e_011313
plain-language theorem explainer
For the six Fin-4 indices (0,1,1,3,1,3), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0$, $b=1$, $c=1$, $d=3$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the integer $m_2$ numerator obtained by folding the coupling list equals $8$ times the explicit kernel value $Z$ at those same indices.
background
In the 4D Regge midpoint analysis the numerator m2Num is the fold of a fixed coupling list: each term contributes an integer depending on six face/edge indices in Fin 4. The companion table explicitZ is a closed-form integer kernel on the same six indices (sample values include $4$, $-2$, and so on).
This module is chunk 1 of a 256-cell decide grid that checks the pointwise identity $m_2\mathrm{Num}=8\cdot Z$ on every index tuple. The local setting is pure integer arithmetic on a finite discrete domain; no continuum limit or curvature hypothesis is in play yet.
Upstream, both m2Num and explicitZ are defined in the kernel certificate module; the present lemma only evaluates them at one concrete point.
proof idea
One-line decide. Lean reduces both sides at the concrete indices $(0,1,1,3,1,3)$ to ground integers (the fold for m2Num and the pattern match for explicitZ) and checks equality by kernel computation. No lemmas are invoked beyond definitional unfolding.
why it matters
The parent theorem m2Num_eq_eight_explicitZ assembles all $4^6$ index cells by nested fin_cases and needs each cell proved. This declaration discharges the single cell $(0,1,1,3,1,3)$.
In the Recognition gravity stack that identity is the algebraic backbone of the exact midpoint M2TT relation used in the discrete curvature bookkeeping. It does not itself touch the forcing chain (T5–T8) or the RCL, but it is infrastructure those continuum-facing gravity claims rely on once the discrete kernel is certified.
Sibling chunk lemmas cover the remaining index tuples; together they close the assemble theorem without sorry.
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