e_010010
plain-language theorem explainer
Pointwise identity: the midpoint Regge mass-squared numerator at multi-index (0,1,0,0,1,0) equals eight times the explicit Z-coupling table entry. Gravity analysts assembling the 4D Regge exact-midpoint M₂TT identity cite these 256 kernel chunks. The proof is a single kernel decide on concrete integers.
Claim. For the multi-index $(0,1,0,0,1,0)\in(\mathbb{F}_4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(0,1,0,0,1,0)$ equals $8$ times the explicit integer table value $Z(0,1,0,0,1,0)$.
background
In the 4D Regge exact-midpoint analysis, the mass-squared numerator is assembled by folding a fixed coupling list: each term contributes an integer depending on six indices in $\mathbb{F}_4$, and the fold starts from zero. The companion table $Z$ is an explicit piecewise integer function on the same six indices (sample values include $\pm 2,4$ on selected diagonal and off-diagonal patterns).
The module is chunk 1 of a 256-way case split: every concrete sextuple is certified separately so the global identity $m_2^{\mathrm{num}}=8Z$ can be reassembled by exhaustive fin_cases. Local setting: pure integer equalities, no analytic estimates.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(0,1,0,0,1,0)$. The left side runs the fold that defines the numerator; the right side looks up the explicit table and multiplies by 8. Both reduce to concrete Int values, so the kernel closes the equality with no lemmas beyond the two definitions.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j\in\mathbb{F}_4$, $m_2^{\mathrm{num}}=8Z$ by six nested fin_cases and discharge of each pointwise chunk. That global identity is the algebraic core of the Regge exact-midpoint M₂TT certificate in the gravity analysis stack. Within Recognition Science this sits in the discrete geometric side of the forcing chain (eight-tick / $D=3$ structure), supplying an exact integer relation rather than a continuum approximation. No open scaffold: the chunk is fully decided.
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