e_020013
plain-language theorem explainer
Pointwise certificate that the folded M2 numerator at multi-index (0,2,0,0,1,3) equals eight times the explicit integer table entry. Gravity analysts assembling the universal Regge midpoint M2–TT identity in 4D cite the forall form built from these chunks; this is one of 256 kernel decides in chunk 2. The proof is a single kernel decision on concrete integers.
Claim. For indices $a=0$, $b=2$, $c=0$, $d=0$, $i=1$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer table value at those indices.
background
In the 4D Regge analysis of the exact midpoint M2–TT identity, two integer maps on sextuples of $\mathrm{Fin},4$ indices are compared. The numerator is a fold over a fixed coupling list: each term contributes an integer at the given indices, and the contributions are summed from zero. The comparison target is an explicit pattern-matched table that records a sparse integer kernel (entries such as $4$, $-2$, and similar values on selected index patterns).
This module is chunk 2 of a 256-case kernel certification that the folded numerator equals eight times the explicit table pointwise. Upstream definitions supply both the fold and the table; the present declaration fixes one concrete sextuple.
proof idea
Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed. The proof is a one-line decide: the kernel normalizes the fold of the coupling list against eight times the matched explicit-table clause and accepts the equality. No intermediate lemmas are invoked beyond the two definitions.
why it matters
The parent assembler proves the universal statement that for every sextuple in $\mathrm{Fin},4$ the folded numerator equals eight times the explicit table, by exhausting indices (and consuming these pointwise certificates). That identity is infrastructure for the Regge exact-midpoint M2–TT analysis in four dimensions: it shows the folded numerator is a pure multiple of the sparse explicit kernel. Within Recognition Science gravity work, such algebraic certificates underwrite discrete curvature bookkeeping before continuum or phenomenological claims are attached. This chunk entry is pure computational glue; it does not invoke the forcing chain (T0–T8), the Recognition Composition Law, or the phi-ladder.
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