e_010032
plain-language theorem explainer
For the Fin-4 index sextuple (0,1,0,0,3,2), the folded numerator coupling equals eight times the tabulated explicit integer. Gravity analysts assembling the global 4D Regge midpoint M2TT identity cite each such kernel case. Discharge is a single native decide on concrete integer arithmetic.
Claim. For indices $a=0$, $b=1$, $c=0$, $d=0$, $i=3$, $j=2$ in $\mathrm{Fin}\,4$, the numerator $m_2$ obtained by folding contributions over the coupling list equals $8$ times the explicit integer table entry at those same indices.
background
This module sits in the 4D Regge gravity analysis that certifies an exact midpoint identity for the $M_2$ numerator. The local setting is chunk 1 of a 256-way kernel split: each chunk discharges a batch of concrete index sextuples by decision procedure.
The numerator map folds a fixed coupling list, adding one integer contribution per term at the six $\mathrm{Fin},4$ slots. The explicit table is a pattern-matched integer function on the same six indices (sample values include $4$, $-2$, and other small constants on the diagonal and off-diagonal patterns).
The claim under audit is one pointwise instance of the global relation numerator $= 8\cdot$ explicit-table.
proof idea
One-line wrapper: by decide. Both sides reduce to concrete Int values (the fold over the coupling list on the left; the pattern match on the right), and the kernel checks equality. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Parent theorem is the assembled identity: for every $a,b,c,d,i,j\in\mathrm{Fin},4$, the numerator equals eight times the explicit table. That assembler runs nested fin_cases on all six indices and lands on kernel facts such as this one.
In the Recognition gravity stack these pointwise certificates underwrite the exact midpoint $M_2$ TT identity in 4D Regge calculus, a computational backbone for discrete curvature bookkeeping. The result is pure finite arithmetic; it does not itself invoke the forcing chain (T0–T8), the J-cost, or the $\varphi$-ladder, but it supplies a verified brick those continuum-facing arguments rely on when they quote the discrete identity.
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