e_201210
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at multi-index (2,0,1,2,1,0) equals eight times the explicit integer kernel value. Gravity analysts assembling the 4D exact midpoint M₂TT identity cite these kernel chunks. The proof is a single computational decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,2,1,0)$ in $\mathrm{Fin}\,4$, the folded 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 one chunk in a 256-case kernel certification that the midpoint Regge mass-squared numerator equals eight times a closed-form integer table on $(\mathrm{Fin},4)^6$. The local slogan is $m_2^{\mathrm{num}}=8\cdot Z$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: accumulate every contribution at the six indices. The explicit kernel $Z$ is a pattern-matched integer table on those same indices (typical entries $\pm 2,,4$, and sparse support).
The ambient setting is 4D Regge calculus analysis: exact algebraic identities for the midpoint mass-squared / TT sector, certified by exhaustive finite checks rather than symbolic expansion.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values (left via the fold definition of the numerator, right via the pattern table for $Z$ and multiplication by 8), and the kernel decision procedure discharges equality.
why it matters
Feeds the assembly theorem that $\forall(a,b,c,d,i,j),, m_2^{\mathrm{num}}=8,Z$, proved by exhaustive fin_cases over $(\mathrm{Fin},4)^6$ and discharged chunkwise. That global identity is the algebraic backbone of the Regge exact midpoint M₂TT certification in 4D.
Within Recognition Science gravity work, such kernel equalities pin discrete curvature/mass-squared bookkeeping before continuum or phenomenological limits. They do not themselves invoke the forcing chain (T5–T8) or the J-cost; they sit in the gravity-analysis layer that must match RS-native discrete structure.
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