e_123033
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (1,2,3,0,3,3) equals eight times the explicit integer table at those indices. Gravity analysts cite it when assembling the full 4D midpoint M2 numerator identity over (Fin 4)^6. The proof is a single kernel decide on concrete integers.
Claim. At the multi-index $(1,2,3,0,3,3)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(1,2,3,0,3,3)=8\,Z_{\mathrm{expl}}(1,2,3,0,3,3)$.
background
In the 4D Regge midpoint analysis, the numerator side of the M2 TT identity is written two ways. One is a fold: sum the local coupling contributions over a fixed list of tensors, yielding an integer for each sextuple of Fin-4 indices. The other is an explicit closed-form table on the same domain, recording a sparse pattern of small integers (typically $\pm 2,\pm 4$, and zeros).
This module is chunk 6 of a 256-way kernel split that discharges the pointwise claim "fold equals eight times table" by concrete decision. The two upstream definitions are exactly that fold and that table; neither carries a proof, only the raw integer data the decide tactic evaluates.
Local setting (module doc): prove $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ one sextuple at a time so the assembler can recombine them.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are literals, so the kernel equality checker closes the goal with no lemmas, rewrites, or case splits inside this declaration.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}4,; m_2^{\mathrm{num}}=8,Z{\mathrm{expl}}$ and is proved by exhaustive fin_cases on the six indices. Each chunk theorem such as this one supplies one concrete cell of that $4^6$ grid.
In the broader gravity stack this identity is bookkeeping for the exact midpoint M2 TT relation in 4D Regge calculus: matching the folded numerator against a sparse closed form clears an algebraic bottleneck before continuum or continuum-limit arguments. It does not itself invoke Recognition landmarks (T5 J-cost, $\phi$, eight-tick), but sits inside the discrete-gravity layer those landmarks later constrain.
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