e_130201
plain-language theorem explainer
For the Fin-4 multi-index (1,3,0,2,0,1), the Regge midpoint M2 numerator equals eight times the explicit kernel value Z at those indices. Gravity analysts certifying the 4D TT-sector identity cite this as one cell of the 256-case exhaustive kernel grid. Proof is a single decide on concrete integer arithmetic after substitution.
Claim. At multi-index $(1,3,0,2,0,1)\in(\mathrm{Fin}\,4)^6$, the integer $M_2$ numerator equals $8$ times the explicit coupling value $Z(1,3,0,2,0,1)$.
background
In the 4D Regge exact-midpoint analysis, the TT-sector identity reduces to a pointwise integer comparison on six indices each ranging over ${0,1,2,3}$. The numerator side is obtained by folding a fixed contribution map over a coupling list; the comparison target is a sparse case table of small integers (typical entries $4$, $-2$, and so on).
This module is chunk 7 of that 256-case kernel certification. The local claim is pure finite evaluation: once the six $\mathrm{Fin},4$ arguments are fixed, both sides are closed terms in $\mathbb{Z}$. Upstream, the numerator fold and the explicit table are the only ingredients.
proof idea
One-line computational discharge via decide. Substituting the six concrete $\mathrm{Fin},4$ values turns both the folded numerator and the explicit table lookup into ground integers; the equality is then a decidable statement in $\mathbb{Z}$ with no residual quantifiers or hypotheses.
why it matters
This cell is consumed by the assembly theorem that asserts $M_2\mathrm{Num}=8\cdot Z$ for every sextuple in $(\mathrm{Fin},4)^6$, proved by exhaustive case split on the six indices. That universal identity is the algebraic core of the Regge exact-midpoint M2 TT-identity certificate in 4D gravity analysis. It sits in the discrete-curvature bookkeeping layer of the Recognition framework (supporting continuum-limit checks), only indirectly tied to the forcing-chain landmarks $D=3$ and the eight-tick octave. Closes one of the 256 decide cells in chunk 7.
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