e_212330
plain-language theorem explainer
For the six Fin-4 indices (2,1,2,3,3,0), the folded midpoint numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one kernel cell of the 256-case grid. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=2$, $b=1$, $c=2$, $d=3$, $i=3$, $j=0$ 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
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification: it asserts cellwise that the folded numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity on a 4-index simplex lattice, with all index arguments ranging over $\mathrm{Fin},4$.
The numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start from $0$ and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The table $\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.). The identity $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ is the algebraic content being certified cell by cell.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values for the fixed sextuple $(2,1,2,3,3,0)$, so the decidable equality checker discharges the goal with no lemmas or case splits inside this declaration.
why it matters
This cell feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and invokes the chunk theorems as the leaves. Closing the numerator identity is a prerequisite step in the Regge exact-midpoint M2TT 4D kernel certificate imported by this analysis stack. In the broader Recognition gravity line, such exact discrete identities underwrite continuum limits and curvature bookkeeping; they do not themselves invoke the T0–T8 forcing chain, $\phi$-ladder masses, or the RCL, but they sit in the gravity-analysis layer those continuum claims rely on for discrete consistency.
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