e_223003
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at multi-index (2,2,3,0,0,3) equals eight times the explicit integer kernel value there. Gravity analysts cite it as one cell of the 4D kernel table. The proof is a single kernel decision on concrete integer arithmetic.
Claim. For indices $(a,b,c,d,i,j)=(2,2,3,0,0,3)$ with each entry in $\{0,1,2,3\}$, the summed coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 10 of a 256-case kernel certification that the midpoint Regge $m_2$ numerator in four dimensions coincides with eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$, i.e. coordinate labels ${0,1,2,3}$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The explicit kernel $Z$ is a pattern-matched integer table on the same six indices (typical nonzero values are $\pm 2,,4$).
The local claim is only the single tuple $(2,2,3,0,0,3)$. Sibling chunks cover the other tuples; an assembly theorem later quantifies over all of $\mathrm{Fin},4^6$.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers. The left side runs the fold that defines the numerator at $(2,2,3,0,0,3)$; the right side multiplies the table value of $Z$ at that tuple by eight. No lemmas are invoked beyond the definitions of the numerator and the explicit kernel.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases on all six indices. Each chunk such as this one discharges one residual goal after case-split, so the full 4D midpoint identity is only as strong as the conjunction of these decides.
In the Recognition gravity stack this identity is bookkeeping infrastructure: it certifies that a summed coupling expression collapses to a sparse explicit kernel, which is the algebraic input later analysis uses when comparing Regge midpoint discretizations to continuum limits. It does not itself touch the forcing chain (T0–T8), $\phi$, or the eight-tick octave; it is pure discrete-gravity arithmetic inside the 4D kernel cert.
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