e_202033
plain-language theorem explainer
For the six-index slot (2,0,2,0,3,3) on Fin 4, the folded M2 numerator equals eight times the explicit kernel integer Z. Gravity analysts cite it when auditing the case-split that proves the global m2Num = 8·explicitZ identity in the 4D Regge midpoint M2TT calculation. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded numerator $m_2^{\mathrm{num}}(2,0,2,0,3,3)$ equals $8$ times the explicit kernel value $Z(2,0,2,0,3,3)$.
background
This module sits inside the 4D Regge-calculus analysis of the midpoint M2TT identity. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer contrib at a six-index slot $(a,b,c,d,i,j)\in(\mathrm{Fin},4)^6$, and the fold starts from 0. The comparison target is explicitZ, a total function on the same six indices that returns a small hardcoded integer (typically $\pm 2$ or $4$ on the nonzero support, else the default).
The local claim of the chunk family is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on every slot. There are $4^6=4096$ formal slots, but the kernel is organized so that 256 representative decides (this file is chunk 8) cover the needed support. Upstream, both $m_2^{\mathrm{num}}$ and $Z$ are pure definitions with no analytic hypotheses: equality is a finite integer computation.
proof idea
One-line proof by decide. Lean reduces both sides at the concrete indices $(2,0,2,0,3,3)$: the left-hand fold over couplingZList evaluates to an integer, the right-hand side is $8$ times the matching explicitZ clause, and the kernel checks numeral equality. No lemmas beyond the two definitions are invoked.
why it matters
The parent theorem m2Num_eq_eight_explicitZ states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal quantifier by exhaustive fin_cases on all six Fin-4 indices. Each chunk theorem such as this one supplies one concrete cell of that case tree (chunk 8 of the 256-decide kernel).
In the broader Recognition gravity stack, the identity certifies that the midpoint-evaluated M2TT numerator collapses to a sparse explicit kernel, which is the algebraic bottleneck before continuum or continuum-limit comparisons. It does not itself invoke the forcing chain (T5–T8) or the J-cost; it is pure discrete tensor bookkeeping on which later curvature and coupling normalizations rest.
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