e_122013
plain-language theorem explainer
Pointwise kernel identity: the Regge midpoint mass-squared numerator at multi-index (1,2,2,0,1,3) equals eight times the explicit Z-table entry. Gravity analysts cite it only as one cell of the 4D midpoint M2TT certificate. The proof is a pure kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,2,0,1,3)$ 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 coupling $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes an integer via a local contribution function, and the fold starts at $0$. The comparison target is an explicit piecewise table $Z$ on the same six indices, with sparse nonzero values such as $\pm 2$ and $4$ on selected patterns.
This module is chunk 6 of a 256-way kernel split: each chunk discharges a block of concrete index tuples by decision procedure, rather than by a single symbolic argument. The local claim is the equality $m_2^{\mathrm{num}}=8Z$ at one fixed tuple.
proof idea
Both sides evaluate to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted. The tactic decide runs the kernel decision procedure on that closed integer equality; no lemmas beyond the definitions of the numerator fold and the explicit table are required.
why it matters
The parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple in $(\mathrm{Fin},4)^6$. Its proof is a nested fin_cases sweep that ultimately lands on these pointwise kernel facts. This declaration is one cell in that sweep (chunk 6), so it is scaffolding for the global midpoint M2TT numerator identity rather than a standalone physical law.
In the broader gravity stack, the identity certifies that the folded coupling numerator is exactly eight times the closed-form Z table, which is the algebraic content needed before continuum or continuum-limit comparisons. It does not itself invoke the RS forcing chain (T5–T8), RCL, or the $\varphi$-ladder; those enter only at higher layers that consume the certified numerator.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.