e_212030
plain-language theorem explainer
Single kernel point check: the folded coupling numerator at multi-index (2,1,2,0,3,0) equals eight times the explicit integer table value at that same index. Gravity analysts assembling the 4D Regge midpoint M2 identity cite it as one of the 256 concrete cases. The proof is a pure `decide` on ground integers.
Claim. Let $N$ be the integer numerator obtained by folding the coupling contribution list, and let $Z$ be the explicit closed-form integer table on $(\mathbb{F}_4)^6$. Then $N(2,1,2,0,3,0)=8\,Z(2,1,2,0,3,0)$.
background
This module sits inside the 4D Regge exact-midpoint analysis for the M2/TT identity. The local goal, stated in the module header, is to certify $N=8Z$ on one chunk of the $4^6=4096$ index space by discharging 256 concrete kernel points.
Upstream, $N(a,b,c,d,i,j)$ is defined as the fold of a fixed coupling list: each term contributes an integer via a local contribution map, and the fold starts at 0. The comparison object $Z$ is an explicit pattern-matched integer table on six $\mathrm{Fin},4$ indices (sample clauses send $(0,0,1,1,2,2)$ to $4$ and $(0,0,1,2,1,2)$ to $-2$).
The present declaration fixes one concrete six-tuple and asserts the scalar identity at that point only.
proof idea
Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted. The tactic decide evaluates the fold that defines $N$ and the pattern match that defines $Z$, then checks integer equality. No lemmas beyond the two defining equations are required; there is no algebraic rewriting.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states the universal claim $\forall a,b,c,d,i,j,,N=8Z$ and discharges it by exhaustive fin_cases on the six indices. Each chunk lemma such as this one supplies one ground instance that the case split must hit, so the 256-point chunk is the computational spine of the identity.
In the broader Recognition gravity stack, the midpoint M2/TT identity is part of the discrete curvature bookkeeping that feeds continuum limits and effective Newtonian response. The factor of 8 is the exact multiplicity relating the summed coupling numerator to the closed table; certifying it pointwise removes a potential source of off-by-constant error in later mass and force extractions.
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