e_132301
plain-language theorem explainer
For the six Fin-4 indices (1,3,2,3,0,1), the integer numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts assembling the 4D Regge midpoint M2 TT identity cite this as one kernel cell of the 256-case chunk. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=1$, $b=3$, $c=2$, $d=3$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the TT-sector mass-squared numerator is built by folding a fixed coupling list: m2Num(a,b,c,d,i,j) sums contrib terms over couplingZList. The companion table explicitZ assigns a sparse pattern of integers (typically 4, -2, or 0) to sextuples of Fin-4 indices; only a handful of patterns are nonzero.
This module is chunk 7 of the exhaustive verification that m2Num = 8·explicitZ on every cell of the 4^6 index grid. The factor eight is the combinatorial multiplicity that converts the folded sum into the closed table. The local setting is pure integer arithmetic on Fin 4, with no floating-point or continuum limit yet.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values for the fixed sextuple (1,3,2,3,0,1), so the equality is discharged by Lean's decision procedure on integer literals. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
The parent theorem m2Num_eq_eight_explicitZ quantifies over all Fin-4 sextuples by nested fin_cases; each leaf is one of these e_* kernel identities. This cell supplies the (1,3,2,3,0,1) case inside that assembly, closing part of chunk 7 of the 256-decide kernel.
In the broader Gravity stack the identity underwrites the exact midpoint M2 TT formula used in the Regge analysis. It is bookkeeping rather than a new physical law, but without the cell-by-cell match the closed-form explicitZ table cannot replace the folded sum in downstream curvature and mass-squared identities.
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