e_132020
plain-language theorem explainer
At multi-index (1,3,2,0,2,0) the folded Regge midpoint numerator equals eight times the explicit integer kernel value. Gravity analysts proving the global 4D TT midpoint identity cite this as one of the 256 kernel point checks. The proof is a single kernel decision on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,2,0,2,0)$ 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 kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 7 of a 256-way case split establishing $m_2^{\mathrm{num}}=8\cdot Z$ on the full $(\mathbb{F}_4)^6$ index space for the exact midpoint Regge identity in 4D.
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 comparison target $Z$ is an explicit integer-valued kernel on the same six $\mathrm{Fin},4$ arguments, tabulated by pattern (e.g. diagonal blocks map to $4$, certain off-diagonal swaps to $-2$).
Both objects live in the kernel certificate module that supplies the closed-form table and the fold definition used throughout the gravity analysis stack.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(1,3,2,0,2,0)$ and checks integer equality. No lemmas are invoked; the kernel reduces the fold defining the numerator and the pattern match defining $Z$ to numerals and compares them.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8\cdot Z$ for all six indices, which exhausts $(\mathbb{F}_4)^6$ by nested fin_cases and lands each point on a chunk identity of this form. That global equality is the algebraic core of the exact midpoint TT identity in 4D Regge analysis: it replaces a folded coupling sum by a sparse explicit kernel, enabling closed-form curvature and graviton-sector identities downstream.
Within Recognition gravity work this is pure certificate scaffolding, not a forcing-chain step (T0–T8), but it underwrites the discrete geometric side of the RS gravity stack where continuum limits and effective $G$ are read off Regge data.
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