e_311213
plain-language theorem explainer
For the concrete multi-index (3,1,1,2,1,3) on Fin 4, the folded M2 numerator equals eight times the explicit Z kernel entry. Gravity analysts cite it as one cell of the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on two fully evaluated integers.
Claim. At multi-index $(a,b,c,d,i,j)=(3,1,1,2,1,3)$ with each coordinate in $\{0,1,2,3\}$, 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
In the 4D Regge exact-midpoint analysis, the M2 numerator is defined by folding a fixed coupling list: each term contributes an integer depending on six Fin-4 indices, and the fold sums those contributions from zero. The companion object is an explicit piecewise integer kernel Z on the same six indices (lookup table with values such as 4, -2, and so on).
The local module is chunk 13 of a 256-cell decide grid whose sole job is to check $m_2^{\mathrm{num}}=8Z$ at one fixed sextuple. Upstream, both the fold definition and the explicit kernel live in the kernel-certificate module; neither is proved here, only evaluated.
The ambient claim is the pointwise identity between that fold and eight times Z, needed before assembling the full M2–TT midpoint identity in four dimensions.
proof idea
One-line proof by decide. Both sides are closed integer terms once the six Fin-4 arguments are substituted: the left side runs the fold of coupling contributions at (3,1,1,2,1,3), the right side multiplies the explicit kernel lookup by 8. The kernel reduces the equality of two concrete Int values with no further lemmas.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and dispatches each cell to a chunk lemma of this form. Without the cell-wise certificates the assembly cannot close.
In the Recognition gravity stack this is bookkeeping for the exact midpoint M2–TT identity in 4D Regge calculus, not a new physical law. It sits downstream of the kernel certificate and upstream of the global numerator identity; no forcing-chain landmark (T5–T8, RCL, phi) is touched directly.
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