e_233200
plain-language theorem explainer
Pointwise identity: the midpoint mass-squared numerator at Fin-4 indices (2,3,3,2,0,0) equals eight times the explicit kernel value Z there. Gravity analysts cite it as one cell of the 256-case kernel that assembles the global m2Num = 8·Z theorem. The proof is a single kernel decide on integer equality.
Claim. For the index sextuple $(a,b,c,d,i,j)=(2,3,3,2,0,0)$ with each index in $\mathbb{F}_4$, the midpoint $m^2$ numerator equals eight times the explicit integer kernel value $Z$ at those indices: $m^2_{\mathrm{num}}(2,3,3,2,0,0)=8\,Z(2,3,3,2,0,0)$.
background
This module is chunk 11 of a 256-cell case split proving that the Regge exact-midpoint mass-squared numerator coincides with eight times an explicit integer kernel on all of $(\mathbb{F}_4)^6$. The ambient setting is 4D discrete gravity analysis: indices run over a four-point label set, and couplings are summed into a numerator polynomial.
The numerator $m^2_{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list, accumulating each term's contribution at the six indices. The comparison object $Z$ is an explicit piecewise-integer function on the same six Fin-4 arguments (sample values include $4$, $-2$, and so on for distinguished patterns).
The local claim is only the single sextuple $(2,3,3,2,0,0)$. Sibling theorems cover the other cells; the assembly theorem quantifies over all indices by finitary case split.
proof idea
One-line kernel proof: decide evaluates both sides as concrete integers (the fold defining the numerator versus eight times the matching clause of the explicit kernel) and closes the equality by computation. No lemmas are invoked beyond the definitions of the numerator and $Z$.
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 the six Fin-4 indices. Each chunk theorem such as this one discharges one concrete cell so the global identity is a pure case glue rather than a symbolic expansion.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge kernel in 4D: once numerator and explicit $Z$ match everywhere, downstream curvature and mass-ladder comparisons can quote a closed integer form instead of a coupling fold. It does not itself invoke the T0–T8 forcing chain, $\phi$, or the eight-tick octave; it is pure discrete-kernel algebra supporting those layers.
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