e_310320
plain-language theorem explainer
For the six-index tuple (3,1,0,3,2,0) in (Fin 4)^6, the folded numerator coupling m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite this as one of the 256 exhaustive kernel cases. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}0$, $d{=}3$, $i{=}2$, $j{=}0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value at those indices.
background
This module is chunk 13 of the 256-case kernel that certifies m2Num = 8·explicitZ pointwise on (Fin 4)^6. The setting is the exact midpoint M2 TT identity in 4D Regge gravity analysis: one must match a folded sum of coupling contributions against a sparse closed-form table.
Upstream, m2Num(a,b,c,d,i,j) is defined as the fold of contrib(·,a,b,c,d,i,j) over couplingZList, starting from 0. The companion explicitZ is a pattern-matched Int-valued table on six Fin-4 indices (nonzero only on a short list of index patterns such as (0,0,1,1,2,2) ↦ 4 and (0,0,1,2,1,2) ↦ −2). The identity claims the fold always recovers eight times that table entry.
Chunk theorems fix one concrete six-tuple and discharge the equality by computation, so the later universal assembler can finish by fin_cases alone.
proof idea
One-line proof by decide. Lean evaluates both sides at the concrete Fin-4 indices (3,1,0,3,2,0): the left-hand fold of contrib over couplingZList and the right-hand 8·explicitZ lookup. Both reduce to the same integer, so the kernel closes the goal with no lemmas or rewriting.
why it matters
Parent theorem m2Num_eq_eight_explicitZ 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 nested fin_cases on all six indices. Each leaf of that case tree is one of these chunk theorems (here e_310320 for the tuple 3,1,0,3,2,0).
Without the pointwise kernel facts, the universal identity cannot be assembled. In the broader Recognition gravity stack this identity is part of the exact midpoint M2 TT analysis that underwrites discrete curvature bookkeeping in 4D; it is computational scaffolding for the Regge side rather than a forcing-chain (T0–T8) landmark, but it is load-bearing for any downstream claim that quotes the closed-form eightfold kernel.
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