e_121301
plain-language theorem explainer
For the six-index tuple (1,2,1,3,0,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2 TT identity in 4D cite this as one of the 256 kernel decides. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=1$, $b=2$, $c=1$, $d=3$, $i=0$, $j=1$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 6 of a 256-case kernel certification that the Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$ on six $\mathrm{Fin},4$ indices. The setting is 4D discrete gravity analysis for an exact midpoint M2 TT identity.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing local contributions at those indices. The companion table $Z$ is a pattern-matched integer function on the same six indices (typical values $\pm 2,,4$, and zero off the listed patterns).
The full identity is $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$. Each chunk theorem discharges one concrete sextuple so the assembler can finish by exhaustive fin_cases.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete indices $(1,2,1,3,0,1)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table; the kernel reduces the fold and the pattern match to numerals and compares them.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states the universal equality for all six $\mathrm{Fin},4$ arguments by casing through every index and invoking the chunk decides. That identity is the algebraic backbone of the Regge exact-midpoint M2 TT certification in 4D gravity analysis inside the monolith.
In the broader Recognition framework this sits in the gravity domain: discrete curvature/coupling bookkeeping that must match closed-form kernel values before continuum or continuum-limit claims are trusted. It does not itself touch the forcing chain (T0–T8), $\varphi$, or the eight-tick octave; it is pure finite combinatorial certification supporting the gravity stack.
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