e_302013
plain-language theorem explainer
For the six-index slot (3,0,2,0,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2 TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3,b{=}0,c{=}2,d{=}0,i{=}1,j{=}3$ in $\mathrm{Fin}\,4$, the summed coupling numerator equals eight times the explicit integer kernel: $N(3,0,2,0,1,3)=8\,Z(3,0,2,0,1,3)$.
background
In the 4D Regge midpoint analysis, the TT-sector mass-squared numerator is built by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and m2Num is that total sum. The companion map explicitZ is a sparse closed-form table of the same integers (nonzero only on a short list of index patterns such as $(0,0,1,1,2,2)\mapsto 4$).
The module is one of the 256-decide chunks that certify the global identity m2Num = 8·explicitZ pointwise. Chunk 12 covers a block of concrete sextuples; the present declaration is the single point $(3,0,2,0,1,3)$ inside that block.
Upstream, both m2Num and explicitZ are pure definitions in the kernel certificate module; no analytic lemma is required beyond evaluating the fold and the pattern match at fixed indices.
proof idea
One-line proof by decide. Lean reduces both sides to concrete integers: the left-hand side by folding couplingZList with the contribution function at the fixed indices, the right-hand side by evaluating the pattern-match definition of explicitZ (or zero if unmatched) and multiplying by 8. Equality of the resulting Int literals is decided by the kernel.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in Fin 4 by exhaustive fin_cases and discharge of each point lemma. That global equality is the algebraic core of the exact midpoint M2 TT identity in 4D Regge gravity inside the Recognition Science gravity stack.
The factor of 8 is the combinatorial normalization tying the folded coupling sum to the sparse kernel table; establishing it pointwise removes any residual gap between the constructive numerator and the closed form used in continuum matching. No open scaffolding remains on this point: the decide closes the integer identity completely.
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