e_032033
plain-language theorem explainer
For the six-index tuple (0,3,2,0,3,3) on Fin 4, the folded numerator coupling equals eight times the explicit integer Z-table value. Gravity analysts cite it when assembling the full pointwise identity m2Num = 8·explicitZ over all 4^6 index combinations. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,0,3,3)$ in $(\mathrm{Fin}\,4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table entry $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of a 256-way kernel split proving the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on all six-tuples of $\mathrm{Fin},4$. The setting is the 4D Regge exact-midpoint analysis of the TT mass-squared numerator.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each list entry at the six indices. The explicit table $Z$ is a closed-form integer function on the same six indices (sample values include $4$, $-2$, and other small integers on diagonal and off-diagonal patterns).
The parent assembly theorem states the identity for every six-tuple and discharges it by exhaustive fin_cases on each coordinate, routing each concrete cell to a chunk lemma of this form.
proof idea
One-line kernel proof: decide. Both sides are closed integer expressions at a fully concrete six-tuple of Fin 4 values, so the decision procedure evaluates the fold that defines the numerator and the matching clause of the explicit $Z$ table and checks equality with $8\cdot Z$.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8\cdot Z$ and proves it by six nested fin_cases over Fin 4. Each chunk lemma such as this one closes one concrete cell so the assembly stays a pure case split with no analytic remainder.
In the gravity analysis stack this identity is bookkeeping infrastructure: it certifies that the folded coupling numerator agrees with the hand-written explicit $Z$ table (scaled by eight) on the full $4^6$ index space. That agreement is a prerequisite for later exact midpoint / TT identities in the 4D Regge sector. It does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law; it is a finite combinatorial certificate inside the gravity kernel.
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