e_201232
plain-language theorem explainer
Pointwise kernel identity: the midpoint mass-matrix numerator at index tuple (2,0,1,2,3,2) equals eight times the explicit integer kernel Z at those same indices. Gravity analysts assembling the 4D Regge midpoint TT identity cite it as one of 256 concrete decides. The proof is a single kernel decision on fully evaluated integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,1,2,3,2)$ in $\{0,1,2,3\}^6$, the midpoint coupling numerator $m_2^{\mathrm{num}}(2,0,1,2,3,2)$ equals $8\,Z(2,0,1,2,3,2)$, where $Z$ is the explicit integer kernel on six $\mathrm{Fin}\,4$ arguments.
background
This module is chunk 8 of the 256-case kernel certification that the 4D Regge midpoint mass-matrix numerator equals eight times a closed-form integer table. The local claim is exactly $m_2^{\mathrm{num}}=8\cdot Z$ at one fixed six-tuple of edge/vertex labels in $\mathrm{Fin},4$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing each contribution at those indices. The companion table $Z$ is an explicit pattern-matched integer function on six $\mathrm{Fin},4$ arguments (typical values $\pm 2,\pm 4$, and zero off the matched patterns). Both live in the kernel-certificate module imported here.
The surrounding analysis targets an exact midpoint TT identity for 4D Regge calculus: once every index tuple satisfies the eightfold relation, the assembled numerator matches the closed form used in the continuum comparison.
proof idea
One-line computational proof: decide. After the six concrete $\mathrm{Fin},4$ arguments are substituted, both sides reduce to closed integers (the fold that defines the numerator versus the pattern match that defines $Z$), and the kernel checks equality. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.
why it matters
Feeds the universal assembly theorem $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$ for all six indices in $\mathrm{Fin},4$. That parent discharges the full identity by exhaustive fin_cases over the $4^6=4096$ tuples, of which this chunk supplies one certified point (the module header: "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)").
In the Recognition gravity stack this is bookkeeping infrastructure for the exact discrete midpoint TT identity, not a new dynamical law. It closes a finite algebraic obligation so later continuum or continuum-limit comparisons can quote a single clean equality rather than a residual case table. No forcing-chain landmark (T5–T8, RCL, $\phi$) is at stake here; the result is pure index algebra inside the Regge analysis layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.