e_313203
plain-language theorem explainer
For the six Fin-4 indices (3,1,3,2,0,3), the folded coupling numerator equals eight times the explicit kernel integer. Gravity analysts certifying the 4D Regge midpoint M2 TT identity cite this as one concrete point in the 256-way kernel check. The proof is a single kernel decide on fixed integer arithmetic.
Claim. For indices $a=3$, $b=1$, $c=3$, $d=2$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the coupling-fold numerator equals eight times the tabulated explicit kernel value at that multi-index.
background
In the 4D Regge midpoint M2 TT analysis, two integer kernels on six $\mathrm{Fin},4$ indices are compared pointwise. The numerator is defined by folding a fixed coupling list and summing a contribution at each multi-index. Its companion is a sparse explicit table of small integers (entries such as $4$, $-2$, and so on).
This module is chunk 13 of a 256-way partition of the kernel identity "numerator $= 8\cdot$ explicit table". Each chunk discharges a block of concrete index tuples by decision procedure, so the universal statement can later be assembled by exhaustive fin_cases.
proof idea
One-line wrapper: by decide. With all six arguments fixed as concrete $\mathrm{Fin},4$ values, both the fold-numerator and the explicit table reduce to closed integer expressions, and Lean's decision procedure checks equality.
why it matters
This point check is consumed by the universal assembly theorem that states the numerator equals eight times the explicit kernel for every six-tuple of $\mathrm{Fin},4$ indices. That assembly runs nested fin_cases and lands each branch on a chunk lemma of this form. The identity is computational scaffolding inside the ReggeExactMidpoint M2 TT certification chain in the gravity layer: it locks discrete curvature bookkeeping rather than stating a new continuum law. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi-ladder mass formula.
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