e_220232
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint coupling numerator at multi-index (2,2,0,2,3,2) equals eight times the explicit integer Z-table entry. Gravity analysts assembling the 4D exact midpoint M2TT identity cite these 256 kernel chunks. The proof is a single computational decide on concrete Fin-4 integers.
Claim. For the multi-index $(a,b,c,d,i,j)=(2,2,0,2,3,2)$ with each coordinate in $\mathbb{F}_4$, 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 10 of a 256-way kernel split proving $m_2^{\mathrm{num}}=8\cdot Z$ at every 4D multi-index. The setting is discrete Regge gravity: midpoint evaluations of a second-moment / TT-sector coupling on a 4-index lattice.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing elementary contributions at six $\mathrm{Fin},4$ slots. The comparison table $Z$ is an explicit pattern-matched integer function on the same six indices (nonzero only on a sparse set of index patterns, with values such as $\pm 2,,4$).
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the kernel certificate module; this chunk only evaluates one concrete sextuple.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed at $(2,2,0,2,3,2)$, so the kernel decision procedure closes the equality with no lemmas or rewriting.
why it matters
Feeds the assembler theorem $m_2^{\mathrm{num}}=8\cdot Z$ for all $a,b,c,d,i,j\in\mathrm{Fin},4$, which proceeds by exhaustive fin_cases and dispatches each cell to a chunk identity of this form. That global identity is the algebraic core of the 4D Regge exact-midpoint M2TT certificate in the Gravity analysis stack.
Within Recognition Science gravity work, these kernel equalities underwrite discrete curvature / coupling bookkeeping that must match the continuum TT sector before continuum limits or phenomenological mass-ladder comparisons are trustworthy. The chunk exists purely to keep the 256-case decide tractable and auditable.
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