e_032012
plain-language theorem explainer
Pointwise identity: the folded M2 numerator at multi-index (0,3,2,0,1,2) equals eight times the explicit integer kernel Z at those same indices. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT numerator identity. The proof is a single kernel decide on concrete Fin-4 integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,2,0,1,2)$ in $(\mathbb{F}_4)^6$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel value $Z(a,b,c,d,i,j)$.
background
This module is chunk 3 of a 256-case kernel certification that the 4D Regge exact-midpoint M2 numerator equals eight times a closed-form integer table. Indices run over $\mathbb{F}_4$ (four discrete directions), matching the four spacetime legs of the TT sector.
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 explicit kernel $Z$ is a pattern-matched integer table on those same six $\mathrm{Fin},4$ arguments (sample entries include $4$, $-2$, and so on). The local claim is one concrete instance of $m_2^{\mathrm{num}}=8Z$.
Upstream, both $m_2^{\mathrm{num}}$ and $Z$ live in the KernelCert module; this chunk only evaluates them at a fixed tuple.
proof idea
One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are fixed at $(0,3,2,0,1,2)$, so the kernel decision procedure discharges equality with no manual algebra.
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 exhausting all $\mathrm{Fin},4$ cases. That global identity is the numerator half of the 4D Regge exact-midpoint M2–TT certification in the Gravity analysis stack.
In the Recognition framework this sits inside the discrete gravity / Regge calculus layer that supports continuum limits and effective Newtonian structure, downstream of the forcing chain’s $D=3$ spatial result (T8) and the eight-tick octave (T7) that fix the discrete causal scaffolding. It does not itself touch $\phi$-ladder masses or $\alpha$; it is pure kernel bookkeeping for the curvature–stress identity.
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