e_110101
plain-language theorem explainer
At multi-index (1,1,0,1,0,1), the folded M2 numerator equals eight times the explicit integer kernel Z. Gravity analysts cite it as one of the 256 kernel cells in the 4D Regge exact-midpoint M2TT identity. The proof is a single `decide` on concrete integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(1,1,0,1,0,1)$ in $(\mathrm{Fin}\,4)^6$, the M2 numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 5 of a 256-cell kernel certification that the folded M2 numerator agrees with eight times an explicit integer table on every 6-tuple of Fin 4 indices. The setting is the 4D Regge exact-midpoint M2TT identity in the gravity analysis layer.
The numerator m2Num is defined by folding a contribution function over a fixed coupling list: start at 0 and add contrib t a b c d i j for each tag t. The comparison target explicitZ is a closed pattern-match table Fin 4^6 → ℤ with sparse nonzero entries (e.g. 4 on diagonal-like pairs, −2 on crossed pairs).
Chunk theorems pin one concrete 6-tuple each so the assembler can discharge the universal statement by exhaustive case split.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers (the fold of m2Num at fixed indices versus 8 * explicitZ at those same indices), and the decidable equality on Int closes the goal with no lemmas beyond the definitions.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states ∀ a b c d i j, m2Num a b c d i j = 8 * explicitZ a b c d i j and proves it by six nested fin_cases over Fin 4. Each chunk cell such as this one supplies one of the 256 residual goals.
In the Recognition gravity stack, that universal identity is the certified algebraic core of the 4D Regge exact-midpoint M2TT relation: the folded coupling numerator is exactly eight copies of the explicit kernel. Without the cell-by-cell decide layer, the assembler cannot close. This is pure discrete kernel bookkeeping, not a continuum GR derivation, but it is the gate the later midpoint-identity theorems rely on.
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