambient setting is exact midpoint analysis for the M2 TT identity in discrete gravity.

The numerator m2Num is defined by folding a fixed coupling list: it sums contrib t a b c d i j over every term t in couplingZList, yielding an integer for each six-tuple of Fin 4 indices. The comparison target explicitZ is a pattern-matched integer table on the same six indices (sample clauses include values 4, -2, and so on for distinguished index patterns).

The local claim is one concrete six-tuple equality inside that table-versus-fold comparison. Upstream, both sides are pure definitions in the kernel certificate module; no analytic hypothesis is carried.

One-line proof by decide. Both sides reduce to concrete integers: the left-hand side by evaluating the fold of contrib over couplingZList at the fixed indices (0,1,2,1,0,3), the right-hand side by unfolding the pattern match for explicitZ and multiplying by 8. Lean’s kernel decision procedure discharges the resulting integer equality with no lemmas and no case splits beyond what decide performs internally.

The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple of Fin 4 indices, proved by exhaustive fin_cases on all six arguments. Each leaf of that case tree is one of these e_* kernel facts; this declaration is the leaf for (0,1,2,1,0,3).

In the Recognition gravity stack, the exact midpoint M2 TT identity is part of the discrete curvature bookkeeping that feeds continuum limits and effective Newtonian potentials. Certifying m2Num = 8 · explicitZ pointwise replaces a symbolic fold by a closed integer table, which keeps later algebraic identities decidable. The module doc frames the whole file as “chunk 1 (256 kernel decides),” so this is pure certification scaffolding for that global equality, not a new physical law.