e_312203
plain-language theorem explainer
For the six-index tuple (3,1,2,2,0,3) in Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=2$, $d=2$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the summed coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-cell case split establishing the algebraic identity m2Num = 8·explicitZ on all six-tuples of Fin 4 indices. The setting is the exact midpoint analysis of the 4D Regge M2TT kernel used in the Recognition gravity stack.
The numerator m2Num(a,b,c,d,i,j) is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison target explicitZ is a sparse integer table on Fin 4^6 (typical nonzero entries are ±2 or 4 on matched index patterns).
Chunk 13 isolates a block of those kernel cells so that each equality can be discharged by a pure integer decision procedure rather than a symbolic expansion of the fold.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six Fin 4 arguments are fixed: the left-hand side evaluates the fold that defines m2Num at (3,1,2,2,0,3); the right-hand side multiplies the table value of explicitZ at the same point by 8. No lemmas beyond the two definitions are required.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple by fin_cases on each coordinate and routes each cell to a chunk equality of this form. That global identity is the certified bridge between the folded coupling numerator and the closed-form kernel used in the exact midpoint M2TT analysis.
In the Recognition gravity pipeline this clears a purely combinatorial obstruction: once every cell matches, downstream curvature and mass-ladder arguments can quote a single closed form instead of a fold. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is infrastructure inside the 4D Regge kernel certification.
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