e_133100
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at indices (1,3,3,1,0,0) equals eight times the explicit integer table. Gravity analysts cite it as one kernel cell in the 4D midpoint M2 TT certification. The proof is a single kernel decide on concrete Fin-4 data.
Claim. For indices $(a,b,c,d,i,j)=(1,3,3,1,0,0)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-cell kernel certification that the 4D Regge midpoint M2 TT numerator matches an explicit closed form. The ambient setting is discrete gravity analysis: index sextuples range over $\mathbb{F}_4$, and integer-valued coupling data are compared two ways.
The folded numerator sums contributions over a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of contrib along that list. The comparison target explicitZ is a pattern-matched integer table on the same six indices (sample clauses include values $4$, $-2$, and so on for distinguished index patterns).
The local claim is only the single cell $(1,3,3,1,0,0)$. Sibling chunk theorems cover the other cells; the universal statement is assembled downstream.
proof idea
One-line computational proof: decide evaluates both sides on the concrete Fin-4 sextuple and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table. The kernel is small enough that Mathlib's decision procedure closes the goal directly.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for every $(a,b,c,d,i,j)\in(\mathbb{F}_4)^6$ and discharges the cases by exhaustive fin_cases. Without the pointwise cells (or an equivalent exhaustive case split), the global numerator-table match does not go through.
In the Recognition gravity stack this is bookkeeping infrastructure for the Regge midpoint M2 TT identity in 4D, not a forcing-chain landmark (T5–T8) by itself. It closes one concrete arithmetic obligation inside the discrete curvature/coupling certification that later continuum or continuum-limit arguments rely on being exact at the integer level.
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