e_013320
plain-language theorem explainer
For the Fin-4 multi-index (0,1,3,3,2,0), the folded numerical M2 coupling equals eight times the explicit integer Z kernel at that point. Gravity analysts certifying the 4D Regge exact-midpoint M2TT identity cite this as one of 256 kernel cells. The proof is a single kernel decision on concrete integer equality.
Claim. For indices $a=0$, $b=1$, $c=3$, $d=3$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the folded numerical second-moment coupling at $(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value at the same multi-index.
background
In the 4D Regge exact-midpoint stack, the numerical second-moment coupling is an integer obtained by folding a fixed contribution map over a coupling list, one value per six-tuple of indices in $\mathrm{Fin},4$. The companion explicit Z table is a sparse closed-form integer function on the same multi-indices (typical nonzero values: $4$ on matched diagonal-type pairs, $-2$ on crossed pairs).
This module is chunk 1 of a 256-cell kernel certification that every numerical coupling equals eight times the matching explicit Z entry. The factor-of-eight identity is pure discrete bookkeeping: evaluate both sides at each multi-index, then assemble.
Upstream, the two sides are exactly those definitions: the fold that builds the numerical coupling, and the pattern-matched explicit Z table.
proof idea
One-line kernel decision. Lean decide evaluates both sides at the concrete multi-index $(0,1,3,3,2,0)$: the fold defining the numerical coupling on the left, and eight times the matched clause of the explicit Z table on the right. Both reduce to integers; propositional equality closes with no further lemmas.
why it matters
This cell is consumed by the assembler that states the identity for every six-tuple in $\mathrm{Fin},4$ and discharges the universal quantifier by exhaustive case splits on each index. The assembled theorem is the bridge from per-cell kernel facts to a single clean equality between the numerical M2 coupling and eight times the explicit Z kernel.
Within Recognition Science gravity analysis, that equality is a required algebraic checkpoint on the path to the exact-midpoint M2TT identity in four dimensions. It is local discrete-gravity certification, not a step in the T0–T8 forcing chain, the Recognition Composition Law, or the continuum constants ladder.
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