e_232010
plain-language theorem explainer
For the six Fin-4 indices (2,3,2,0,1,0), the folded coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts certifying the exact midpoint M2 TT identity in 4D cite this as one cell of the exhaustive case split. Proof is a single kernel decide on concrete Int arithmetic.
Claim. At multi-index $(2,3,2,0,1,0)\in(\mathrm{Fin}\,4)^6$, the coupling-fold numerator equals eight times the corresponding explicit integer kernel value.
background
In the Regge-exact midpoint analysis for the 4D M2 TT identity, two integer kernels on six Fin 4 indices are compared pointwise. The numerator is defined by folding a fixed coupling-contribution list: each term adds an integer contrib at the given multi-index, starting from zero. The explicit kernel is a sparse lookup table on the same six indices, returning small integers (for example 4 on certain paired diagonals, -2 on selected off-diagonal swaps).
This module is chunk 11 of a partitioned decide campaign whose goal is the global identity numerator = 8 · explicit table. The setting is pure finite integer arithmetic; no metric, continuum limit, or Recognition forcing hypothesis enters these cells.
proof idea
One-line wrapper discharged by decide. Both sides of the equality reduce, via the definitions of the fold-numerator and the explicit table, to concrete Int values at the fixed indices (2,3,2,0,1,0). The kernel decision procedure checks integer equality; no intermediate lemmas are invoked.
why it matters
Supplies one discharged cell to the assembler m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in (Fin 4)^6 by nested fin_cases and consumes the chunk theorems. That global algebraic identity is part of the certified core of the Regge-exact midpoint M2 TT identity in the 4D gravity analysis stack. It is discrete bookkeeping rather than a T0–T8 forcing step, but it closes a verification obligation that later geometric or continuum claims rest on.
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