e_013302
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at multi-index (0,1,3,3,0,2) equals eight times the explicit Z-table entry. Gravity analysts cite it as one kernel cell in the 4D midpoint M2TT identity. The proof is a single kernel decide on a closed integer equality.
Claim. For indices $(a,b,c,d,i,j)=(0,1,3,3,0,2)$ in $\mathrm{Fin}\,4$, 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
This module is chunk 1 of the kernel certification that the midpoint mass-squared numerator equals eight times an explicit integer table on all six-tuples in $(\mathrm{Fin},4)^6$. The ambient setting is 4D Regge-calculus analysis of the exact midpoint M2TT identity.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term at the six indices. The table $Z$ is an explicit case-split function $\mathrm{Fin},4^6\to\mathbb{Z}$ with sparse nonzero entries (e.g. $4$ on matched diagonal pairs, $-2$ on certain crossed pairs).
The full quantified statement is assembled downstream by exhausting all index combinations; each chunk theorem such as this one discharges one concrete cell.
proof idea
One-line computational proof: decide evaluates both sides as closed integer expressions (the fold of contributions versus eight times the matching explicitZ clause) and accepts the equality. No lemmas are invoked beyond the kernel decision procedure on Int.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ and proves it by nested fin_cases over the six indices. That global identity is the algebraic core of the 4D Regge exact-midpoint M2TT certification in the Gravity analysis stack. Within Recognition Science it supports the discrete geometric side of the forcing chain's dimensional and causal structure (T7 eight-tick octave, T8 $D=3$ spatial), by locking the midpoint mass-squared combinatorics to an explicit integer kernel rather than an opaque sum.
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