e_211120
plain-language theorem explainer
For the six-index slot (2,1,1,1,2,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D Regge midpoint kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2,b=1,c=1,d=1,i=2,j=0$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 9 of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a sparse explicit integer table. The ambient setting is discrete gravity analysis: couplings on oriented 4-simplices reduced to six Fin-4 indices.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison object explicitZ is a pattern-matched integer table on six Fin-4 arguments (typical nonzero values $\pm 2,\pm 4$).
The global claim is $\forall a,b,c,d,i,j:, m_2^{\mathrm{num}}=8,Z$. Because the domain is finite of size $4^6=4096$ but the proof is split into decide chunks, each chunk theorem pins one concrete sextuple.
proof idea
One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at the fixed indices $(2,1,1,1,2,0)$, the right by looking up explicitZ (or defaulting to zero if unmatched) and multiplying by 8. Lean’s kernel decision procedure discharges the integer equality with no lemmas or case splits in this file.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which introduces six Fin-4 variables and runs fin_cases across all coordinates, invoking one chunk equality per cell. Without these pointwise facts the global midpoint identity does not close.
In the Recognition gravity stack this identity is bookkeeping for the exact discrete curvature/mass kernel at the Regge midpoint, not a new dynamical law. It supports later continuum or continuum-limit comparisons once the numerator is replaced by the closed form $8Z$. No T0–T8 forcing step is proved here; the result is pure finite combinatorial certification inside the gravity analysis layer.
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