e_010100
plain-language theorem explainer
Pointwise check that the folded M2 numerator coupling equals eight times the explicit Z-table entry at multi-index (0,1,0,1,0,0). Gravity analysts assembling the 4D Regge midpoint M2–TT identity cite this as one of 256 kernel cells. The proof is a single kernel decide on concrete integers.
Claim. At indices $(a,b,c,d,i,j)=(0,1,0,1,0,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib kernel, summed from zero. The companion map $Z$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that sum divided by eight.
This module is chunk 1 of the 256-cell kernel that discharges $m_2^{\mathrm{num}}=8Z$ by exhaustive evaluation. The ambient goal is the exact midpoint M2–TT identity used in the gravity sector of the Recognition calculus, where discrete curvature couplings must match a sparse explicit pattern before continuum limits are taken.
proof idea
Both sides are closed integer expressions at a fixed six-tuple of $\mathrm{Fin},4$ values. The tactic decide evaluates the fold that defines the numerator and the matching clause of the explicit $Z$ table, then checks integer equality. No lemmas beyond the two definitions are invoked; the kernel does the arithmetic.
why it matters
The parent theorem m2Num_eq_eight_explicitZ quantifies over all six indices and proves $m_2^{\mathrm{num}}=8Z$ by nested fin_cases, each leaf a chunk identity of this form. Without the pointwise cells, the assembly cannot close. In the gravity stack this identity certifies that the discrete M2 numerator is exactly eight times a sparse explicit coupling table, a prerequisite for the Regge midpoint M2–TT comparison in four dimensions. It is pure computational scaffolding inside the gravity analysis, not a forcing-chain landmark (T0–T8), but it hardens the discrete curvature bookkeeping those continuum claims rest on.
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