e_010011
plain-language theorem explainer
Pointwise identity: the folded Regge numerator at index sextuple (0,1,0,0,1,1) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one of 256 kernel cells assembling the global m2Num = 8·explicitZ statement. Proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,1,0,0,1,1)$ in $(\mathrm{Fin}\,4)^6$, 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 1 of a 256-cell kernel certification that the folded Regge midpoint numerator equals eight times a closed-form integer table on every sextuple of Fin 4 indices. The ambient setting is 4D Regge calculus analysis for the exact midpoint M2TT identity.
The numerator m2Num is defined by folding a fixed coupling list: start at 0 and add a contribution term for each coupling entry at the six indices. The table explicitZ is a pattern-matched integer function on six Fin 4 arguments (sample clauses include values 4, -2, and so on). The claim is pure integer arithmetic at one fixed point of the six-dimensional discrete domain.
Upstream, both sides are pure defs from the kernel certificate module; no analytic hypotheses are carried.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are substituted: the fold in the numerator becomes a finite sum of contributions, and explicitZ hits a single pattern clause (or the default). Lean’s kernel decision procedure checks the resulting Int equality. No lemmas are invoked beyond the two definitions.
why it matters
This cell is one of the twelve siblings in chunk 01 (and one of 256 overall) that the assembly theorem m2Num_eq_eight_explicitZ consumes. That parent states the universal identity $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by nested fin_cases over all six arguments, discharging each leaf by the matching e_* decide theorem.
In the Recognition gravity stack, the identity certifies that the discrete midpoint curvature numerator collapses to a sparse explicit table, which is the algebraic backbone for exact 4D Regge midpoint identities used downstream in the gravity analysis chain. It does not itself touch T0–T8 or the J-cost forcing chain; it is infrastructure inside the gravity numerics layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.